Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5

Depinning of charge density waves of different dimensionalities in 1T-TiSe2and NbSe3

Nonlinear transport behavior is one of the signatures of the formation of electronic crystals such as charge density wave (CDW), as it provides evidence for their collective motion. Such experimental evidence has been widely reported in quasi-one-dimensional (1D) materials but is rarely studied in 2D systems. Only a few studies on the RTe3materials have been previsouly reported. Here we report for the first time the observation of CDW depinning and sliding in the layered 1T-TiSe2compound, based on the observation of (1) nonlinear voltage-current characteristics and (2) the electrical noise, which are associated with the CDW depinning and sliding process. Similar measurements are also conducted on quasi-1D system NbSe3. The depinning behavior of the CDWs with different dimensionalities in these two systems are compared. It is found that the threshold electric field (ET) increases linearly with decreasing temperature for the 2D case, consistent with previous results on RTe3, while it demonstrated an activated behavior in 1D, as expected within the weak-pinning Fukuyama-Lee-Rice framework. Such a distinction of the threshold behavior in CDW systems of different dimensions therefore indicates a possible strong pinning picture in higher-dimensional CDW systems in general.ETis found to be much higher in 1T-TiSe2, consistent with a strong pinning picture, and could account for the scarcity of the depinning study in these 2D systems. Our results thus pave the way for a unified understanding of the CDW collective motion in different dimensionalities.

Switchable Quantized Signal between Longitudinal Conductance and Hall Conductance in Dual Quantum Spin Hall Insulator TaIrTe4

Topological insulating states in 2-dimensional (2D) materials are ideal systems to study different types of quantized response signals due to their in gap metallic states. Very recently, the quantum spin Hall effect was discovered in monolayer TaIrTe4 via the observation of quantized longitudinal conductance that rarely exists in other 2D topological insulators. The nontrivial Z 2 topological charges can exist at both charge neutrality point and the van Hove singularity point with correlation-effect-induced bandgap. On the basis of this model 2D material, we studied the switch of quantized signals between longitudinal conductance and transversal Hall conductance via tuning external magnetic field. In Z 2 topological phase of monolayer TaIrTe4, the zero Chern number can be understood as 1 - 1 = 0 from the double band inversion from spin-up and spin-down channels. After applying a magnetic field perpendicular to the plane, the Zeeman split changes the band order for one branch of the band inversion from spin-up and spin-down channels, along with a sign charge of the Berry phase. Then, the net Chern number of 1 - 1 = 0 is tuned to 1 + 1 = 2 or -1 - 1 = -2 depending on the orientation of the magnetic field. The quantized signal not only provides another effective method for the verification of topological state in monolayer TaIrTe4 but also offers a strategy for the utilization of the new quantum topological states based on switchable quantized responses.

Observation of 3D acoustic quantum Hall states

Quantum Hall effect, the quantized transport phenomenon of electrons under strong magnetic fields, remains one of the hottest research topics in condensed matter physics since its discovery in 2D electronic systems. Recently, as a great advance in the research of quantum Hall effects, the quantum Hall effect in 3D systems, despite its big challenge, has been achieved in the bulk ZrTe5 and Cd3As2 materials. Interestingly, Cd3As2 is a Weyl semimetal, and quantum Hall effect is hosted by the Fermi arc states on opposite surfaces via the Weyl nodes of the bulk, and induced by the unique edge states on the boundaries of the opposite surfaces. However, such intriguing edge state distribution has not yet been experimentally observed. Here, we aim to reveal experimentally the unusual edge states of Fermi arcs in acoustic Weyl system with the aid of pseudo-magnetic field. Benefiting from the macroscopic nature of acoustic crystals, the pseudo-magnetic field is introduced by elaborately designed the gradient on-site energy, and the edge states of Fermi arcs on the boundaries of the opposite surfaces are unambiguously demonstrated in experiments. Our system serves as an ideal and highly tunable platform to explore the Hall physics in 3D system, and has the potential in the application of new acoustic devices.

Electronic instability in pressured black phosphorus under strong magnetic field

In this paper, we have systematically studied the electronic instability of pressured black phosphorous (BP) under strong magnetic field. We first present an effective model Hamiltonian for pressured BP near theLifshitzpoint. Then we show that when the magnetic field exceeds a critical value, the nodal-line semimetal (NLSM) state of BP with a small band overlap re-enters the semiconductive phase by re-opening a small gap. This results in a narrow-bandgap semiconductor with a partially flat valence band edge. Moreover, we demonstrate that above this critical magnetic field, two possible instabilities, i.e. charge density wave phase and excitonic insulator (EI) phase, are predicted as the ground state for high and low doping concentrations, respectively. By comparing our results with the experiment (Sunet al2018Sci. Bull.631539), we suggest that the field-induced instability observed experimentally corresponds to an EI. Furthermore, we propose that the semimetallic BP under pressure with small band overlaps may provide a good platform to study the magneto-exciton insulators. Our findings bring the first insight into the electronic instability of topological NLSM in the quantum limit.

Spontaneous Inversion Symmetry Breaking and Emergence of Berry Curvature and Orbital Magnetization in Topological ZrTe_{5} Films

ZrTe_{5} has recently attracted much attention due to the observation of intriguing nonreciprocal transport responses which necessitate the lack of inversion symmetry (I). However, there has been debate on the exact I-asymmetric structure and the underlying I-breaking mechanism. Here, we report a spontaneous I breaking in ZrTe_{5} films, which initiates from interlayer sliding and is stabilized by subtle intralayer distortion. Moreover, we predict significant nonlinear anomalous Hall effect (NAHE) and kinetic magnetoelectric effect (KME), which are attributed to the emergence of Berry curvature and orbital magnetization in the absence of I symmetry. We also explicitly manifest the direct coupling between sliding ferroelectricity, NAHE, and KME based on a sliding-dependent k·p model. By studying the subsurface sliding in ZrTe_{5} multilayers, we speculate that surface nonlinear Hall current and magnetization would emerge on the natural cleavage surface. Our findings elucidate the sliding-induced I-broken mechanism in ZrTe_{5} films and open new avenues for tuning nonreciprocal transport properties in Van der Waals layered materials.

