Observation of the nonlinear Hall effect under time-reversal-symmetric conditions

Photon-drag photovoltaic effects and quantum geometric nature

The bulk photovoltaic effect (BPVE) generates a direct current photocurrent under uniform irradiation and is a nonlinear optical effect traditionally studied in noncentrosymmetric materials. The two main origins of BPVE are the shift and injection currents, arising from transitions in electron position and electron velocity during optical excitation, respectively. Recently, it was proposed that photon-drag effects could unlock BPVE in centrosymmetric materials. However, experimental progress remains limited. In this work, we provide a comprehensive theoretical analysis of photon-drag effects inducing BPVE (photon-drag BPVE). Notably, we find that photon-drag BPVE can be directly linked to quantum geometric tensors. Additionally, we propose that photon-drag shift currents can be fully isolated from other current contributions in nonmagnetic centrosymmetric materials. We apply our theory explicitly to the 2D topological insulator 1T'-WTe2. Furthermore, we investigate photon-drag BPVE in a centrosymmetric magnetic Weyl semimetal, where we demonstrate that linearly polarized light generates photon-drag shift currents.

Quantum geometry in condensed matter

One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of electrons in condensed matter can be characterized by the geometry of the Hilbert space of their wave functions. Such quantum geometry, comprising Berry curvature and the quantum metric, can thus exert profound influences on various properties of materials. The dipoles of both Berry curvature and the quantum metric produce nonlinear transport. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. The uniformly distributed momentum-space quantum geometry stabilizes the fractional Chern insulators and results in the fractional quantum anomalous Hall effect. Here we review in detail quantum geometry in condensed matter, paying close attention to its effects on nonlinear transport, superconductivity and topological properties. Possible future research directions in this field are also envisaged.

Intrinsic Nonlinear Spin Hall Effect and Manipulation of Perpendicular Magnetization

We propose an intrinsic nonlinear spin Hall effect, which enables the generation of collinearly polarized spin current in a large class of nonmagnetic materials with the corresponding linear response being symmetry forbidden. This opens a new avenue for field-free switching of perpendicular magnetization, which is required for the next-generation information storage technology. We develop the microscopic theory of this effect and clarify its quantum origin in band geometric quantities which can be enhanced by topological nodal features. Combined with first-principles calculations, we predict pronounced effects at room temperature in topological metals PbTaSe_{2} and PdGa. Our work establishes a fundamental nonlinear response in spin transport and opens the door to exploring spintronic applications based on nonlinear spin Hall effect.

Metallic Electro-optic Effect in Gapped Bilayer Graphene

Electro-optic (EO) modulation is a critical device action in photonics. Recently, the non-Drude dynamics induced by the Berry curvature dipole (BCD) in metals have attracted attention as a potential candidate for terahertz EO modulation. However, such BCD-induced EO effects can be challenging to realize, often requiring flat bands and complex materials such as a strained magic-angle twisted bilayer graphene on hexagonal boron nitride. Here, we argue that metallic EO can be achieved with a much simpler material, gapped bilayer graphene, with EO coefficients comparable to that in flat-band materials in the terahertz range. In particular, we find metallic EO can be realized without a Berry curvature dipole; we identify skew-scattering and a "snap" (third-order derivative of velocity) can readily produce pronounced Pockels and Kerr EO effects in clean metals. These yield nonreciprocal and field-activated birefringence and field-induced modulations to transmission and reflection, essential components for terahertz EO modulators.

Terahertz Radiation Driven Nonlinear Transport Phenomena in Two-Dimensional Tellurene

Nonlinear electron transport induced by polarized terahertz radiation is studied in two-dimensional tellurene at room temperature. A direct current, quadratic in the radiation's electric field, is observed. Contributions sensitive to radiation helicity and polarization orientation as well as polarization independent current are found. We show that these contributions can be modified by the magnitude of the external gate potential. We demonstrate that this terahertz-driven electric current arises from the Berry curvature dipole and the side-jump microscopic mechanisms.

Probing quantum geometry through optical conductivity and magnetic circular dichroism

Probing ground-state quantum geometry and topology through optical responses is not only of fundamental interest, but it can also offer several practical advantages. Here, using first-principles calculations on thin films of the antiferromagnetic topological insulator MnBi2Te4, we demonstrate how the generalized optical weight arising from the absorptive part of the optical conductivity can be used to probe the ground-state quantum geometry and topology. We show that three-septuple-layer MnBi2Te4 film exhibit an enhanced, almost-perfect magnetic circular dichroism for a narrow photon energy window in the infrared region. We calculate the quantum weight in this MnBi2Te4 film and show that it far exceeds the lower bound provided by the Chern number. Our results suggest that the well-known optical methods are powerful tools for probing the ground-state quantum geometry and topology.

First-principles study of electron dynamics of MoS2 under femtosecond laser irradiation from deep ultraviolet to near-infrared wavelengths

Time-dependent density functional theory was employed to investigate the electron dynamics of MoS2 following femtosecond pulse irradiation. The study concerned the effects of laser wavelength, intensities, and polarization and elucidated the ionization mechanisms across the intensity range of 1010-1014 W/cm2. As laser intensity increases, MoS2 irradiated with an infrared (IR) laser (800 nm) deviates from single-photon absorption at lower intensities compared to that subjected to an ultraviolet (UV) laser (266 nm), and nonlinear effects in the current arise at lower intensities for the 800 nm laser. At a wavelength of 266 nm, MoS2 irradiated with an a-axis polarized laser deposited more energy and generated more electron-hole pairs compared to c-axis polarization. Rate equations were used to estimate the total number of excited electrons in MoS2 and the corresponding plasma frequency. Simulation results indicate that the damage threshold of the UV laser is higher than that of the IR laser, which contradicts the experimental results. This outcome suggests that the mechanism of material damage induced by the UV femtosecond laser near the damage threshold is independent of optical breakdown. The findings of this research are significant for enhancing the performance of MoS2-based photodetectors and optimizing their stability and reliability in high-power, short-wavelength laser applications.

