Nonlinear anomalous Hall effect in few-layer WTe2.

Under broken time reversal symmetry such as in the presence of external magnetic field or internal magnetization, a transverse voltage can be established in materials perpendicular to both longitudinal current and applied magnetic field, known as classical Hall effect. However, this symmetry constraint can be relaxed in the nonlinear regime, thereby enabling nonlinear anomalous Hall current in time-reversal invariant materials – an underexplored realm with exciting new opportunities beyond classical linear Hall effect. Here, using group theory and first-principles theory, we demonstrate a remarkable ferroelectric nonlinear anomalous Hall effect in time-reversal invariant few-layer WTe2 where nonlinear anomalous Hall current switches in odd-layer WTe2 except 1T′ monolayer while remaining invariant in even-layer WTe2 upon ferroelectric transition. This even-odd oscillation of ferroelectric nonlinear anomalous Hall effect was found to originate from the absence and presence of Berry curvature dipole reversal and shift dipole reversal due to distinct ferroelectric transformation in even and odd-layer WTe2. Our work not only treats Berry curvature dipole and shift dipole on an equal footing to account for intraband and interband contributions to nonlinear anomalous Hall effect, but also establishes Berry curvature dipole and shift dipole as new order parameters for noncentrosymmetric materials. The present findings suggest that ferroelectric metals and Weyl semimetals may offer unprecedented opportunities for the development of nonlinear quantum electronics.

The electrical Hall effect is the production, upon the application of an electric field, of a transverse voltage under an out-of-plane magnetic field. Studies of the Hall effect have led to important breakthroughs, including the discoveries of Berry curvature and topological Chern invariants1,2. The internal magnetization of magnets means that the electrical Hall effect can occur in the absence of an external magnetic field2; this 'anomalous' Hall effect is important for the study of quantum magnets2-7. The electrical Hall effect has rarely been studied in non-magnetic materials without external magnetic fields, owing to the constraint of time-reversal symmetry. However, only in the linear response regime-when the Hall voltage is linearly proportional to the external electric field-does the Hall effect identically vanish as a result of time-reversal symmetry; the Hall effect in the nonlinear response regime is not subject to such symmetry constraints8-10. Here we report observations of the nonlinear Hall effect10 in electrical transport in bilayers of the non-magnetic quantum material WTe2 under time-reversal-symmetric conditions. We show that an electric current in bilayer WTe2 leads to a nonlinear Hall voltage in the absence of a magnetic field. The properties of this nonlinear Hall effect are distinct from those of the anomalous Hall effect in metals: the nonlinear Hall effect results in a quadratic, rather than linear, current-voltage characteristic and, in contrast to the anomalous Hall effect, the nonlinear Hall effect results in a much larger transverse than longitudinal voltage response, leading to a nonlinear Hall angle (the angle between the total voltage response and the applied electric field) of nearly 90 degrees. We further show that the nonlinear Hall effect provides a direct measure of the dipole moment10 of the Berry curvature, which arises from layer-polarized Dirac fermions in bilayer WTe2. Our results demonstrate a new type of Hall effect and provide a way of detecting Berry curvature in non-magnetic quantum materials.

Generally, the dissipationless Hall effect in solids requires time-reversal symmetry breaking (TRSB), where TRSB induced by external magnetic field results in the ordinary Hall effect, while TRSB caused by spontaneous magnetization gives rise to the anomalous Hall effect (AHE) which scales with the net magnetization. The AHE is therefore not expected in antiferromagnets with vanishing small magnetization. However, large AHE was recently observed in certain antiferromagnets with noncollinear spin structure and nonvanishing Berry curvature. Here, we report another origin of AHE in a layered antiferromagnet EuAl_{2}Si_{2}, namely, the domain wall (DW) skew scattering with Weyl points near the Fermi level, in experiments for the first time. Interestingly, the DWs form a unique periodic stripe structure with controllable periodicity by external magnetic field, which decreases nearly monotonically from 975nm at 0T to 232nm at 4T. Electrons incident on DW with topological bound states experience strong asymmetric scattering, leading to a giant AHE, with the DW Hall conductivity (DWHC) at 2K and 1.2T reaching a record value of ∼1.51×10^{4} Scm^{-1} among bulk systems and being 2 orders of magnitude larger than the intrinsic anomalous Hall conductivity. The observation not only sets a new paradigm for exploration of large anomalous Hall effect, but also provides potential applications in spintronic devices.