Anisotropic structural, vibrational, electronic, optical, and elastic properties of single-layer hafnium pentatelluride: an ab initio study

Motivated by the highly anisotropic nature of bulk hafnium pentatelluride (HfTe5), the structural, vibrational, electronic, optical, and elastic properties of single-layer two-dimensional (2D) HfTe5 were investigated by performing density functional theory (DFT)-based first-principles calculations. Total energy and geometry optimizations reveal that the 2D single-layer form of HfTe5 exhibits in-plane anisotropy. The phonon band structure shows dynamic stability of the free-standing layer and the predicted Raman spectrum displays seven characteristic Raman-active phonon peaks. In addition to its dynamic stability, HfTe5 is shown to exhibit thermal stability at room temperature, as confirmed by quantum molecular dynamics simulations. Moreover, the obtained elastic stiffness tensor elements indicate the mechanical stability of HfTe5 with its orientation-dependent soft nature. The electronic band structure calculations show the indirect-gap semiconducting behavior of HfTe5 with a narrow electronic band gap energy. The optical properties of HfTe5, in terms of its imaginary dielectric function, absorption coefficient, reflectance, and transmittance, are shown to exhibit strong in-plane anisotropy. Furthermore, structural analysis of several point defects and their oxidized structures was performed by means of simulated STM images. Among the considered vacancy defects, namely , , VTeout, VTein, , and VHf, the formation of VTeout is revealed to be the most favorable defect. While and VHf defects lead to local magnetism, only the oxygen-substituted VHf structure possesses magnetism among the oxidized defects. Moreover, it is found that all the bare and oxidized vacant sites can be distinguished from each other through the STM images. Overall, our study indicates not only the fundamental anisotropic features of single-layer HfTe5, but also shows the signatures of feasible point defects and their oxidized structures, which may be useful for future experiments on 2D HfTe5.

Realization of monolayer ZrTe5 topological insulators with wide band gaps

Two-dimensional topological insulators hosting the quantum spin Hall effect have application potential in dissipationless electronics. To observe the quantum spin Hall effect at elevated temperatures, a wide band gap is indispensable to efficiently suppress bulk conduction. Yet, most candidate materials exhibit narrow or even negative band gaps. Here, via elegant control of van der Waals epitaxy, we have successfully grown monolayer ZrTe5 on a bilayer graphene/SiC substrate. The epitaxial ZrTe5 monolayer crystalizes in two allotrope isomers with different intralayer alignments of ZrTe3 prisms. Our scanning tunneling microscopy/spectroscopy characterization unveils an intrinsic full band gap as large as 254 meV and one-dimensional edge states localized along the periphery of the ZrTe5 monolayer. First-principles calculations further confirm that the large band gap originates from strong spin-orbit coupling, and the edge states are topologically nontrivial. These findings thus provide a highly desirable material platform for the exploration of the high-temperature quantum spin Hall effect.

Non-centrosymmetric topological phase probed by non-linear Hall effect

Non-centrosymmetric topological material has attracted intense attention due to its superior characteristics as compared with the centrosymmetric one, although probing the local quantum geometry in non-centrosymmetric topological material remains challenging. The non-linear Hall (NLH) effect provides an ideal tool to investigate the local quantum geometry. Here, we report a non-centrosymmetric topological phase in ZrTe5, probed by using the NLH effect. The angle-resolved and temperature-dependent NLH measurement reveals the inversion and ab-plane mirror symmetries breaking at <30 K, consistently with our theoretical calculation. Our findings identify a new non-centrosymmetric phase of ZrTe5 and provide a platform to probe and control local quantum geometry via crystal symmetries.

Rashba-splitting-induced topological flat band detected by anomalous resistance oscillations beyond the quantum limit in ZrTe5

Topological flat bands - where the kinetic energy of electrons is quenched - provide a platform for investigating the topological properties of correlated systems. Here, we report the observation of a topological flat band formed by polar-distortion-assisted Rashba splitting in the three-dimensional Dirac material ZrTe5. The polar distortion and resulting Rashba splitting on the band are directly detected by torque magnetometry and the anomalous Hall effect, respectively. The local symmetry breaking further flattens the band, on which we observe resistance oscillations beyond the quantum limit. These oscillations follow the temperature dependence of the Lifshitz-Kosevich formula but are evenly distributed in B instead of 1/B at high magnetic fields. Furthermore, the cyclotron mass gets anomalously enhanced about 102 times at fields ~ 20 T. Our results provide an intrinsic platform without invoking moiré or order-stacking engineering, which opens the door for studying topologically correlated phenomena beyond two dimensions.

Surface potential-adjusted surface states in 3D topological photonic crystals

Surface potential in a topological matter could unprecedentedly localize the waves. However, this surface potential is yet to be exploited in topological photonic systems. Here, we demonstrate that photonic surface states can be induced and controlled by the surface potential in a dielectric double gyroid (DG) photonic crystal. The basis translation in a unit cell enables tuning of the surface potential, which in turn regulates the degree of wave localization. The gradual modulation of DG photonic crystals enables the generation of a pseudomagnetic field. Overall, this study shows the interplay between surface potential and pseudomagnetic field regarding the surface states. The physical consequences outlined herein not only widen the scope of surface states in 3D photonic crystals but also highlight the importance of surface treatments in a photonic system.

Three-dimensional flat Landau levels in an inhomogeneous acoustic crystal

When electrons moving in two dimensions (2D) are subjected to a strong uniform magnetic field, they form flat bands called Landau levels (LLs). LLs can also arise from pseudomagnetic fields (PMFs) induced by lattice distortions. In three-dimensional (3D) systems, there has been no experimental demonstration of LLs  as a type of flat band thus far. Here, we report the experimental realization of a flat 3D LL in an acoustic crystal. Starting from a lattice whose bandstructure exhibits a nodal ring, we design an inhomogeneous distortion corresponding to a specific pseudomagnetic vector potential (PVP). This distortion causes the nodal ring states to break up into LLs, including a zeroth LL that is flat along all three directions. These findings suggest the possibility of using nodal ring materials to generate 3D flat bands, allowing access to strong interactions and other attractive physical regimes in 3D.

