Third-order nonlinear Hall effect induced by the Berry-connection polarizability tensor

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.

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.

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 Valley Hall Effect

The valley Hall effect arises from valley-contrasting Berry curvature and requires inversion symmetry breaking. Here, we propose a nonlinear mechanism to generate a valley Hall current in systems with both inversion and time-reversal symmetry, where the linear and second-order charge Hall currents vanish along with the linear valley Hall current. We show that a second-order valley Hall signal emerges from the electric field correction to the Berry curvature, provided a valley-contrasting anisotropic dispersion is engineered. We demonstrate the nonlinear valley Hall effect in tilted massless Dirac fermions in strained graphene and organic semiconductors. Our Letter opens up the possibility of controlling the valley degree of freedom in inversion symmetric systems via nonlinear valleytronics.

Giant Third-Order Nonlinear Hall Effect in Misfit Layer Compound (SnS)1.17(NbS2)3

The nonlinear Hall effect (NLHE) holds immense significance in recognizing the band geometry and its potential applications in current rectification. Recent discoveries have expanded the study from second-order to third-order nonlinear Hall effect (THE), which is governed by an intrinsic band geometric quantity called the Berry Connection Polarizability tensor. Here we demonstrate a giant THE in a misfit layer compound, (SnS)1.17(NbS2)3. While the THE is prohibited in individual NbS2 and SnS due to the constraints imposed by the crystal symmetry and their band structures, a remarkable THE emerges when a superlattice is formed by introducing a monolayer of SnS. The angular-dependent THE and its scaling relationship indicate that the phenomenon could be correlated to the band geometry modulation, concurrently with the symmetry breaking. The resulting strength of THE is orders of magnitude higher compared to recent studies. Our work illuminates the modulation of structural and electronic geometries for novel quantum phenomena through interface engineering.

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.

Gate-Tunable Berry Curvature Dipole Polarizability in Dirac Semimetal Cd_{3}As_{2}

We reveal the gate-tunable Berry curvature dipole polarizability in Dirac semimetal Cd_{3}As_{2} nanoplates through measurements of the third-order nonlinear Hall effect. Under an applied electric field, the Berry curvature exhibits an asymmetric distribution, forming a field-induced Berry curvature dipole, resulting in a measurable third-order Hall voltage with a cubic relationship to the longitudinal electric field. Notably, the magnitude and polarity of this third-order nonlinear Hall effect can be effectively modulated by gate voltages. Furthermore, our scaling relation analysis demonstrates that the sign of the Berry curvature dipole polarizability changes when tuning the Fermi level across the Dirac point, in agreement with theoretical calculations. The results highlight the gate control of nonlinear quantum transport in Dirac semimetals, paving the way for promising advancements in topological electronics.

Nonlinear Topological Magnon Spin Hall Effect

When a magnon passes through two-dimensional magnetic textures, it will experience a fictitious magnetic field originating from the 3×3 skew-symmetric gauge fields. To date, only one of the three independent components of the gauge fields has been found to play a role in generating the fictitious magnetic field, while the other two are perfectly hidden. In this Letter, we show that they are concealed in the nonlinear magnon transport in magnetic textures. Without loss of generality, we theoretically study the nonlinear magnon-skyrmion interaction in antiferromagnets. By analyzing the scattering features of three-magnon processes between the circularly polarized incident magnon and breathing skyrmion, we predict a giant Hall angle of both the confluence and splitting modes. Furthermore, we find that the Hall angle reverses its sign when one switches the handedness of the incident magnons. We dub this the nonlinear topological magnon spin Hall effect. Our findings are deeply rooted in the bosonic nature of magnons that the particle number is not conserved, which has no counterpart in low-energy fermionic systems and may open the door for probing gauge fields by nonlinear means.