In-Plane Anomalous Hall Effect Associated with Orbital Magnetization: Measurements of Low-Carrier Density Films of a Magnetic Weyl Semimetal

For over a century, the Hall effect, a transverse effect under an out-of-plane magnetic field or magnetization, has been a cornerstone for magnetotransport studies and applications. Modern theoretical formulation based on the Berry curvature has revealed the potential that even an in-plane magnetic field can induce an anomalous Hall effect, but its experimental demonstration has remained difficult due to its potentially small magnitude and strict symmetry requirements. Here, we report observation of the in-plane anomalous Hall effect by measuring low-carrier density films of magnetic Weyl semimetal EuCd_{2}Sb_{2}. Anomalous Hall resistance exhibits distinct threefold rotational symmetry for changes in the in-plane field component, and this can be understood in terms of out-of-plane Weyl points splitting or orbital magnetization induced by the in-plane field, as also confirmed by model calculation. Our findings demonstrate the importance of the in-plane field to control the Hall effect, accelerating materials development and further exploration of various in-plane field-induced phenomena.

Colossal room-temperature non-reciprocal Hall effect

Non-reciprocal charge transport has gained significant attention due to its potential in exploring quantum symmetry and its promising applications. Traditionally, non-reciprocal transport has been observed in the longitudinal direction, with non-reciprocal resistance being a small fraction of the ohmic resistance. Here we report a transverse non-reciprocal transport phenomenon featuring a quadratic current-voltage characteristic and divergent non-reciprocity, termed the non-reciprocal Hall effect. This effect is observed in microscale Hall devices fabricated from platinum (Pt) deposited by a focused ion beam on silicon substrates. The transverse non-reciprocal Hall effect arises from the geometrically asymmetric scattering of textured Pt nanoparticles within the focused-ion-beam-deposited Pt structures. Notably, the non-reciprocal Hall effect generated in focused-ion-beam-deposited Pt electrodes can propagate to adjacent conductors such as Au and NbP through Hall current injection. Additionally, this pronounced non-reciprocal Hall effect facilitates broadband frequency mixing. These findings not only validate the non-reciprocal Hall effect concept but also open avenues for its application in terahertz communication, imaging and energy harvesting.

Nonlinear Longitudinal Current of Band-Geometric Origin in Wires of Finite Thickness

The miniaturization of integrated circuits is facing an obstruction due to the escalating electrical resistivity of conventional copper interconnects. The underlying reason for this problem was unveiled by Fuchs and Sondheimer, who showed that thinner wires are more resistive because current- carrying electrons encounter the rough surfaces of the wire more frequently therein. Here, we present a generalization of the Fuchs-Sondheimer theory to Dirac and Weyl materials, which are candidates for next-generation interconnects. We predict a nonlinear longitudinal electric current originating from the combined action of the Berry curvature and nonspecular surface scattering.

Skyrmion Hall Effect in Altermagnets

It is widely believed that the skyrmion Hall effect is absent in antiferromagnets because of the vanishing topological charge. However, the Aharonov-Casher theory indicates the possibility of topological effects for neutral particles. In this Letter, we predict the skyrmion Hall effect in emerging altermagnets with zero net magnetization and zero skyrmion charge. We first show that the neutral skyrmion manifests as a magnetic quadrupole in altermagnets. We reveal a hidden gauge field from the magnetic quadrupole, which induces the skyrmion Hall effect when driven by spin transfer torque. Interestingly, we identify a sign change of the Hall angle when one swaps the anisotropic exchange couplings in altermagnets. Furthermore, we demonstrate that both the velocity and Hall angle of altermagnetic skyrmions sensitively depend on the current direction. Our findings real the critical role of magnetic quadrupole in driving the skyrmion Hall effect with vanishing charge, and pave the way to discovering new Hall effect of neutral quasiparticles beyond magnetic skyrmions.

Nonlinear edge transport in a quantum Hall system

Nonlinear transport phenomena in condensed matter reflect the geometric nature, quantum coherence, and many-body correlation of electronic states. Electric currents in solids are classified into (i) ohmic current, (ii) supercurrent, and (iii) geometric or topological current. While the nonlinear current-voltage (I-V) characteristics of the former two categories have been extensive research topics recently, those of the last category remains unexplored. Among them, the quantum Hall current is a representative example. Realized in two-dimensional electronic systems under a strong magnetic field, the topological protection quantizes the Hall conductance in the unit of e2/h (e, elementary charge; and h, Planck constant), of which the edge transport picture gives a good account. Here, we theoretically study the nonlinear I-VH characteristic of the edge transport up to third order in VH. We find that nonlinearity arises in the Hall response from electron-electron interaction between the counterpropagating edge channels with the nonlinear energy dispersions. We also discuss possible experimental observations.

Quantum Rectification Based on Room Temperature Multidirectional Nonlinearity in Bi2Te3

Recent interest in quantum nonlinearity has spurred the development of rectifiers for harvesting energy from ambient radiofrequency waves. However, these rectifiers face efficiency and bandwidth limitations at room temperature. We address these challenges by exploring Bi2Te3, a time-reversal symmetric topological quantum material. Bi2Te3 exhibits robust room temperature second-order voltage generation in both the longitudinal and transverse directions. We harness these coexisting nonlinearities to design a multidirectional quantum rectifier that can simultaneously extract energy from various components of an input signal. We demonstrate the efficacy of Bi2Te3-based rectifiers across a broad frequency range, spanning from existing Wi-Fi bands (2.45 GHz) to frequencies relevant to next-generation 5G technology (27.4 GHz). Our Bi2Te3-based rectifier surpasses previous limitations by achieving a high rectification efficiency and operational frequency, alongside a low operational threshold and broadband functionality. These findings enable practical topological quantum rectifiers for high-frequency electronics and energy conversion, advancing wireless energy harvesting for next-generation communication.