Nonlinear anomalous (NLA) Hall effect is the Berry curvature dipole induced second-order Hall voltage or temperature difference induced by a longitudinal electric field or temperature gradient. These are the prominent Hall responses in time-reversal symmetric systems. These band-geometry induced responses in recently realized twistronic platforms can probe their novel electronic band structure and topology. Here, we investigate the family (electrical, thermoelectric, and thermal) of second-order NLA Hall effects in the moiré system of twisted double bilayer graphene (TDBG). We combine the semiclassical transport framework with the continuum model of TDBG to demonstrate that the NLA Hall signals can probe topological phase transitions in moiré systems. We show that the whole family of NLA Hall responses undergo a sign reversal across a topological phase transition. Our study establishes a deeper connection between valley topology and nonlinear Hall effects in time-reversal symmetric systems.

The interplay between symmetry breaking and topological electronic structure is crucial to design anomalous transport properties in materials. Materials with strong or quantum electromagnetic responses have an extensive impact on the development of data storage, information processing, energy conversion, etc. In magnetic materials, the anomalous transport of anomalous Hall effect, anomalous Nernst effect, and magneto-optical effect et al. can be understood from the Berry curvature of the electronic band structures. Two typical band structures of Weyl points and nodal line band structures host strong local Berry curvature. Since the Berry curvature is time-reversal symmetry odd, such strong Berry curvatures can lead to strongly enhanced anomalous transport signals. With this guiding principle, we studied the anomalous Hall effect in magnetic Weyl semimetal Co3Sn2S2 [1-3] and magnetic nodal line semimetal in Heusler compound Co2Mn(Ga/Al) [4].With a mirror symmetry, the inverted band structure forms a nodal loop in the absence of spin-orbital coupling. This nodal line can be broken by spin-orbital coupling and a bandgap opens, which generates non-zero Berry curvature in the bandgap and forms a hot loop, see Figure 1a-b. Such strong Berry curvature in the magnetic system can lead to a strongly enhance or even quantized anomalous Hall effect. We applied this idea to real materials of magnetic Heusler compounds Co2Mn(Ga/Al). Protected by mirror symmetries the band inversion between the bands with opposite mirror eigenvalue forms three gapless nodal lines in the kx=0, ky=0, and kz=0 mirror planes, respectively. With spin-orbital coupling, the symmetry of the system is reduced. Taking magnetic along z, the mirror symmetries in kx=0 and ky=0 planes are broken, which leads to band anti-crossings with strong local Berry curvature locating in the opened bandgap around original nodal lines, see Figure 1d. Integral of the Berry curvature in the whole k-space gives a large intrinsic anomalous Hall conductivity reaching ~1500 to ~2000 S/cm [4].Weyl points is another typical band structure and present as the Berry curvature monopole, and therefore naturally results in a strong anomalous Hall effect. In ideal models with only one pair of Weyl points locating at the Fermi level, the intrinsic anomalous Hall conductivity can be presented as the combination distance of Weyl points and the quantized anomalous Hall conductance. Inspired by these excellent relations, we studied the anomalous Hall effect in Co3Sn2S2, and a new record of three-dimensional anomalous Hall angel (~20%) was observed, which offers the 1st three-dimensional material with both strong anomalous Hall conductivity and anomalous Hall angle [1]. It indeed shows as a Weyl semimetal from electronic band structure analysis. One crucial symmetry in Co3Sn2S2 is the three mirror planes parallel to the c direction, which results in three pairs of nodal lines connected by a c3z rotation symmetry. Because the magnetization is aligned along the z-direction, the mirror symmetries are broken by spin-orbital coupling. Meanwhile, one pair of Weyl points with opposite chirality remains along each of the former nodal lines, leading to the so large anomalous Hall effect.Though the strong anomalous Hall effect provides a promising signature for the existence of Weyl and nodal line band structure. Our transport work about Co3Sn2S2 and Co2MnGa/Al inspired the direct band structure detection by ARPES and STM [5-7], and they are in turn became the 1st experimentally verified magnetic Weyl semimetal and nodal line semimetal, respectively.Applying temperature gradient instead of the electrical field, the Weyl points and nodal lines induced Berry curvature can also lead to strongly enhanced anomalous Nernst effect. From our theoretical calculations and experimental measurements, the anomalous Nernst conductivity can reach around 3 and 6 A/(m-K) in Co3Sn2S2 [8] and Co2MnGa [9], respectively, with Co2MnGa keeping the record. Owing to the large anisotropy, Co3Sn2S2 is, so far, the only material with a large anomalous Nernst effect with zero magnetic fields. In addition, very recently, a giant magneto-optical response was observed in Co3Sn2S2 with the applied field from polarized light [10].Very recently, a strong interest in antiferromagnets is rising. In an antiferromagnet without such kind of joint TO symmetry to reverse Berry curvature, it allows the existence of anomalous Hall effect, anomalous Nernst effect, magneto-optical responses, and special spin current, etc. The nonzero anomalous Hall effect in antiferromagnets was proposed as early as 2001 in distorted non-linear magnetic structures [11]. However, its experimental realization was not successful until 2015 [12-16]. This understanding can be further expanded into collinear antiferromagnets. Different from non-linear antiferromagnets, the collinear antiferromagnetic structure can be usually understood from two sublattices connected by translation of inversion operation. Therefore, there are mainly two ways to break the joint symmetry, to replace the magnetic atoms connected by the joint TO symmetry, or change the the nonmagnetic sites. With this understanding, we predicted the anomalous Hall and Nernst effect in anti-Heusler Weyl semimetal Ti2MnAl [17-18]. **