Quantum-metric-induced nonlinear transport in a topological antiferromagnet

The Berry curvature and quantum metric are the imaginary part and real part, respectively, of the quantum geometric tensor, which characterizes the topology of quantum states1. The Berry curvature is known to generate a number of important transport phenomena, such as the quantum Hall effect and the anomalous Hall effect2,3; however, the consequences of the quantum metric have rarely been probed by transport measurements. Here we report the observation of quantum-metric-induced nonlinear transport, including both a nonlinear anomalous Hall effect and a diode-like non-reciprocal longitudinal response, in thin films of a topological antiferromagnet, MnBi2Te4. Our observations reveal that the transverse and longitudinal nonlinear conductivities reverse signs when reversing the antiferromagnetic order, diminish above the Néel temperature and are insensitive to disorder scattering, thus verifying their origin in the band-structure topology. They also flip signs between electron- and hole-doped regions, in agreement with theoretical calculations. Our work provides a means to probe the quantum metric through nonlinear transport and to design magnetic nonlinear devices.

Emergent layer stacking arrangements in c-axis confined MoTe2

The layer stacking order in 2D materials strongly affects functional properties and holds promise for next-generation electronic devices. In bulk, octahedral MoTe2 possesses two stacking arrangements, the ferroelectric Weyl semimetal Td phase and the higher-order topological insulator 1T' phase. However, in thin flakes of MoTe2, it is unclear if the layer stacking follows the Td, 1T', or an alternative stacking sequence. Here, we use atomic-resolution scanning transmission electron microscopy to directly visualize the MoTe2 layer stacking. In thin flakes, we observe highly disordered stacking, with nanoscale 1T' and Td domains, as well as alternative stacking arrangements not found in the bulk. We attribute these findings to intrinsic confinement effects on the MoTe2 stacking-dependent free energy. Our results are important for the understanding of exotic physics displayed in MoTe2 flakes. More broadly, this work suggests c-axis confinement as a method to influence layer stacking in other 2D materials.

Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure

Quantum geometry in condensed matter physics has two components: the real part quantum metric and the imaginary part Berry curvature. Whereas the effects of Berry curvature have been observed through phenomena such as the quantum Hall effect in 2D electron gases and the anomalous Hall effect (AHE) in ferromagnets, quantum metric has rarely been explored. Here, we report a nonlinear Hall effect induced by quantum metric dipole by interfacing even-layered MnBi₂Te₄ with black phosphorus. The quantum metric nonlinear Hall effect switches direction upon reversing the AFM spins and exhibits distinct scaling that is independent of the scattering time. Our results open the door to discovering quantum metric responses predicted theoretically and pave the way for applications that bridge nonlinear electronics with AFM spintronics.

Time-Reversal-Even Nonlinear Current Induced Spin Polarization

We propose a time-reversal-even spin generation in second order of electric fields, which dominates the current induced spin polarization in a wide class of centrosymmetric nonmagnetic materials, and leads to a novel nonlinear spin-orbit torque in magnets. We reveal a quantum origin of this effect from the momentum space dipole of the anomalous spin polarizability. First-principles calculations predict sizable spin generations in several nonmagnetic hcp metals, in monolayer TiTe_{2}, and in ferromagnetic monolayer MnSe_{2}, which can be detected in experiment. Our work opens up the broad vista of nonlinear spintronics in both nonmagnetic and magnetic systems.

Time-reversal even charge hall effect from twisted interface coupling

Under time-reversal symmetry, a linear charge Hall response is usually deemed to be forbidden by the Onsager relation. In this work, we discover a scenario for realizing a time-reversal even linear charge Hall effect in a non-isolated two-dimensional crystal allowed by time reversal symmetry. The restriction by Onsager relation is lifted by interfacial coupling with an adjacent layer, where the overall chiral symmetry requirement is fulfilled by a twisted stacking. We reveal the underlying band geometric quantity as the momentum-space vorticity of layer current. The effect is demonstrated in twisted bilayer graphene and twisted homobilayer transition metal dichalcogenides with a wide range of twist angles, which exhibit giant Hall ratios under experimentally practical conditions, with gate voltage controlled on-off switch. This work reveals intriguing Hall physics in chiral structures, and opens up a research direction of layertronics that exploits the quantum nature of layer degree of freedom to uncover exciting effects.