Ferroelectric Semimetals with α-Bi/SnSe van der Waals Heterostructures and Their Topological Currents

We show that proximity effects can be utilized to engineer van der Waals heterostructures (vdWHs) displaying semimetallic spin-ferroelectricity locking, where ferroelectricity and semimetallic spin states are confined to different layers, but are correlated by means of proximity effects. Our findings are supported by first principles calculations involving α-Bi/SnSe bilayers. We show that such systems support ferroelectrically switchable nonlinear anomalous Hall effect originating from large Berry curvature dipoles as well as direct and inverse spin Hall effects with giant bulk spin-charge interconversion efficiencies. The giant efficiencies are consequences of the proximity-induced semimetallic nature of low energy electron states, which are shown to behave as two-dimensional pseudo-Weyl fermions by means of symmetry analysis and first principles calculations as well as direct angle-resolved photoemission spectroscopy measurements.

Opto-twistronic Hall effect in a three-dimensional spiral lattice

Studies of moiré systems have explained the effect of superlattice modulations on their properties, demonstrating new correlated phases1. However, most experimental studies have focused on a few layers in two-dimensional systems. Extending twistronics to three dimensions, in which the twist extends into the third dimension, remains underexplored because of the challenges associated with the manual stacking of layers. Here we study three-dimensional twistronics using a self-assembled twisted spiral superlattice of multilayered WS2. Our findings show an opto-twistronic Hall effect driven by structural chirality and coherence length, modulated by the moiré potential of the spiral superlattice. This is an experimental manifestation of the noncommutative geometry of the system. We observe enhanced light-matter interactions and an altered dependence of the Hall coefficient on photon momentum. Our model suggests contributions from higher-order quantum geometric quantities to this observation, providing opportunities for designing quantum-materials-based optoelectronic lattices with large nonlinearities.

Third-order nonlinear Hall effect in a quantum Hall system

In two-dimensional systems, perpendicular magnetic fields can induce a bulk band gap and chiral edge states, which gives rise to the quantum Hall effect. The quantum Hall effect is characterized by zero longitudinal resistance (Rxx) and Hall resistance (Rxy) plateaus quantized to h/(υe2) in the linear response regime, where υ is the Landau level filling factor, e is the elementary charge and h is Planck's constant. Here we explore the nonlinear response of monolayer graphene when tuned to a quantum Hall state. We observe a third-order Hall effect that exhibits a nonzero voltage plateau scaling cubically with the probe current. By contrast, the third-order longitudinal voltage remains zero. The magnitude of the third-order response is insensitive to variations in magnetic field (down to ~5 T) and in temperature (up to ~60 K). Moreover, the third-order response emerges in graphene devices with a variety of geometries, different substrates and stacking configurations. We term the effect third-order nonlinear response of the quantum Hall state and propose that electron-electron interaction between the quantum Hall edge states is the origin of the nonlinear response of the quantum Hall state.

Probing Majorana Wave Functions in Kitaev Honeycomb Spin Liquids with Second-Order Two-Dimensional Spectroscopy

Two-dimensional coherent terahertz spectroscopy (2DCS) emerges as a valuable tool to probe the nature, couplings, and lifetimes of excitations in quantum materials. It thus promises to identify unique signatures of spin liquid states in quantum magnets by directly probing properties of their exotic fractionalized excitations. Here, we calculate the second-order 2DCS of the Kitaev honeycomb model and demonstrate that distinct spin liquid fingerprints appear already in this lowest-order nonlinear response χ_{yzx}^{(2)}(ω_{1},ω_{2}) when using crossed light polarizations. We further relate the off-diagonal 2DCS peaks to the localized nature of the matter Majorana excitations trapped by Z_{2} flux excitations and show that 2DCS thus directly probes the inverse participation ratio of Majorana wave functions. By providing experimentally observable features of spin liquid states in the 2D spectrum, our Letter can guide future 2DCS experiments on Kitaev magnets.

Intrinsic dipole Hall effect in twisted MoTe2: magnetoelectricity and contact-free signatures of topological transitions

We discover an intrinsic dipole Hall effect in a variety of magnetic insulating states at integer fillings of twisted MoTe2 moiré superlattice, including topologically trivial and nontrivial ferro-, antiferro-, and ferri-magnetic configurations. The dipole Hall current, in linear response to in-plane electric field, generates an in-plane orbital magnetization M∥ along the field, through which an AC field can drive magnetization oscillation up to THz range. Upon the continuous topological phase transitions from trivial to quantum anomalous Hall states in both ferromagnetic and antiferromagnetic configurations, the dipole Hall current and M∥ have an abrupt sign change, enabling contact-free detection of the transitions through the magnetic stray field. In configurations where the linear response is forbidden by symmetry, the dipole Hall current and M∥ appear as a crossed nonlinear response to both in-plane and out-of-plane electric fields. These magnetoelectric phenomena showcase fascinating functionalities of insulators from the interplay between magnetism, topology, and electrical polarization.