Symmetries and broken symmetries play important roles in physics. In particular, they constrain the electromagnetic response of materials and the allowed light-matter interactions. In such a context, symmetry breaking can lead to unique and peculiar physical phenomena. The Hall effects, for example, result from broken symmetries. In the usual linear Hall effects, the time-reversal symmetry is broken by a magnetic bias. Alternatively, nonlinear Hall effects can occur in systems with a broken inversion symmetry rather than broken time-reversal symmetry. Here we explore several physical platforms that can enable nonreciprocal and non-Hermitian responses based on the nonlinear Hall effect and 2D material layers biased with a static electric field. It will be shown that the electric field bias may create unique physical responses, including regimes of loss and gain controlled by the wave-polarization, asymmetric responses, and others. In this talk, we will present an overview of our on-going work on this topic.

The Hall effect is one of the best known effects in (solid-state) physics. Conventionally, this phenomenon describes the occurrence of charge currents that are perpendicular to an externally applied electric field due to a time-reversal symmetry breaking magnetic field. Besides, in ferromagnetic systems, the net magnetization can break time-reversal symmetry even in the absence of a magnetic field which allows the so-called anomalous Hall effect. This effect originates from extrinsic and intrinsic contributions that are both related to the existence of spin-orbit coupling [1]. Moreover, another contribution to the Hall effect, which is known as the topological Hall effect, can exist even if spin-orbit coupling is negligible. It may occur in certain noncollinear noncoplanar magnetic textures with a nonzero scalar spin chirality like skyrmions [2, 3].However, recent works [4, 5] reported the occurrence of an anomalous Hall effect in several compensated kagome magnets (cf. Fig. 1). These materials are coplanar antiferromagnets with vanishing net magnetization, and still, a group theoretical analysis allows the existence of the effect. The large conductivities obtained via first-principle calculations have been confirmed in experiments for Mn3Sn [6] and Mn3Ge [7]. However, a straightforward microscopic picture for this phenomenon was still missing.In this talk, we present an explanation on a microscopic level based on tight-binding calculations and analytical considerations [8]. For coplanar kagome magnets, we show the equivalence of spin-orbit coupling and an out-of-plane tilting of the magnetic moments. The existence of spin-orbit interaction does not only break a combined time-reversal and mirror symmetry of the Hamiltonian but can be transformed to a magnetic texture that is virtually canted, whereas, the original texture remains coplanar [cf. Fig. 2(a)]. Consequently, the ‘new’ anomalous Hall effect can be interpreted as a combination of an effective anomalous and topological Hall effect due to the net magnetic moment and the net scalar spin chirality of this virtual magnetic texture, respectively.Furthermore, as we demonstrate, a noncoplanar kagome magnet with spin-orbit coupling is able to behave like a system that is virtually coplanar and with compensated spin-orbit coupling [cf. Fig. 2(b)]. In this case, the combination of mirror and time-reversal symmetry of the Hamiltonian that was broken before has been restored. A critical out-of-plane tilting angle of the real texture can be found, where the virtual texture is coplanar and the Hall effect is absent for all energies. As we show in detail, the electronic properties are determined by this virtual texture that is hidden in the Hamiltonian.In consequent investigations, the calculations have been repeated for other transport quantities like the spin Hall effect where charge currents are converted into spin currents. These results can again be related to the virtual spin texture which has, however, different consequences for the spin Hall effect. Besides, in order to simulate the experimental situation, the investigated model was extended from a two-dimensional kagome lattice, as considered here, to a more realistic model including d-orbitals and kagome planes that are stacked along the out-of-plane direction. **