Giant Magnetochiral Anisotropy in Weyl Semimetal WTe_{2} Induced by Diverging Berry Curvature

The concept of Berry curvature is essential for various transport phenomena. However, an effect of the Berry curvature on magnetochiral anisotropy, i.e., nonreciprocal magnetotransport, is still elusive. Here, we report that the Berry curvature induces the large magnetochiral anisotropy. In Weyl semimetal WTe_{2}, we observe the strong enhancement of the magnetochiral anisotropy when the Fermi level is located near the Weyl points. Notably, the maximal figure of merit γ[over ¯] reaches 1.2×10^{-6}  m^{2} T^{-1} A^{-1}, which is the largest ever reported in bulk materials. Our semiclassical calculation shows that the diverging Berry curvature at the Weyl points strongly enhances the magnetochiral anisotropy.

Intrinsic Nonlinear Planar Hall Effect

We propose an intrinsic nonlinear planar Hall effect, which is of band geometric origin, independent of scattering, and scales with the second order of electric field and first order of magnetic field. We show that this effect is less symmetry constrained compared with other nonlinear transport effects and is supported in a large class of nonmagnetic polar and chiral crystals. Its characteristic angular dependence provides an effective way to control the nonlinear output. Combined with first-principles calculations, we evaluate this effect in the Janus monolayer MoSSe and report experimentally measurable results. Our work reveals an intrinsic transport effect, which offers a new tool for material characterization and a new mechanism for nonlinear device application.

Control over Berry Curvature Dipole with Electric Field in WTe_{2}

Berry curvature dipole plays an important role in various nonlinear quantum phenomena. However, the maximum symmetry allowed for nonzero Berry curvature dipole in the transport plane is a single mirror line, which strongly limits its effects in materials. Here, via probing the nonlinear Hall effect, we demonstrate the generation of Berry curvature dipole by applied dc electric field in WTe_{2}, which is used to break the symmetry constraint. A linear dependence between the dipole moment of Berry curvature and the dc electric field is observed. The polarization direction of the Berry curvature is controlled by the relative orientation of the electric field and crystal axis, which can be further reversed by changing the polarity of the dc field. Our Letter provides a route to generate and control Berry curvature dipole in broad material systems and to facilitate the development of nonlinear quantum devices.

Giant photon momentum locked THz emission in a centrosymmetric Dirac semimetal

Strong second-order optical nonlinearities often require broken material centrosymmetry, thereby limiting the type and quality of materials used for nonlinear optical devices. Here, we report a giant and highly tunable terahertz (THz) emission from thin polycrystalline films of the centrosymmetric Dirac semimetal PtSe₂. Our PtSe₂ THz emission is turned on at oblique incidence and locked to the photon momentum of the incident pump beam. Notably, we find an emitted THz efficiency that is giant: It is two orders of magnitude larger than the standard THz-generating nonlinear crystal ZnTe and has values approaching that of the noncentrosymmetric topological material TaAs. Further, PtSe₂ THz emission displays THz sign and amplitude that is controlled by the incident pump polarization and helicity state even as optical absorption is only weakly polarization dependent and helicity independent. Our work demonstrates how photon drag can activate pronounced optical nonlinearities that are available even in centrosymmetric Dirac materials.

Resonant Second-Harmonic Generation as a Probe of Quantum Geometry

Nonlinear responses are actively studied as probes of topology and band geometric properties of solids. Here, we show that second harmonic generation serves as a probe of the Berry curvature, quantum metric, and quantum geometric connection. We generalize the theory of second harmonic generation to include Fermi surface effects in metallic systems, and finite scattering timescale. In doped materials the Fermi surface and Fermi sea cause all second harmonic terms to exhibit resonances, and we identify two novel contributions to the second harmonic signal: a double resonance due to the Fermi surface and a higher-order pole due to the Fermi sea. We discuss experimental observation in the monolayer of time reversal symmetric Weyl semimetal WTe_{2} and the parity-time reversal symmetric topological antiferromagnet CuMnAs.