Quantum Geometry Induced Nonlinear Transport in Altermagnets

Quantum geometry plays a pivotal role in the second-order response of PT-symmetric antiferromagnets. Here we study the nonlinear response of 2D altermagnets protected by C_{n}T symmetry and show that their leading nonlinear response is third order. Furthermore, we show that the contributions from the quantum metric and Berry curvature enter separately: the longitudinal response for all planar altermagnets only has a contribution from the quantum metric quadrupole (QMQ), while transverse responses in general have contributions from both the Berry curvature quadrupole (BCQ) and QMQ. We show that for the well-known example of d-wave altermagnets the Hall response is dominated by the BCQ. Both longitudinal and transverse responses are strongly dependent on the crystalline anisotropy. While altermagnets are strictly defined in the limit of vanishing spin orbit coupling (SOC), real altermagnets exhibit weak SOC, which is essential to observe this response. Specifically, SOC gaps the spin-group protected nodal line, generating a response peak that is sharpest when SOC is weak. Two Dirac nodes also contribute a divergence to the nonlinear response, whose scaling changes as a function of SOC. Finally, we apply our results to thin films of the 3D altermagnet RuO_{2}. Our work uncovers distinct features of altermagnets in nonlinear transport, providing experimental signatures as well as a guide to disentangling the different components of their quantum geometry.

Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4

The study of quantum geometry effects in materials has been one of the most important research directions in recent decades. The quantum geometry of a material is characterized by the quantum geometric tensor of the Bloch states. The imaginary part of the quantum geometry tensor gives rise to the Berry curvature while the real part gives rise to the quantum metric. While Berry curvature has been well studied in the past decades, the experimental investigation on the quantum metric effects is only at its infancy stage. In this work, we measure the nonlinear transport of bulk MnBi2Te4, which is a topological anti-ferromagnet. We found that the second order nonlinear responses are negligible as required by inversion symmetry, the third-order nonlinear responses are finite. The measured third-harmonic longitudinal (V x x 3 ω ) and transverse (V x y 3 ω ) voltages with frequency 3 ω , driven by an a.c. current with frequency ω , show an intimate connection with magnetic transitions of MnBi2Te4 flakes. Their magnitudes change abruptly as MnBi2Te4 flakes go through magnetic transitions from an antiferromagnetic state to a canted antiferromagnetic state and to a ferromagnetic state. In addition, the measuredV x x 3 ω is an even function of the applied magnetic field B whileV x y 3 ω is odd in B. Amazingly, the field dependence of the third-order responses as a function of the magnetic field suggests thatV x x 3 ω is induced by the quantum metric quadrupole andV x y 3 ω is induced by the Berry curvature quadrupole. Therefore, the quadrupoles of both the real and the imaginary part of the quantum geometry tensor of bulk MnBi2Te4 are revealed through the third order nonlinear transport measurements. This work greatly advanced our understanding on the connections between the higher order moments of quantum geometry and nonlinear transport.

Gate Modulation of Dissipationless Nonlinear Quantum Geometric Current in 2D Te

Two-dimensional trigonal tellurium (2D Te), a narrow-bandgap semiconductor with a bandgap of approximately 0.3 eV, hosts Weyl points near the band edge and exhibits a narrow, strong Berry curvature dipole (BCD). By applying a back-gate bias to align the Fermi level with the BCD, a sharp increase in the dissipationless transverse nonlinear Hall response is observed in 2D Te. Gate modulation of the BCD demonstrates an on/off ratio of 104 and a responsivity of nearly 106 V/W, while the longitudinal current induced by band modulation reaches an on/off ratio of about 10. This current is sustained up to 200 K, exhibiting a change of 3 orders of magnitude. The inclusion of both transistor action and rectification enhances the temperature sensitivity of the dissipationless Hall current, offering potential applications in electrothermal detectors and sensors and highlighting the significance of topological properties in advancing electronic applications.

Observation of giant non-reciprocal charge transport from quantum Hall states in a topological insulator

Symmetry breaking in quantum materials is of great importance and can lead to non-reciprocal charge transport. Topological insulators provide a unique platform to study non-reciprocal charge transport due to their surface states, especially quantum Hall states under an external magnetic field. Here we report the observation of non-reciprocal charge transport mediated by quantum Hall states in devices composed of the intrinsic topological insulator Sn-Bi1.1Sb0.9Te2S, which is attributed to asymmetric scattering between quantum Hall states and Dirac surface states. A giant non-reciprocal coefficient of up to 2.26 × 105 A-1 is found. Our work not only reveals the properties of non-reciprocal charge transport of quantum Hall states in topological insulators but also paves the way for future electronic devices.

Nonlinear Electrical Transport Unveils Fermi Surface Malleability in a Moiré Heterostructure

Van Hove singularities enhance many-body interactions and induce collective states of matter ranging from superconductivity to magnetism. In magic-angle twisted bilayer graphene, van Hove singularities appear at low energies and are malleable with density, leading to a sequence of Lifshitz transitions and resets observable in Hall measurements. However, without a magnetic field, linear transport measurements have limited sensitivity to the band's topology. Here, we utilize nonlinear longitudinal and transverse transport measurements to probe these unique features in twisted bilayer graphene at zero magnetic field. We demonstrate that the nonlinear responses, induced by the Berry curvature dipole and extrinsic scattering processes, intricately map the Fermi surface reconstructions at various fillings. Importantly, our experiments highlight the intrinsic connection of these features with the moiré bands. Beyond corroborating the insights from linear Hall measurements, our findings establish nonlinear transport as a pivotal tool for probing band topology and correlated phenomena.