Magic Momenta and Three-Dimensional Landau Levels from a Three-Dimensional Graphite Moiré Superlattice

In this Letter, we theoretically explore the physical properties of a new type of three-dimensional graphite moiré superlattice, the bulk alternating twisted graphite (ATG) system with homogeneous twist angle, which is grown by in situ chemical vapor decomposition method. Compared to twisted bilayer graphene (TBG), the bulk ATG system is bestowed with an additional wave vector degree of freedom due to the extra dimensionality. As a result, when the twist angle of bulk ATG is smaller than twice of the magic angle of TBG, there always exist "magic momenta" which host topological flat bands with vanishing in-plane Fermi velocities. Most saliently, when the twist angle is relatively large, a dispersionless three-dimensional zeroth Landau level would emerge in the bulk ATG, which may give rise to robust three-dimensional quantum Hall effects and unusual quantum-Hall physics over a large range of twist angles.

Controllable strain-driven topological phase transition and dominant surface-state transport in HfTe5

The fine-tuning of topologically protected states in quantum materials holds great promise for novel electronic devices. However, there are limited methods that allow for the controlled and efficient modulation of the crystal lattice while simultaneously monitoring the changes in the electronic structure within a single sample. Here, we apply significant and controllable strain to high-quality HfTe5 samples and perform electrical transport measurements to reveal the topological phase transition from a weak topological insulator phase to a strong topological insulator phase. After applying high strain to HfTe5 and converting it into a strong topological insulator, we found that the resistivity of the sample increased by 190,500% and that the electronic transport was dominated by the topological surface states at cryogenic temperatures. Our results demonstrate the suitability of HfTe5 as a material for engineering topological properties, with the potential to generalize this approach to study topological phase transitions in van der Waals materials and heterostructures.

Asymmetric Monolayer Mesoporous Nanosheets of Regularly Arranged Semi-Opened Pores via a Dual-Emulsion-Directed Micelle Assembly

Constructing asymmetric two-dimensional (2D) mesoporous nanomaterials with new pore structure, tunable monolayer architectures, and especially anisotropic surfaces remains a great challenge in materials science. Here, we report a dual-emulsion directed micelle assembly approach to fabricate a novel type of asymmetric monolayer mesoporous organosilica nanosheet for the first time. In this asymmetric 2D structure, numerous quasi-spherical semiopened mesopores (∼20 nm in diameter, 24 nm in opening size) were regularly arranged on a plane, endowing the porous nanosheets (several micrometers in size) with a typical surface anisotropy on two sides. Meanwhile, lots of triangular intervoids (4.0-5.0 nm in size) can also be found among each three semiopened mesopores, enabling the nanosheet to be interconnected. Vitally, such interconnected, anisotropic porous nanosheets exhibit ultrahigh accessible surface area (∼714 m2 g-1) and good lipophilicity properties owing to the abundant semiopened mesopores. Additionally, besides the nanosheet, the configuration of the asymmetric porous structure can also be transformed into a microcapsule when controlling the emulsification size via a facile ultrasonic treatment. As a demonstration, we show that the asymmetric microcapsule shows a high demulsification efficiency (>98%) and cyclic stability (>6 recycle times). Our protocol opens up a new avenue for developing next-generation asymmetric mesoporous materials for various applications.

Nonlinear Transport due to Magnetic-Field-Induced Flat Bands in the Nodal-Line Semimetal ZrTe_{5}

The Dirac material ZrTe_{5} at very low carrier density was recently found to be a nodal-line semimetal, where ultraflat bands are expected to emerge in magnetic fields parallel to the nodal-line plane. Here, we report that in very low carrier-density samples of ZrTe_{5}, when the current and the magnetic field are both along the crystallographic a axis, the current-voltage characteristics presents a pronounced nonlinearity which tends to saturate in the ultra quantum limit. The magnetic-field dependence of the nonlinear coefficient is well explained by the Boltzmann theory for flat-band transport, and we argue that this nonlinear transport is likely due to the combined effect of flat bands and charge puddles; the latter appear due to very low carrier densities.

Photonic helicoid-like surface states in chiral metamaterials

We investigate the photonic topological phases in chiral metamaterials characterized by the magnetoelectric tensors with diagonal chirality components. The underlying medium is considered a photonic analogue of the topological semimetal featured with a Weyl cone and a cylindrical surface in the frequency-wave vector space. As the 'spin'-degenerate condition is satisfied, the photonic system can be rearranged as two hybrid modes that are completely decoupled. By introducing the pseudospin states as the basis for the hybrid modes, the photonic system is described by two subsystems in the form of spin-orbit Hamiltonians of spin 1, which result in nonzero spin Chern numbers that determine the topological properties. Surface modes at the interface between vacuum and the chiral metamaterial exist in their common gap in the wave vector space, which are analytically formulated by algebraic equations. In particular, the surface modes form a pair of spiral surface sheets wrapping around the Weyl cone, resembling the helicoid surface states that occur in topological semimetals. At the Weyl frequency, the surface modes contain two Fermi arc-like states that concatenate to yield a straight line segment.

High-energy photoemission final states beyond the free-electron approximation

Three-dimensional (3D) electronic band structure is fundamental for understanding a vast diversity of physical phenomena in solid-state systems, including topological phases, interlayer interactions in van der Waals materials, dimensionality-driven phase transitions, etc. Interpretation of ARPES data in terms of 3D electron dispersions is commonly based on the free-electron approximation for the photoemission final states. Our soft-X-ray ARPES data on Ag metal reveals, however, that even at high excitation energies the final states can be a way more complex, incorporating several Bloch waves with different out-of-plane momenta. Such multiband final states manifest themselves as a complex structure and added broadening of the spectral peaks from 3D electron states. We analyse the origins of this phenomenon, and trace it to other materials such as Si and GaN. Our findings are essential for accurate determination of the 3D band structure over a wide range of materials and excitation energies in the ARPES experiment.

Gate-Tunable Multiband Transport in ZrTe5 Thin Devices

Interest in ZrTe₅ has been reinvigorated in recent years owing to its potential for hosting versatile topological electronic states and intriguing experimental discoveries. However, the mechanism of many of its unusual transport behaviors remains controversial: for example, the characteristic peak in the temperature-dependent resistivity and the anomalous Hall effect. Here, through employing a clean dry-transfer fabrication method in an inert environment, we successfully obtain high-quality ZrTe₅ thin devices that exhibit clear dual-gate tunability and ambipolar field effects. Such devices allow us to systematically study the resistance peak as well as the Hall effect at various doping densities and temperatures, revealing the contribution from electron-hole asymmetry and multiple-carrier transport. By comparing with theoretical calculations, we suggest a simplified semiclassical two-band model to explain the experimental observations. Our work helps to resolve the longstanding puzzles on ZrTe₅ and could potentially pave the way for realizing novel topological states in the two-dimensional limit.