Intrinsic Nonlinear Electric Spin Generation in Centrosymmetric Magnets

We propose an intrinsic nonlinear electric spin generation effect, which can dominate in centrosymmetric magnets. We reveal the band geometric origin of this effect and clarify its symmetry characters. As an intrinsic effect, it is determined solely by the material's band structure and represents a material characteristic. Combining our theory with first-principle calculations, we predict sizable nonlinear spin generation in single-layer MnBi_{2}Te_{4}, which can be detected in experiment. Our theory opens a new route for all-electric controlled spintronics in centrosymmetric magnets which reside outside of the current paradigm based on linear spin response.

Non-linear Hall effect in multi-Weyl semimetals

In the presence of time reversal symmetry, a non-linear Hall effect can occur in systems without an inversion symmetry. One of the prominent candidates for detection of such Hall signals are Weyl semimetals. In this article, we investigate the Berry curvature induced second and third order Hall effect in multi-Weyl semimetals with topological chargesn=1,2,3. We use low energy effective models to obtain general analytical expressions and discover the presence of a large Berry curvature dipole (BCD) in multi-Weyl semimetals, compared to usual (n = 1) Weyl semimetals. We also study the BCD in a realistic tight-binding lattice model and observe two different kinds of variation with increasing topological charge-these can be attributed to different underlying Berry curvature components. We provide estimates of the signatures of second harmonic of Hall signal in multi-Weyl semimetals, which can be detected experimentally. Furthermore, we predict the existence of a third order Hall signal in multi-Weyl semimetals. We derive the analytical expressions of Berry connection polarizability tensor, which is responsible for third order effects, using a low energy model and estimate the measurable conductivity. Our work can help guide experimental discovery of Berry curvature multipole physics in multi-Weyl semimetals.

Room-temperature third-order nonlinear Hall effect in Weyl semimetal TaIrTe4

The second-order nonlinear Hall effect observed in the time-reversal symmetric system has not only shown abundant physical content, but also exhibited potential application prospects. Recently, a third-order nonlinear Hall effect has been observed in MoTe₂ and WTe₂. However, few-layer MoTe₂ and WTe₂ are usually unstable in air and the observed third-order nonlinear Hall effect can be measured only at low temperature, which hinders further investigation as well as potential application. Thus, exploring new air-stable material systems with a sizable third-order nonlinear Hall effect at room temperature is an urgent task. Here, in type-II Weyl semimetal TaIrTe₄, we observed a pronounced third-order nonlinear Hall effect, which can exist at room temperature and remain stable for months. The third-order nonlinear Hall effect is connected to the Berry-connection polarizability tensor instead of the Berry curvature. The possible mechanism of the observation of the third-order nonlinear Hall effect in TaIrTe₄ at room temperature has been discussed. Our findings will open an avenue towards exploring room-temperature nonlinear devices in new quantum materials.

Intrinsic Second-Order Anomalous Hall Effect and Its Application in Compensated Antiferromagnets

Response properties that are purely intrinsic to physical systems are of paramount importance in physics research, as they probe fundamental properties of band structures and allow quantitative calculation and comparison with experiment. For anomalous Hall transport in magnets, an intrinsic effect can appear at the second order to the applied electric field. We show that this intrinsic second-order anomalous Hall effect is associated with an intrinsic band geometric property-the dipole moment of Berry-connection polarizability (BCP) in momentum space. The effect has scaling relation and symmetry constraints that are distinct from the previously studied extrinsic contributions. Particularly, in antiferromagnets with PT symmetry, the intrinsic effect dominates. Combined with first-principles calculations, we demonstrate the first quantitative evaluation of the effect in the antiferromagnet Mn_{2}Au. We show that the BCP dipole and the resulting intrinsic second-order conductivity are pronounced around band near degeneracies. Importantly, the intrinsic response exhibits sensitive dependence on the Néel vector orientation with a 2π periodicity, which offers a new route for electric detection of the magnetic order in PT-invariant antiferromagnets.