Stacking Faults Enabled Second Harmonic Generation in Centrosymmetric van der Waals RhI3

Second harmonic generation (SHG) in van der Waals (vdW) materials has garnered significant attention due to its potential for integrated nonlinear optical and optoelectronic applications. Stacking faults in vdW materials are a typical kind of planar defect that introduces a degree of freedom to modulate the crystal symmetry and resultant SHG response. However, the physical origin and tunability of stacking-fault-governed SHG in vdW materials remain unclear. Here, taking the intrinsically centrosymmetric vdW RhI3 as an example, we theoretically reveal the origin of stacking-fault-governed SHG response, where the SHG response comes from the energetically favorable AC̅ stacking fault of which the electrical transitions along the high-symmetry paths Γ-M and Γ-K in the Brillion zone play the dominant role at 810 nm. Such a stacking-fault-governed SHG response is further confirmed via structural characterizations and SHG measurements. Furthermore, by applying hydrostatic pressure on RhI3, the correlation between structural evolution and SHG response is revealed with SHG enhancement up to 6.9 times, where the decreased electronic transition energies and higher momentum matrix elements due to the stronger interlayer interactions upon compression magnify the SHG susceptibility. This study develops a promising foundation for nonlinear nano-optics applications through the strategic design of stacking faults.

Giant nonlinear Hall and wireless rectification effects at room temperature in the elemental semiconductor tellurium

The second-order nonlinear Hall effect (NLHE) in non-centrosymmetric materials has recently drawn intense interest, since its inherent rectification could enable various device applications such as energy harvesting and wireless charging. However, previously reported NLHE systems normally suffer from relatively small Hall voltage outputs and/or low working temperatures. In this study, we report the observation of a pronounced NLHE in tellurium (Te) thin flakes at room temperature. Benefiting from the semiconductor nature of Te, the obtained nonlinear response can be readily enhanced through electrostatic gating, leading to a second-harmonic output at 300 K up to 2.8 mV. By utilizing such a giant NLHE, we further demonstrate the potential of Te as a wireless Hall rectifier within the radiofrequency range, which is manifested by the remarkable and tunable rectification effect also at room temperature. Extrinsic scattering is then revealed to be the dominant mechanism for the NLHE in Te, with symmetry breaking on the surface playing a key role. As a simple elemental semiconductor, Te provides an appealing platform to advance our understanding of nonlinear transport in solids and to develop NLHE-based electronic devices.

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.

Room temperature ferroelectricity and an electrically tunable Berry curvature dipole in III-V monolayers

Two-dimensional ferroelectric monolayers are promising candidates for compact memory devices and flexible electronics. Here, through first-principles calculations, we predict room temperature ferroelectricity in AB-type monolayers comprising group III (A = Al, In, Ga) and group V (B = As, P, Sb) elements. We show that their spontaneous polarization, oriented out-of-plane, ranges from 9.48 to 13.96 pC m-1, outperforming most known 2D ferroelectrics. We demonstrate an electric field tunable Berry curvature dipole and nonlinear Hall current in these monolayers. Additionally, we highlight their applicability in next-generation memory devices by forming efficient ferroelectric tunnel junctions, especially in InP, which supports high tunneling electroresistance. Our findings motivate further exploration of these monolayers for studying the interplay between the Berry curvature and ferroelectricity and for integrating these ferroelectric monolayers in next-generation electronic devices.

Magnetic parity violation and parity-time-reversal-symmetric magnets

Parity-time-reversal symmetry (PTsymmetry), a symmetry for the combined operations of space inversion (P) and time reversal (T), is a fundamental concept of physics and characterizes the functionality of materials as well asPandTsymmetries. In particular, thePT-symmetric systems can be found in the centrosymmetric crystals undergoing the parity-violating magnetic order which we call the odd-parity magnetic multipole order. While this spontaneous order leavesPTsymmetry intact, the simultaneous violation ofPandTsymmetries gives rise to various emergent responses that are qualitatively different from those allowed by the nonmagneticP-symmetry breaking or by the ferromagnetic order. In this review, we introduce candidates hosting the intriguing spontaneous order and overview the characteristic physical responses. Various off-diagonal and/or nonreciprocal responses are identified, which are closely related to the unusual electronic structures such as hidden spin-momentum locking and asymmetric band dispersion.

Two-dimensional topological semimetals: an emerging candidate for terahertz detectors and on-chip integration

The importance of terahertz (THz) detection lies in its ability to provide detailed information in a non-destructive manner, making it a valuable tool across various domains including spectroscopy, communication, and security. The ongoing development of THz detectors aims to enhance their sensitivity, resolution and integration into compact and portable devices such as handheld scanners or integrated communication chips. Generally, two-dimensional (2D) materials are considered potential candidates for device miniaturization but detecting THz radiation using 2D semiconductors is generally difficult due to the ultra-small photon energy. However, this challenge is being addressed by the advent of topological semimetals (TSM) with zero-bandgap characteristics. These semimetals offer low-energy excitations in proximity to the Dirac point, which is particularly important for applications requiring a broad detection range. Their distinctive band structures with linear energy-momentum dispersion near the Fermi level also lead to high electron mobility and low effective mass. The presence of topologically protected dissipationless conducting channels and self-powered response provides a basis for low-energy integration. In order to establish paradigms for semimetal-based THz detectors, this review initially offers an analytical summary of THz detection principles. Then, the review demonstrates the distinct design of devices, the excellent performance derived from the topological surface state and unique band structures in TSM. Finally, we outline the prospective avenues for on-chip integration of TSM-based THz detectors. We believe this review can promote further research on the new generation of THz detectors and facilitate advancements in THz imaging, spectroscopy, and communication systems.

Microscale Chiral Rectennas for Energy Harvesting

Wireless radiofrequency rectifiers have the potential to power the billions of "Internet of Things" (IoT) devices currently in use by effectively harnessing ambient electromagnetic radiation. However, the current technology relies on the implementation of rectifiers based on Schottky diodes, which exhibit limited capabilities for high-frequency and low-power applications. Consequently, they require an antenna to capture the incoming signal and amplify the input power, thereby limiting the possibility of miniaturizing devices to the millimeter scale. Here, the authors report wireless rectification at the GHz range in a microscale device built on single chiral tellurium with extremely low input powers. By studying the crystal symmetry and the temperature dependence of the rectification, the authors demonstrate that its origin is the intrinsic nonlinear conductivity of the material. Additionally, the unprecedented ability to modulate the rectification output by an electrostatic gate is shown. These results open the path to developing tuneable microscale wireless rectifiers with a single material.