Quantum-Hall physics and three dimensions

The discovery of the quantum Hall effect (QHE) in 1980 marked a turning point in condensed matter physics: given appropriate experimental conditions, the Hall conductivityσxyof a two-dimensional electron system is exactly quantized. But what happens to the QHE in three dimensions (3D)? Experiments over the past 40 years showed that some of the remarkable physics of the QHE, in particular plateau-like Hall conductivitiesσxyaccompanied by minima in the longitudinal resistivityρxx, can also be found in 3D materials. However, since typicallyρxxremains finite and a quantitative relation betweenσxyand the conductance quantume2/hcould not be established, the role of quantum Hall physics in 3D remains unsettled. Following a recent series of exciting experiments, the QHE in 3D has now returned to the center stage. Here, we summarize the leap in understanding of 3D matter in magnetic fields emerging from these experiments.

Observation of Dirac Charge-Density Waves in Bi2Te2Se

While parallel segments in the Fermi level contours, often found at the surfaces of topological insulators (TIs), would imply "strong" nesting conditions, the existence of charge-density waves (CDWs)-periodic modulations of the electron density-has not been verified up to now. Here, we report the observation of a CDW at the surface of the TI Bi2Te2Se(111), below ≈350K, by helium-atom scattering and, thus, experimental evidence for a CDW involving Dirac topological electrons. Deviations of the order parameter observed below 180K, and a low-temperature break of time reversal symmetry, suggest the onset of a spin-density wave with the same period as the CDW in the presence of a prominent electron-phonon interaction, originating from Rashba spin-orbit coupling.

Superconductivity in single-crystalline ZrTe3-x (x ≤ 0.5) nanoplates

Superconductivity with an unusual filamented character below 2 K has been reported in bulk ZrTe₃ crystals, a well-known charge density wave (CDW) material, but still lacks in its nanostructures. Here, we systemically investigated the transport properties of controllable chemical vapor transport synthesized ZrTe_3-x nanoplates. Intriguingly, superconducting behavior is found at T _c = 3.4 K and can be understood by the suppression of CDW due to the atomic disorder formed by Te vacancies. Magnetic field and angle dependent upper critical field revealed that the superconductivity in the nanoplates exhibits a large anisotropy and two-dimensional character. This two-dimensional nature of superconductivity was further satisfactorily described using the Berezinsky-Kosterlitz-Thouless transition. Our results not only demonstrate the critical role of Te vacancies for superconductivity in ZrTe_3-x nanoplates, but also provide a promising platform to explore the exotic physics in the nanostructure devices.

Topological Lifshitz transition and one-dimensional Weyl mode in HfTe5

Landau band crossings typically stem from the intra-band evolution of electronic states in magnetic fields and enhance the interaction effect in their vicinity. Here in the extreme quantum limit of topological insulator HfTe₅, we report the observation of a topological Lifshitz transition from inter-band Landau level crossings using magneto-infrared spectroscopy. By tracking the Landau level transitions, we demonstrate that band inversion drives the zeroth Landau bands to cross with each other after 4.5 T and forms a one-dimensional Weyl mode with the fundamental gap persistently closed. The unusual reduction of the zeroth Landau level transition activity suggests a topological Lifshitz transition at 21 T, which shifts the Weyl mode close to the Fermi level. As a result, a broad and asymmetric absorption feature emerges due to the Pauli blocking effect in one dimension, along with a distinctive negative magneto-resistivity. Our results provide a strategy for realizing one-dimensional Weyl quasiparticles in bulk crystals.

Photonic topological phases in Tellegen metamaterials

We investigate the photonic topological phases in Tellegen metamaterials characterized by the antisymmetric magnetoelectric tensors with real-valued quantities. The underlying medium is considered a photonic analogue of the topological semimetal featured with a displaced Weyl cone in the frequency-wave vector space. As the 'spin'-degenerate condition is satisfied, the photonic system consists of two hybrid modes that are completely decoupled. By introducing the pseudospin states as the basis for the hybrid modes, the photonic system is described by two subsystems in terms of the spin-orbit Hamiltonians with spin 1, which result in nonzero spin Chern numbers that determine the topological properties. Surface modes at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter exist at their common gap in the wave vector space, which are analytically formulated by algebraic equations. In particular, two types of surface modes are tangent to or wrapping around the Weyl cones, which form a pair of bended and a pair of twisted surface sheets. At the Weyl frequency, the surface modes contain a typical and two open Fermi arc-like states that concatenate to yield an infinite straight line. Topological features of the Tellegen metamaterials are further illustrated with the robust transport of surface modes at an irregular boundary.

Axial-Bonding-Driven Dimensionality Effect on the Charge-Density Wave in NbSe2

2H-NbSe₂ is a prototypical charge-density-wave (CDW) system, exhibiting such a symmetry-breaking quantum ground state in its bulk and down to a single-atomic-layer limit. However, how this state depends on dimensionality and what governs the dimensionality effect remain controversial. Here, we experimentally demonstrate a robust 3 × 3 CDW phase in both freestanding and substrate-supported bilayer NbSe₂, far above the bulk transition temperature. We exclude environmental effects and reveal a strong temperature and thickness dependence of Raman intensity from an axially vibrating A_1g phonon mode, involving Se ions. Using first-principles calculations, we show that these result from a delicate but profound competition between the intra- and interlayer bonding formed between Se-p_z orbitals. Our results suggest the crucial role of Se out-of-plane displacement in driving the CDW distortion, revealing the Se-dominated dimensionality effect and establishing a new perspective on the chemical bonding and mechanical stability in layered CDW materials.