Two-dimensional semi-Dirac materials, with quadratic dispersion in one direction and linear dispersion in the orthogonal direction, provide a route to formation of the Chern-insulating states in solids. Within the framework of the Floquet theory, we investigate the photon-modulated linear and nonlinear anomalous Hall effect (AHE) in type-II semi-Dirac semimetals and find that rich topological phases, as well as interesting phenomena related to the topological transitions, can be realized and manipulated by circularly polarized light (CPL). By tuning the CPL parameters, we can control the local inverted gaps independently and switch the system between different topological phases optionally. We also demonstrate that, when the Fermi level locates outside the gap, the Drude component of the conductivity can contribute to the AHE as well, because of the anisotropy of the electronic structure. Besides, the nonlinear Hall effect due to the Berry curvature dipole can be observable when the inversion symmetry between opposite valleys is broken. Our findings are helpful to understand the topological properties of the emergent type-II semi-Dirac semimetals.

The anomalous Hall effect in time-reversal symmetry broken systems is underpinned by the concept of Berry curvature in band theory. However, recent experiments reveal that the nonlinear Hall effect (NHE) can be observed in non-magnetic systems without applying an external magnetic field. The emergence of NHE under time-reversal symmetric conditions can be explained in terms of non-vanishing Berry curvature dipole (BCD) arising from inversion symmetry breaking. In this work, we availed realistic tight-binding models, first-principles calculations, and symmetry analyses to explore the combined effect of transverse electric field and strain, which leads to a giant BCD in the elemental buckled honeycomb lattices—silicene, germanene, and stanene. The external electric field breaks the inversion symmetry of these systems, while strain helps to attain an asymmetrical distribution of Berry curvature of a single valley. Furthermore, the topology of the electronic wavefunction switches from the band inverted quantum spin Hall state to normal insulating one at the gapless point. This band gap closing at the critical electric field strength is accompanied by an enhanced Berry curvature and concomitantly a giant BCD at the Fermi level. Our results predict the occurrence of an electrically switchable nonlinear electrical and thermal Hall effect in a new class of elemental systems that can be experimentally verified.

The anomalous Hall effect (AHE) is a nonlinear Hall effect appearing in magnetic conductors, boosted by internal magnetism beyond what is expected from the ordinary Hall effect. With the recent discovery of the quantized version of the AHE, the quantum anomalous Hall effect (QAHE), in Cr- or V-doped topological insulator (TI) (Sb,Bi)2Te3 thin films, the AHE in magnetic TIs has been attracting significant interest. However, one of the puzzles in this system has been that while Cr- or V-doped (Sb,Bi)2Te3 and V-doped Bi2Se3 exhibit AHE, Cr-doped Bi2Se3 has failed to exhibit even ferromagnetic AHE, the expected predecessor to the QAHE, though it is the first material predicted to exhibit the QAHE. Here, we have successfully implemented ferromagnetic AHE in Cr-doped Bi2Se3 thin films by utilizing a surface state engineering scheme. Surprisingly, the observed ferromagnetic AHE in the Cr-doped Bi2Se3 thin films exhibited only a positive slope regardless of the carrier type. We show that this sign problem can be explained by the intrinsic Berry curvature of the system as calculated from a tight-binding model combined with a first-principles method.

Antiferromagnetic (AFM) spintronics exploits the Néel vector as a state variable for novel spintronic devices. Recent studies have shown that the fieldlike and antidamping spin-orbit torques (SOTs) can be used to switch the Néel vector in antiferromagnets with proper symmetries. However, the precise detection of the Néel vector remains a challenging problem. In this Letter, we predict that the nonlinear anomalous Hall effect (AHE) can be used to detect the Néel vector in most compensated antiferromagnets supporting the antidamping SOT. We show that the magnetic crystal group symmetry of these antiferromagnets combined with spin-orbit coupling produce a sizable Berry curvature dipole and hence the nonlinear AHE. As a specific example, we consider the half-Heusler alloy CuMnSb, in which the Néel vector can be switched by the antidamping SOT. Based on density-functional theory calculations, we show that the nonlinear AHE in CuMnSb results in a measurable Hall voltage under conventional experimental conditions. The strong dependence of the Berry curvature dipole on the Néel vector orientation provides a new detection scheme of the Néel vector based on the nonlinear AHE. Our predictions enrich the material platform for studying nontrivial phenomena associated with the Berry curvature and broaden the range of materials useful for AFM spintronics.