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.

Engineering the Band Topology in a Rhombohedral Trilayer Graphene Moiré Superlattice

Moiré superlattices have become a fertile playground for topological Chern insulators, where the displacement field can tune the quantum geometry and Chern number of the topological band. However, in experiments, displacement field engineering of spontaneous symmetry-breaking Chern bands has not been demonstrated. Here in a rhombohedral trilayer graphene moiré superlattice, we use a thermodynamic probe and transport measurement to monitor the Chern number evolution as a function of the displacement field. At a quarter filling of the moiré band, a novel Chern number of three is unveiled to compete with the well-established number of two upon turning on the electric field and survives when the displacement field is sufficiently strong. The transition can be reconciled by a nematic instability on the Fermi surface due to the pseudomagnetic vector field potentials associated with moiré strain patterns. Our work opens more opportunities to active control of Chern numbers in van der Waals moiré systems.

Effective Manipulation of a Colossal Second-Order Transverse Response in an Electric-Field-Tunable Graphene Moiré System

The second-order nonlinear transport illuminates a frequency-doubling response emerging in quantum materials with a broken inversion symmetry. The two principal driving mechanisms, the Berry curvature dipole and the skew scattering, reflect various information including ground-state symmetries, band dispersions, and topology of electronic wave functions. However, effective manipulation of them in a single system has been lacking, hindering the pursuit of strong responses. Here, we report on the effective manipulation of the two mechanisms in a single graphene moiré superlattice, AB-BA stacked twisted double bilayer graphene. Most saliently, by virtue of the high tunability of moiré band structures and scattering rates, a record-high second-order transverse conductivity ∼ 510 μm S V-1 is observed, which is orders of magnitude higher than any reported values in the literature. Our findings establish the potential of electrically tunable graphene moiré systems for nonlinear transport manipulations and applications.

Quantum-Geometric Origin of Out-of-Plane Stacking Ferroelectricity

Stacking ferroelectricity (SFE) has been discovered in a wide range of van der Waals materials and holds promise for applications, including photovoltaics and high-density memory devices. We show that the microscopic origin of out-of-plane stacking ferroelectric polarization can be generally understood as a consequence of a nontrivial Berry phase borne out of an effective Su-Schrieffer-Heeger model description with broken sublattice symmetry, thus elucidating the quantum-geometric origin of polarization in the extremely nonperiodic bilayer limit. Our theory applies to known stacking ferroelectrics such as bilayer transition-metal dichalcogenides in 3R and T_{d} phases, as well as general AB-stacked honeycomb bilayers with staggered sublattice potential. Our explanatory and self-consistent framework based on the quantum-geometric perspective establishes quantitative understanding of out-of-plane SFE materials beyond symmetry principles.

Spin-orbit-splitting-driven nonlinear Hall effect in NbIrTe4

The Berry curvature dipole (BCD) serves as a one of the fundamental contributors to emergence of the nonlinear Hall effect (NLHE). Despite intense interest due to its potential for new technologies reaching beyond the quantum efficiency limit, the interplay between BCD and NLHE has been barely understood yet in the absence of a systematic study on the electronic band structure. Here, we report NLHE realized in NbIrTe4 that persists above room temperature coupled with a sign change in the Hall conductivity at 150 K. First-principles calculations combined with angle-resolved photoemission spectroscopy (ARPES) measurements show that BCD tuned by the partial occupancy of spin-orbit split bands via temperature is responsible for the temperature-dependent NLHE. Our findings highlight the correlation between BCD and the electronic band structure, providing a viable route to create and engineer the non-trivial Hall effect by tuning the geometric properties of quasiparticles in transition-metal chalcogen compounds.

Engineering Symmetry Breaking in Twisted MoS2-MoSe2 Heterostructures for Optimal Thermoelectric Performance

Engineering symmetry breaking in thermoelectric materials holds promise for achieving an optimal thermoelectric efficiency. van der Waals (vdW) layered transition metal dichalcogenides (TMDCs) provide critical opportunities for manipulating the intrinsic symmetry through in-plane symmetry breaking interlayer twists and out-of-plane symmetry breaking heterostructures. Herein, the symmetry-dependent thermoelectric properties of MoS2 and MoSe2 obtained via first-principles calculations are reported, yielding an advanced ZT of 2.96 at 700 K. The underlying mechanisms reveal that the in-plane symmetry breaking results in a lowest thermal conductivity of 1.96 W·m-1·K-1. Additionally, the electric properties can be significantly modulated through band flattening and bandgap alteration, stemming directly from the modified interlayer electronic coupling strength owing to spatial repulsion effects. In addition, out-of-plane symmetry breaking induces band splitting, leading to a decrease in the degeneracy and complex band structures. Consequently, the power factor experiences a notable enhancement from ∼1.32 to 1.71 × 10-2 W·m-1·K-2, which is attributed to the intricate spatial configuration of charge densities and the resulting intensified intralayer electronic coupling. Upon simultaneous implementation of in-plane and out-of-plane symmetry breaking, the TMDCs exhibit an indirect bandgap to direct bandgap transition compared to the pristine structure. This work demonstrates an avenue for optimizing thermoelectric performance of TMDCs through the implementation of symmetry breaking.