Signatures of a magnetic-field-induced Lifshitz transition in the ultra-quantum limit of the topological semimetal ZrTe5

The quantum limit (QL) of an electron liquid, realised at strong magnetic fields, has long been proposed to host a wealth of strongly correlated states of matter. Electronic states in the QL are, for example, quasi-one dimensional (1D), which implies perfectly nested Fermi surfaces prone to instabilities. Whereas the QL typically requires unreachably strong magnetic fields, the topological semimetal ZrTe5 has been shown to reach the QL at fields of only a few Tesla. Here, we characterize the QL of ZrTe5 at fields up to 64 T by a combination of electrical-transport and ultrasound measurements. We find that the Zeeman effect in ZrTe5 enables an efficient tuning of the 1D Landau band structure with magnetic field. This results in a Lifshitz transition to a 1D Weyl regime in which perfect charge neutrality can be achieved. Since no instability-driven phase transitions destabilise the 1D electron liquid for the investigated field strengths and temperatures, our analysis establishes ZrTe5 as a thoroughly understood platform for potentially inducing more exotic interaction-driven phases at lower temperatures.

Photonic Weyl semimetals in pseudochiral metamaterials

We investigate the photonic topological phases in pseudochiral metamaterials characterized by the magnetoelectric tensors with symmetric off-diagonal chirality components. The underlying medium is considered a photonic analogue of the type-II Weyl semimetal featured with two pairs of tilted Weyl cones in the frequency-wave vector space. As the ’spin’-degenerate condition is satisfied, the photonic system consists of two hybrid modes that are completely decoupled. By introducing the pseudospin states as the basis for the hybrid modes, the photonic system is described by two subsystems in terms of the spin-orbit Hamiltonians with spin 1, which result in nonzero spin Chern numbers that determine the topological properties. Surface modes at the interface between vacuum and the pseudochiral metamaterial exist in their common gap in the wave vector space, which are analytically formulated by algebraic equations. In particular, the surface modes are tangent to both the vacuum light cone and the Weyl cones, which form two pairs of crossing surface sheets that are symmetric about the transverse axes. At the Weyl frequency, the surface modes that connect the Weyl points form four Fermi arc-like states as line segments. Topological features of the pseudochiral metamaterials are further illustrated with the robust transport of surface modes at an irregular boundary.

Magnetic field-induced non-linear transport in HfTe5

The interplay of electron correlations and topological phases gives rise to various exotic phenomena including fractionalization, excitonic instability and axionic excitation. Recently discovered transition-metal pentatellurides can reach the ultra-quantum limit in low magnetic fields and serve as good candidates for achieving such a combination. Here, we report evidence of density wave and metal-insulator transition in HfTe5 induced by intense magnetic fields. Using the non-linear transport technique, we detect a distinct non-linear conduction behavior in the longitudinal resistivity within the a-c plane, corresponding to the formation of a density wave induced by magnetic fields. In high fields, the onset of non-linear conduction in the Hall resistivity indicates an impurity-pinned magnetic freeze-out as the possible origin of the insulating behavior. These frozen electrons can be gradually reactivated into mobile states above a threshold of electric field. This experimental evidence calls for further investigation into the underlying mechanism of the bulk quantum Hall effect and field-induced phase transitions in pentatellurides.

Topological Chern vectors in three-dimensional photonic crystals

The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator^1-5, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector^6,7, similar to the three-dimensional (3D) quantum Hall effect^8-13. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers are determined by the Chern vectors. We demonstrate a sample with surface states forming a (2, 2) torus link or Hopf link in the surface Brillouin zone, which is topologically distinct from the surface states of other 3D topological phases. These results establish the Chern vector as an intrinsic bulk topological invariant in 3D topological materials, with surface states possessing unique topological characteristics.

Topological Crystalline Insulator Candidate ErAsS with Hourglass Fermion and Magnetic-Tuned Topological Phase Transition

Topological crystalline insulators (TCIs) with hourglass fermion surface state have attracted a lot of attention and are further enriched by crystalline symmetries and magnetic order. Here, the emergence of hourglass fermion surface state and exotic phases in the newly discovered, air-stable ErAsS single crystals are shown. In the paramagnetic phase, ErAsS is expected to be a TCI with hourglass fermion surface state protected by the nonsymmorphic symmetry. Dirac-cone-like bands and nearly linear dispersions in large energy range are experimentally observed, consistent well with theoretical calculations. Below T_N  ≈ 3.27 K, ErAsS enters a collinear antiferromagnetic state, which is a trivial insulator breaking the time-reversal symmetry. An intermediate incommensurate magnetic state appears in a narrow temperature range (3.27-3.65 K), exhibiting an abrupt change in magnetic coupling. The results reveal that ErAsS is an experimentally available TCI candidate and provide a unique platform to understand the formation of hourglass fermion surface state and explore magnetic-tuned topological phase transitions.

Protonation enhanced superconductivity in PdTe2

Electrochemical ionic liquid gating is an effective way to intercalate ions into layered materials and modulate the properties. Here we report an enhanced superconductivity in a topological superconductor candidate PdTe₂through electrochemical gating procedure. The superconducting transition temperature was increased to approximately 3.2 K by ionic gating induced protonation at room temperature. Moreover, a further enhanced superconductivity of both superconducting transition temperature and superconducting volume fraction was observed after the gated samples were placed in a glove box for 2 months. This may be caused by the diffusion of protons in the gated single crystals, which is rarely reported in electrochemical ionic liquid gating experiments. Our results further the superconducting study of PdTe₂and may reveal a common phenomenon in the electrochemical gating procedure.

Quasi-1D van der Waals Antiferromagnetic CrZr4 Te14 with Large In-Plane Anisotropic Negative Magnetoresistance

The discovery of 2D van der Waals (vdW) magnetic materials is of great significance to explore intriguing 2D magnetic physics and develop innovative spintronic devices. In this work, a new quasi-1D vdW layered compound CrZr₄ Te₁₄ is successfully synthesized. Owing to the existence of 1D [CrTe₂ ] and [ZrTe₃ ] chains along the b-axis, CrZr₄ Te₁₄ crystals show strong anisotropy of phonon vibrations, electrical transport, and magnetism. Density functional theory calculations reveal the ferromagnetic (FM) coupling within the [CrTe₂ ] chain, while the interchain and interlayer couplings are both weakly antiferromagnetic (AF). Notably, a large intrinsic negative magnetoresistance (nMR) of -56% is achieved at 2 K under 9 T, and the in-plane anisotropic factor of nMR can reach up to 8.2 in the CrZr₄ Te₁₄ device. The 1D FM chains and anisotropic nMR effect make CrZr₄ Te₁₄ an interesting platform for exploring novel polarization-sensitive spintronics.