The anomalous Hall effect (AHE), conventionally associated with time-reversal symmetry breaking in ferromagnetic materials, has recently been observed in nonmagnetic topological materials, raising questions about its origin. We unravel the unconventional Hall response in the nonmagnetic Dirac material ZrTe5, known for its massive Dirac bands and unique electronic and transport properties. Using the Kubo-Streda formula within the Landau level framework, we explore the interplay of quantum effects induced by the magnetic field (B) and disorder across the semiclassical and quantum regimes. In the semiclassical regime, the Hall resistivity remains linear in the magnetic field, but the Hall coefficient will be renormalized by the quantum geometric effects and electron-hole coherence, especially at low carrier densities where the disorder scattering dominates. In quantum limit, the Hall conductivity exhibits an unsaturating 1/B scaling. As a result, the transverse conductivity dominates transport in the ultra-quantum limit, and the Hall resistivity crosses over from B to B−1 dependence as the system transitions from the semiclassical regime to the quantum limit. This work elucidates the mechanisms underlying the unconventional Hall effect in ZrTe5 and provides insights into the AHE in other nonmagnetic Dirac materials as well. ZrTe5 has received significant attention for it’s non-trivial topological band structure and reports of a large anomalous Hall effect despite being a nonmagnetic material. Here, using the Kubo-Streda formula the authors investigate the origins of the unconventional Hall response of ZrTe5 in low and high magnetic fields.

We find that the Hall effect in a single crystal of UCoGe varies as a function of the angle  between the applied magnetic field and the easy magnetic axis up to fields of 18 T at 0.2 K, i.e. in the region where both superconductivity and ferromagnetic order coexist. Instead of following the conventional cos dependence the two components that com-prise the total Hall resistance, the anomalous and ordinary Hall effect, exhibit quite an unusual behavior with the field direction. The anomalous Hall effect is found to be determined by the parallel component of the magnetization. We sug-gest that the field induced changes in magnetization due to the field rotation play an important role in the observed unu-sual behavior. The ordinary Hall effect cannot be described by the simple relation to the perpendicular component of the magnetic field implying that this component of the Hall effect may be also affected by the variations in magnetization at the characteristic field (kink field). A field induced moment polarization is also observed in Hall effect as in magnetore-sistance, which advances previous findings in UCoGe. The Hall effect slope reverses sign at the kink field indicative of small but possible Fermi surface reconstruction around this field. Our findings show that in UCoGe multiple mecha-nisms contribute to the observed field induced moment polarization at the kink field.

The Hall effect, in which a current flows perpendicular to an electrical bias, has been prominent in the history of condensed matter physics. Appearing variously in classical, relativistic and quantum guises, the Hall effect has — among other roles — contributed to the establishment of the band theory of solids, to research on new phases of interacting electrons and to the phenomenology of topological condensed matter. The dissipationless Hall current requires time-reversal symmetry breaking. When this symmetry breaking is due to an externally applied magnetic field, the effect is referred to as the ordinary Hall effect; when it is due to a non-zero internal magnetization (ferromagnetism), it is referred to as the anomalous Hall effect. The Hall effect has not usually been associated with antiferromagnetic order. More recently, however, theoretical predictions and experimental observations have identified large Hall effects in some compensated magnetic crystals, governed by neither of the global magnetic-dipole symmetry-breaking mechanisms mentioned above. The goal of this Review is to systematically organize the present understanding of anomalous antiferromagnetic materials that generate a Hall effect — which we call anomalous Hall antiferromagnets — and to discuss this class of materials in a broader fundamental and applied research context. Our motivation is twofold: first, because Hall effects that are not governed by magnetic-dipole symmetry breaking are at odds with the traditional understanding of the phenomenon, the topic deserves attention on its own. Second, this new incarnation of the Hall effect has placed it again in the middle of an emerging field in physics, at the intersection of multipole magnetism, topological condensed matter and spintronics.

The anomalous transverse transport properties, including the anomalous Hall and Nernst effects, are crucial for probing the topological band features in magnetic materials. In this study, we present a comprehensive investigation of the magnetic, electrical, and thermal transport, as well as the electronic band structure, of the noncollinear ferromagnet ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$. Our findings reveal that ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ exhibits prominent anomalous Hall and Nernst effects. Scaling analysis suggests that the anomalous Hall effect is predominantly influenced by the intrinsic Berry curvature contribution. Additionally, the large anomalous Nernst signal exceeds the magnetization scaling relations observed in conventional ferromagnets, and the calculated anomalous Nernst conductivity exhibits a $T\text{ln}T$ relation. These observations align with the behavior seen in other magnetic topological materials, further suggesting a substantial intrinsic Berry curvature effect. Complemented by detailed theoretical calculations, we clarify the electronic band properties and confirm a substantial net Berry curvature near the Fermi level in this compound. Our work elucidates that the large anomalous Hall and Nernst effects of ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ are dominated by an intrinsic mechanism and presents it as a probable magnetic topological material with a significant Berry phase effect in the momentum space.