Broken Inversion Symmetry in Van Der Waals Topological Ferromagnetic Metal Iron Germanium Telluride

Inversion symmetry breaking is critical for many quantum effects and fundamental for spin-orbit torque, which is crucial for next-generation spintronics. Recently, a novel type of gigantic intrinsic spin-orbit torque is established in the topological van der Waals (vdW) magnet iron germanium telluride. However, it remains a puzzle because no clear evidence exists for interlayer inversion symmetry breaking. Here, the definitive evidence of broken inversion symmetry in iron germanium telluride directly measured by the second harmonic generation (SHG) technique is reported. The data show that the crystal symmetry reduces from centrosymmetric P63/mmc to noncentrosymmetric polar P3m1 space group, giving the threefold SHG pattern with dominant out-of-plane polarization. Additionally, the SHG response evolves from an isotropic pattern to a sharp threefold symmetry upon increasing Fe deficiency, mainly due to the transition from random defects to ordered Fe vacancies. Such SHG response is robust against temperature, ensuring unaltered crystalline symmetries above and below the ferromagnetic transition temperature. These findings add crucial new information to the understanding of this interesting vdW metal, iron germanium telluride: band topology, intrinsic spin-orbit torque, and topological vdW polar metal states.

Defect-induced helicity dependent terahertz emission in Dirac semimetal PtTe2 thin films

Nonlinear transport enabled by symmetry breaking in quantum materials has aroused considerable interest in condensed matter physics and interdisciplinary electronics. However, achieving a nonlinear optical response in centrosymmetric Dirac semimetals via defect engineering has remained a challenge. Here, we observe the helicity dependent terahertz emission in Dirac semimetal PtTe2 thin films via the circular photogalvanic effect under normal incidence. This is activated by a controllable out-of-plane Te-vacancy defect gradient, which we unambiguously evidence with electron ptychography. The defect gradient lowers the symmetry, which not only induces the band spin splitting but also generates the giant Berry curvature dipole responsible for the circular photogalvanic effect. We demonstrate that the THz emission can be manipulated by the Te-vacancy defect concentration. Furthermore, the temperature evolution of the THz emission features a minimum in the THz amplitude due to carrier compensation. Our work provides a universal strategy for symmetry breaking in centrosymmetric Dirac materials for efficient nonlinear transport.

Orbital Magneto-Nonlinear Anomalous Hall Effect in Kagome Magnet Fe_{3}Sn_{2}

It has been theoretically predicted that perturbation of the Berry curvature by electromagnetic fields gives rise to intrinsic nonlinear anomalous Hall effects that are independent of scattering. Two types of nonlinear anomalous Hall effects are expected. The electric nonlinear Hall effect has recently begun to receive attention, while very few studies are concerned with the magneto-nonlinear Hall effect. Here, we combine experiment and first-principles calculations to show that the kagome ferromagnet Fe_{3}Sn_{2} displays such a magneto-nonlinear Hall effect. By systematic field angular and temperature-dependent transport measurements, we unambiguously identify a large anomalous Hall current that is linear in both applied in-plane electric and magnetic fields, utilizing a unique in-plane configuration. We clarify its dominant orbital origin and connect it to the magneto-nonlinear Hall effect. The effect is governed by the intrinsic quantum geometric properties of Bloch electrons. Our results demonstrate the significance of the quantum geometry of electron wave functions from the orbital degree of freedom and open up a new direction in Hall transport effects.

Nonlinear Hall Effect from Long-Lived Valley-Polarizing Relaxons

The nonlinear Hall effect has attracted much attention due to the famous, widely adopted interpretation in terms of the Berry curvature dipole in momentum space. Using ab initio Boltzmann transport equations, we find a 60% enhancement in the nonlinear Hall effect of n-doped GeTe and its noticeable frequency dependence, qualitatively different from the predictions based on the Berry curvature dipole. The origin of these differences is long-lived valley polarization in the electron distribution arising from electron-phonon scattering. Our findings await immediate experimental confirmation.

Quantifying the photocurrent fluctuation in quantum materials by shot noise

The DC photocurrent can detect the topology and geometry of quantum materials without inversion symmetry. Herein, we propose that the DC shot noise (DSN), as the fluctuation of photocurrent operator, can also be a diagnostic of quantum materials. Particularly, we develop the quantum theory for DSNs in gapped systems and identify the shift and injection DSNs by dividing the second-order photocurrent operator into off-diagonal and diagonal contributions, respectively. Remarkably, we find that the DSNs can not be forbidden by inversion symmetry, while the constraint from time-reversal symmetry depends on the polarization of light. Furthermore, we show that the DSNs also encode the geometrical information of Bloch electrons, such as the Berry curvature and the quantum metric. Finally, guided by symmetry, we apply our theory to evaluate the DSNs in monolayer GeS and bilayer MoS2 with and without inversion symmetry and find that the DSNs can be larger in centrosymmetric phase.

Magnetochiral tunneling in paramagnetic Co1/3NbS2

Electric currents have the intriguing ability to induce magnetization in nonmagnetic crystals with sufficiently low crystallographic symmetry. Some associated phenomena include the non-linear anomalous Hall effect in polar crystals and the nonreciprocal directional dichroism in chiral crystals when magnetic fields are applied. In this work, we demonstrate that the same underlying physics is also manifested in the electronic tunneling process between the surface of a nonmagnetic chiral material and a magnetized scanning probe. In the paramagnetic but chiral metallic compound Co1/3NbS2, the magnetization induced by the tunneling current is shown to become detectable by its coupling to the magnetization of the tip itself. This results in a contrast across different chiral domains, achieving atomic-scale spatial resolution of structural chirality. To support the proposed mechanism, we used first-principles theory to compute the chirality-dependent current-induced magnetization and Berry curvature in the bulk of the material. Our demonstration of this magnetochiral tunneling effect opens up an avenue for investigating atomic-scale variations in the local crystallographic symmetry and electronic structure across the structural domain boundaries of low-symmetry nonmagnetic crystals.