Chirality locking charge density waves in a chiral crystal

In Weyl semimetals, charge density wave (CDW) order can spontaneously break the chiral symmetry, gap out the Weyl nodes, and drive the material into the axion insulating phase. Investigations have however been limited since CDWs are rarely seen in Weyl semimetals. Here, using scanning tunneling microscopy/spectroscopy (STM/S), we report the discovery of a novel unidirectional CDW order on the (001) surface of chiral crystal CoSi - a unique Weyl semimetal with unconventional chiral fermions. The CDW is incommensurate with both lattice momentum and crystalline symmetry directions, and exhibits an intra unit cell π phase shift in the layer stacking direction. The tunneling spectrum shows a particle-hole asymmetric V-shaped energy gap around the Fermi level that modulates spatially with the CDW wave vector. Combined with first-principle calculations, we identify that the CDW is locked to the crystal chirality and is related by a mirror reflection between the two enantiomers of the chiral crystal. Our findings reveal a novel correlated topological quantum state in chiral CoSi crystals and raise the potential for exploring the unprecedented physical behaviors of unconventional chiral fermions.

Controllable Synthesis Quadratic-Dependent Unsaturated Magnetoresistance of Two-Dimensional Nonlayered Fe7S8 with Robust Environmental Stability

Two-dimensional (2D) iron chalcogenides (FeX, X = S, Se, Te) are emerging as an appealing class of materials for a wide range of research topics, including electronics, spintronics, and catalysis. However, the controlled syntheses and intrinsic property explorations of such fascinating materials still remain daunting challenges, especially for 2D nonlayered Fe₇S₈ with mixed-valence states and high conductivity. Herein, we design a general and temperature-mediated chemical vapor deposition (CVD) approach to synthesize ultrathin and large-domain Fe₇S₈ nanosheets on mica substrates, with the thickness down to ∼4.4 nm (2 unit-cell). Significantly, we uncover a quadratic-dependent unsaturated magnetoresistance (MR) with out-of-plane anisotropy in 2D Fe₇S₈, thanks to its ultrahigh crystalline quality and high conductivity (∼2.7 × 10⁵ S m^-1 at room temperature and ∼1.7 × 10⁶ S m^-1 at 2 K). More interestingly, the CVD-synthesized 2D Fe₇S₈ nanosheets maintain robust environmental stability for more than 8 months. These results hereby lay solid foundations for synthesizing 2D nonlayered iron chalcogenides with mixed-valence states and exploring fascinating quantum phenomena.

Gigantic Magnetochiral Anisotropy in the Topological Semimetal ZrTe_{5}

Topological materials with broken inversion symmetry can give rise to nonreciprocal responses, such as the current rectification controlled by magnetic fields via magnetochiral anisotropy. Bulk nonreciprocal responses usually stem from relativistic corrections and are always very small. Here we report our discovery that ZrTe_{5} crystals in proximity to a topological quantum phase transition present gigantic magnetochiral anisotropy, which is the largest ever observed to date. We argue that a very low carrier density, inhomogeneities, and a torus-shaped Fermi surface induced by breaking of inversion symmetry in a Dirac material are central to explain this extraordinary property.

Suspended semiconductor nanostructures: physics and technology

The current state of research on quantum and ballistic electron transport in semiconductor nanostructures with a two-dimensional electron gas separated from the substrate and nanoelectromechanical systems is reviewed. These nanostructures fabricated using the surface nanomachining technique have certain unexpected features in comparison to their non-suspended counterparts, such as additional mechanical degrees of freedom, enhanced electron-electron interaction and weak heat sink. Moreover, their mechanical functionality can be used as an additional tool for studying the electron transport, complementary to the ordinary electrical measurements. The article includes a comprehensive review of spin-dependent electron transport and multichannel effects in suspended quantum point contacts, ballistic and adiabatic transport in suspended nanostructures, as well as investigations on nanoelectromechanical systems. We aim to provide an overview of the state-of-the-art in suspended semiconductor nanostructures and their applications in nanoelectronics, spintronics and emerging quantum technologies.

Anisotropic Three-Dimensional Quantum Hall Effect and Magnetotransport in Mesoscopic Weyl Semimetals

Weyl semimetals are emerging to become a new class of quantum-material platform for various novel phenomena. Especially, the Weyl orbit made from surface Fermi arcs and bulk relativistic states is expected to play a key role in magnetotransport, leading even to a three-dimensional quantum Hall effect (QHE). It is experimentally and theoretically important although yet unclear whether it bears essentially the same phenomenon as the conventional two-dimensional QHE. We discover an unconventional fully three-dimensional anisotropy in the quantum transport under a magnetic field. Strong suppression and even disappearance of the QHE occur when the Hall-bar current is rotated away from being transverse to parallel with respect to the Weyl point alignment, which is attributed to a peculiar absence of conventional bulk-boundary correspondence. Besides, transport along the magnetic field can exhibit a remarkable reversal from negative to positive magnetoresistance. These results establish the uniqueness of this QHE system as a novel three-dimensional quantum matter.

Anisotropic large diamagnetism in Dirac semimetals ZrTe5and HfTe5

Dirac semimetals, e.g., ZrTe₅and HfTe₅, have been widely investigated and have exhibited various exotic physical properties. Nevertheless, several properties of these compounds, including diamagnetism, are still unclear. In this study, we measured the temperature- and field-dependent diamagnetism of ZrTe₅and HfTe₅along all three crystallographic axes (a-,b-, andc-axis). The temperature-dependent magnetization shows an anomaly, which is a characteristic of Dirac crossing. Diamagnetic signal reaches the highest value of 17.3 × 10^-4emu mol^-1Oe^-1along the van der Waals layers, i.e., theb-axis. However, the diamagnetism remains temperature-independent along the other two axes. The field-dependent diamagnetic signal grows linearly without any sign of saturation and maintains a large value along theb-axis. Interestingly, the observed diamagnetism is anisotropic like other physical properties of these compounds and is strongly related to the effective mass, indicating the dominating contribution of orbital diamagnetism in Dirac semimetals induced by interband effects. ZrTe₅and HfTe₅show one of the largest diamagnetic value among previously reported state-of-the-art topological semimetals. Our present study adds another important experimental aspect to characterize nodal crossing and search for other topological materials with large magnetic susceptibility.