Angle-resolved transport non-reciprocity and spontaneous symmetry breaking in twisted trilayer graphene

The identification and characterization of spontaneous symmetry breaking is central to our understanding of strongly correlated two-dimensional materials. In this work, we utilize the angle-resolved measurements of transport non-reciprocity to investigate spontaneous symmetry breaking in twisted trilayer graphene. By analysing the angular dependence of non-reciprocity in both longitudinal and transverse channels, we are able to identify the symmetry axis associated with the underlying electronic order. We report that a hysteretic rotation in the mirror axis can be induced by thermal cycles and a large current bias, supporting the spontaneous breaking of rotational symmetry. Moreover, the onset of non-reciprocity with decreasing temperature coincides with the emergence of orbital ferromagnetism. Combined with the angular dependence of the superconducting diode effect, our findings uncover a direct link between rotational and time-reversal symmetry breaking. These symmetry requirements point towards exchange-driven instabilities in momentum space as a possible origin for transport non-reciprocity in twisted trilayer graphene.

Interfacial magnetic spin Hall effect in van der Waals Fe3GeTe2/MoTe2 heterostructure

The spin Hall effect (SHE) allows efficient generation of spin polarization or spin current through charge current and plays a crucial role in the development of spintronics. While SHE typically occurs in non-magnetic materials and is time-reversal even, exploring time-reversal-odd (T-odd) SHE, which couples SHE to magnetization in ferromagnetic materials, offers a new charge-spin conversion mechanism with new functionalities. Here, we report the observation of giant T-odd SHE in Fe3GeTe2/MoTe2 van der Waals heterostructure, representing a previously unidentified interfacial magnetic spin Hall effect (interfacial-MSHE). Through rigorous symmetry analysis and theoretical calculations, we attribute the interfacial-MSHE to a symmetry-breaking induced spin current dipole at the vdW interface. Furthermore, we show that this linear effect can be used for implementing multiply-accumulate operations and binary convolutional neural networks with cascaded multi-terminal devices. Our findings uncover an interfacial T-odd charge-spin conversion mechanism with promising potential for energy-efficient in-memory computing.

Orbital Origin of the Intrinsic Planar Hall Effect

Recent experiments reported an antisymmetric planar Hall effect, where the Hall current is odd in the in plane magnetic field and scales linearly with both electric and magnetic fields applied. Existing theories rely exclusively on a spin origin, which requires spin-orbit coupling to take effect. Here, we develop a general theory for the intrinsic planar Hall effect (IPHE), highlighting a previously unknown orbital mechanism and connecting it to a band geometric quantity-the anomalous orbital polarizability (AOP). Importantly, the orbital mechanism does not request spin-orbit coupling, so sizable IPHE can occur and is dominated by an orbital contribution in systems with weak spin-orbit coupling. Combined with first-principles calculations, we demonstrate our theory with quantitative evaluation for bulk materials TaSb_{2}, NbAs_{2}, and SrAs_{3}. We further show that AOP and its associated orbital IPHE can be greatly enhanced at topological band crossings, offering a new way to probe topological materials.

Odd Nonlinear Conductivity under Spatial Inversion in Chiral Tellurium

Electrical transport in noncentrosymmetric materials departs from the well-established phenomenological Ohm's law. Instead of a linear relation between current and electric field, a nonlinear conductivity emerges along specific crystallographic directions. This nonlinear transport is fundamentally related to the lack of spatial inversion symmetry. However, the experimental implications of an inversion symmetry operation on the nonlinear conductivity remain to be explored. Here, we report on a large, nonlinear conductivity in chiral tellurium. By measuring samples with opposite handedness, we demonstrate that the nonlinear transport is odd under spatial inversion. Furthermore, by applying an electrostatic gate, we modulate the nonlinear output by a factor of 300, reaching the highest reported value excluding engineered heterostructures. Our results establish chiral tellurium as an ideal compound not just to study the fundamental interplay between crystal structure, symmetry operations and nonlinear transport; but also to develop wireless rectifiers and energy-harvesting chiral devices.

Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry

The quantum geometry has significant consequences in determining transport and optical properties in quantum materials. Here, we use a semiclassical formalism coupled with perturbative corrections unifying the nonlinear anomalous Hall effect and nonreciprocal magnetoresistance (longitudinal resistance) from the quantum geometry. In the dc limit, both transverse and longitudinal nonlinear conductivities include a term due to the normalized quantum metric dipole. The quantum metric contribution is intrinsic and does not scale with the quasiparticle lifetime. We demonstrate the coexistence of a nonlinear anomalous Hall effect and nonreciprocal magnetoresistance in films of the doped antiferromagnetic topological insulator MnBi_{2}Te_{4}. Our work indicates that both longitudinal and transverse nonlinear transport provide a sensitive probe of the quantum geometry in solids.

Light-Induced Ideal Weyl Semimetal in HgTe via Nonlinear Phononics

Interactions between light and matter allow the realization of out-of-equilibrium states in quantum solids. In particular, nonlinear phononics is one of the most efficient approaches to realizing the stationary electronic state in nonequilibrium. Herein, by an extended ab initio molecular dynamics method, we identify that long-lived light-driven quasistationary geometry could stabilize the topological nature in the material family of HgTe compounds. We show that coherent excitation of the infrared-active phonon mode results in a distortion of the atomic geometry with a lifetime of several picoseconds. We show that four Weyl points are located exactly at the Fermi level in this nonequilibrium geometry, making it an ideal long-lived metastable Weyl semimetal. We propose that such a metastable topological phase can be identified by photoelectron spectroscopy of the Fermi arc surface states or ultrafast pump-probe transport measurements of the nonlinear Hall effect.