High-Performance Memristors Based on Ultrathin 2D Copper Chalcogenides

Copper chalcogenides represent a class of materials with unique crystal structures, high electrical conductivity, and earth abundance, and are recognized as promising candidates for next-generation green electronics. However, their 2D structures and the corresponding electronic properties have rarely been touched. Herein, a series of ultrathin copper chalcogenide nanosheets with thicknesses down to two unit cells are successfully synthesized, including layered Cu₂ Te, as well as nonlayered CuSe and Cu₉ S₅ , via van der Waals epitaxy, and their nonvolatile memristive behavior is investigated for the first time. Benefiting from the highly active Cu ions with low migration barriers, the memristors based on ultrathin 2D copper chalcogenide crystals exhibit relatively small switching voltage (≈0.4 V), fast switching speed, high switching uniformity, and wide operating temperature range (from 80 to 420 K), as well as stable retention and good cyclic endurance. These results demonstrate their tangible applications in future low-power, cryogenic, and high temperature harsh electronics.

Quasi-quantized Hall response in bulk InAs

The quasi-quantized Hall effect (QQHE) is the three-dimensional (3D) counterpart of the integer quantum Hall effect (QHE), exhibited only by two-dimensional (2D) electron systems. It has recently been observed in layered materials, consisting of stacks of weakly coupled 2D platelets that are yet characterized by a 3D anisotropic Fermi surface. However, it is predicted that the quasi-quantized 3D version of the 2D QHE should occur in a much broader class of bulk materials, regardless of the underlying crystal structure. Here, we compare the observation of quasi-quantized plateau-like features in the Hall conductivity of the n-type bulk semiconductor InAs with the predictions for the 3D QQHE in presence of parabolic electron bands. InAs takes form of a cubic crystal without any low-dimensional substructure. The onset of the plateau-like feature in the Hall conductivity scales with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{2/3}k_{\text {F}}^{z}/\pi$$\end{document}2/3kFz/π in units of the conductance quantum and is accompanied by a Shubnikov–de Haas minimum in the longitudinal resistivity, consistent wit the results of calculations. This confirms the suggestion that the 3D QQHE may be a generic effect directly observable in materials with small Fermi surfaces, placed in sufficiently strong magnetic fields.

Gate-Tunable Transport in Quasi-One-Dimensional α-Bi4I4 Field Effect Transistors

Bi₄I₄ belongs to a novel family of quasi-one-dimensional (1D) topological insulators (TIs). While its β phase was demonstrated to be a prototypical weak TI, the α phase, long thought to be a trivial insulator, was recently predicted to be a rare higher order TI. Here, we report the first gate tunable transport together with evidence for unconventional band topology in exfoliated α-Bi₄I₄ field effect transistors. We observe a Dirac-like longitudinal resistance peak and a sign change in the Hall resistance; their temperature dependences suggest competing transport mechanisms: a hole-doped insulating bulk and one or more gate-tunable ambipolar boundary channels. Our combined transport, photoemission, and theoretical results indicate that the gate-tunable channels likely arise from novel gapped side surface states, two-dimensional (2D) TI in the bottommost layer, and/or helical hinge states of the upper layers. Markedly, a gate-tunable supercurrent is observed in an α-Bi₄I₄ Josephson junction, underscoring the potential of these boundary channels to mediate topological superconductivity.

Vergniory M, García-Etxarri A. Cubic 3D Chern photonic insulators with orientable large Chern vectors

Time Reversal Symmetry (TRS) broken topological phases provide gapless surface states protected by topology, regardless of additional internal symmetries, spin or valley degrees of freedom. Despite the numerous demonstrations of 2D topological phases, few examples of 3D topological systems with TRS breaking exist. In this article, we devise a general strategy to design 3D Chern insulating (3D CI) cubic photonic crystals in a weakly TRS broken environment with orientable and arbitrarily large Chern vectors. The designs display topologically protected chiral and unidirectional surface states with disjoint equifrequency loops. The resulting crystals present the following characteristics: First, by increasing the Chern number, multiple surface states channels can be supported. Second, the Chern vector can be oriented along any direction simply changing the magnetization axis, opening up larger 3D CI/3D CI interfacing possibilities as compared to 2D. Third, by lowering the TRS breaking requirements, the system is ideal for realistic photonic applications where the magnetic response is weak.

Synthetic Three-Dimensional Z×Z_{2} Topological Insulator in an Elastic Metacrystal

We report a three-dimensional (3D) topological insulator (TI) formed by stacking identical layers of Chern insulators in a hybrid real-synthetic space. By introducing staggered interlayer hopping that respects mirror symmetry, the bulk bands possess an additional Z_{2} topological invariant along the stacking dimension, which, together with the nontrivial Chern numbers, endows the system with a Z×Z_{2} topology. A 4-tuple topological index characterizes the system's bulk bands. Consequently, two distinct types of topological surface modes (TSMs) are found localized on different surfaces. Type-I TSMs are gapless and are protected by Chern numbers, whereas type-II gapped TSMs are protected by Z_{2} bulk polarization in the stacking direction. Remarkably, each type-II TSM band is also topologically nontrivial, giving rise to second-order topological hinge modes (THMs). Both types of TSMs and the THMs are experimentally observed in an elastic metacrystal.

Magneto-transport evidence for strong topological insulator phase in ZrTe5

The identification of a non-trivial band topology usually relies on directly probing the protected surface/edge states. But, it is difficult to achieve electronically in narrow-gap topological materials due to the small (meV) energy scales. Here, we demonstrate that band inversion, a crucial ingredient of the non-trivial band topology, can serve as an alternative, experimentally accessible indicator. We show that an inverted band can lead to a four-fold splitting of the non-zero Landau levels, contrasting the two-fold splitting (spin splitting only) in the normal band. We confirm our predictions in magneto-transport experiments on a narrow-gap strong topological insulator, zirconium pentatelluride (ZrTe5), with the observation of additional splittings in the quantum oscillations and also an anomalous peak in the extreme quantum limit. Our work establishes an effective strategy for identifying the band inversion as well as the associated topological phases for future topological materials research.

Quantum Oscillations in Ferromagnetic (Sb, V)2 Te3 Topological Insulator Thin Films

An effective way of manipulating 2D surface states in magnetic topological insulators may open a new route for quantum technologies based on the quantum anomalous Hall effect. The doping-dependent evolution of the electronic band structure in the topological insulator Sb2- x Vx Te3 (0 ≤ x ≤ 0.102) thin films is studied by means of electrical transport. Sb2- x Vx Te3 thin films were prepared by molecular beam epitaxy, and Shubnikov-de Hass (SdH) oscillations are observed in both the longitudinal and transverse transport channels. Doping with the 3d element, vanadium, induces long-range ferromagnetic order with enhanced SdH oscillation amplitudes. The doping effect is systematically studied in various films depending on thickness and bottom gate voltage. The angle-dependence of the SdH oscillations reveals their 2D nature, linking them to topological surface states as their origin. Furthermore, it is shown that vanadium doping can efficiently modify the band structure. The tunability by doping and the coexistence of the surface states with ferromagnetism render Sb2- x Vx Te3 thin films a promising platform for energy band engineering. In this way, topological quantum states may be manipulated to crossover from quantum Hall effect to quantum anomalous Hall effect, which opens an alternative route for the design of quantum electronics and spintronics.

Engineering Three-Dimensional Moiré Flat Bands

Twisting two adjacent layers of van der Waals materials with respect to each other can lead to flat two-dimensional electronic bands which enables a wealth of physical phenomena. Here, we generalize this concept of so-called moiré flat bands to engineer flat bands in all three spatial dimensions controlled by the twist angle. The basic concept is to stack the material such that the large spatial moiré interference patterns are spatially shifted from one twisted layer to the next. We exemplify the general concept by considering graphitic systems, boron nitride, and WSe2, but the approach is applicable to any two-dimensional van der Waals material. For hexagonal boron nitride, we develop an ab initio fitted tight binding model that captures the corresponding three-dimensional low-energy electronic structure. We outline that interesting three-dimensional correlated phases of matter can be induced and controlled following this route, including quantum magnets and unconventional superconducting states.

Field-Tunable One-Sided Higher-Order Topological Hinge States in Dirac Semimetals

Recently, higher-order topological matter and 3D quantum Hall effects have attracted a great amount of attention. The Fermi-arc mechanism of the 3D quantum Hall effect proposed to exist in Weyl semimetals is characterized by the one-sided hinge states, which do not exist in all the previous quantum Hall systems, and more importantly, pose a realistic example of the higher-order topological matter. The experimental effort so far is in the Dirac semimetal Cd_{3}As_{2}, where, however, time-reversal symmetry leads to hinge states on both sides of the top and bottom surfaces, instead of the aspired one-sided hinge states. We propose that under a tilted magnetic field, the hinge states in Cd_{3}As_{2}-like Dirac semimetals can be one sided, highly tunable by field direction and Fermi energy, and robust against weak disorder. Furthermore, we propose a scanning tunneling Hall measurement to detect the one-sided hinge states. Our results will be insightful for exploring not only the quantum Hall effects beyond two dimensions, but also other higher-order topological insulators in the future.

Two-Dimensional Quantum Hall Effect and Zero Energy State in Few-Layer ZrTe5

Topological matter plays a central role in today's condensed matter research. Zirconium pentatelluride (ZrTe5) has attracted attention as a Dirac semimetal at the boundary of weak and strong topological insulators (TI). Few-layer ZrTe5 is anticipated to exhibit the quantum spin Hall effect due to topological states inside the band gap, but sample degradation inflicted by ambient conditions and processing has so far hampered the fabrication of high quality devices. The quantum Hall effect (QHE), serving as the litmus test for 2D systems to be considered of high quality, has not been observed so far. Only a 3D variant on bulk was reported. Here, we succeeded in preserving the intrinsic properties of thin films lifting the carrier mobility to ∼3500 cm2 V-1 s-1, sufficient to observe the integer QHE and a bulk band gap related zero-energy state. The magneto-transport results offer evidence for the gapless topological states within this gap.

Theory for Magnetic-Field-Driven 3D Metal-Insulator Transitions in the Quantum Limit

Metal-insulator transitions driven by magnetic fields have been extensively studied in 2D, but a 3D theory is still lacking. Motivated by recent experiments, we develop a scaling theory for the metal-insulator transitions in the strong-magnetic-field quantum limit of a 3D system. By using a renormalization-group calculation to treat electron-electron interactions, electron-phonon interactions, and disorder on the same footing, we obtain the critical exponent that characterizes the scaling relations of the resistivity to temperature and magnetic field. By comparing the critical exponent with those in a recent experiment [F. Tang et al., Nature (London) 569, 537 (2019)NATUAS0028-083610.1038/s41586-019-1180-9], we conclude that the insulating ground state was not only a charge-density wave driven by electron-phonon interactions but also coexisting with strong electron-electron interactions and backscattering disorder. We also propose a current-scaling experiment for further verification. Our theory will be helpful for exploring the emergent territory of 3D metal-insulator transitions under strong magnetic fields.

Orbit topology analyzed from π phase shift of magnetic quantum oscillations in three-dimensional Dirac semimetal

With the emergence of Dirac fermion physics in the field of condensed matter, magnetic quantum oscillations (MQOs) have been used to discern the topology of orbits in Dirac materials. However, many previous researchers have relied on the single-orbit Lifshitz-Kosevich (LK) formula, which overlooks the significant effect of degenerate orbits on MQOs. Since the single-orbit LK formula is valid for massless Dirac semimetals with small cyclotron masses, it is imperative to generalize the method applicable to a wide range of Dirac semimetals, whether massless or massive. This report demonstrates how spin-degenerate orbits affect the phases in MQOs of three-dimensional massive Dirac semimetal, NbSb2 With varying the direction of the magnetic field, an abrupt π phase shift is observed due to the interference between the spin-degenerate orbits. We investigate the effect of cyclotron mass on the π phase shift and verify its close relation to the phase from the Zeeman coupling. We find that the π phase shift occurs when the cyclotron mass is half of the electron mass, indicating the effective spin gyromagnetic ratio as g s = 2. Our approach is not only useful for analyzing MQOs of massless Dirac semimetals with a small cyclotron mass but also can be used for MQOs in massive Dirac materials with degenerate orbits, especially in topological materials with a sufficiently large cyclotron mass. Furthermore, this method provides a useful way to estimate the precise g s value of the material.