Nonlinear Hall Effect: Basics & Advances

• Nonlinear Hall Effect (NLHE) is a second-order transverse electrical response arising in inversion-broken, time-reversal symmetric systems, allowing Hall currents without a magnetic field.
• The effect is primarily driven by Berry curvature dipole mechanisms, with additional contributions from disorder, skew scattering, and quantum metric effects.
• NLHE has been observed in platforms like bilayer WTe2 and TaIrTe4, offering new experimental avenues in quantum transport and potential spintronic applications.

The nonlinear Hall effect (NLHE) is a transverse electrical response that appears at second order in the applied electric field, conventionally written as $j_a^{(2)}=\chi_{abc}E_bE_c$. Unlike the linear Hall effect, it can occur at zero magnetic field and can survive in time-reversal-symmetric systems, provided inversion symmetry is broken; under an AC drive it naturally produces both a rectified DC component and a second-harmonic response at $2\omega$ (Ma et al., 2018). Since its first transport observation in bilayer WTe$_2$, the subject has expanded from Berry-curvature-dipole physics in nonmagnetic metals to disorder-dominated regimes, oxide interfaces, magnetic topological systems, real-space texture-driven responses, and finite-frequency nonlinear Hall transport in insulators (Ma et al., 2018, Du et al., 2021, He et al., 2024).

1. Response theory and experimental definition

The NLHE is formulated by expanding the current density in powers of the electric field,

$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$

where $\sigma_{ij}$ is the linear conductivity tensor and $\chi_{ijk}$ is the second-order conductivity tensor (Ma et al., 2018). In the Hall geometry, the defining component is transverse to the drive, for example $j_y^{(2)}=\chi_{yxx}E_x^2$. For an AC field $E_x(t)=E_0\sin\omega t$, the quadratic response contains

$$E_x^2(t)=\frac{E_0^2}{2}\left[1-\cos(2\omega t)\right],$$

so the nonlinear Hall voltage has both a DC rectified part and a second-harmonic part (Zhang et al., 19 Jul 2025).

Transport measurements therefore use lock-in detection at $2\omega$. The canonical signatures are a transverse voltage $2\omega$0, reversal under current or Hall-probe exchange, and weak dependence on the drive frequency in the low-frequency transport regime (Zhang et al., 19 Jul 2025). In bilayer WTe$2\omega$1, the second-order transverse response strongly dominated the nonlinear longitudinal response, yielding a nonlinear Hall angle of about $2\omega$2 (Ma et al., 2018). This immediately distinguishes NLHE from ordinary linear Hall transport and from contact rectification.

A recurring misconception is that zero-field Hall transport must vanish in any time-reversal-symmetric conductor. That statement applies to the linear Hall conductivity, not to the second-order tensor $2\omega$3. The NLHE is therefore not a weak variant of the anomalous Hall effect, but a distinct transport channel with different symmetry content and a different frequency structure (Ma et al., 2018, Du et al., 2021).

2. Symmetry constraints and microscopic mechanisms

The canonical intrinsic mechanism is the Berry curvature dipole (BCD). In the semiclassical theory, the BCD is a Fermi-surface quantity obtained from the asymmetric distribution of Berry curvature over occupied states, and the second-order Hall conductivity is proportional to it (Ma et al., 2018). In inversion-broken metals with preserved time-reversal symmetry, the total Berry curvature still integrates to zero, but its first moment need not vanish. That distinction is the basis of the intrinsic NLHE.

Broken inversion symmetry is necessary for any second-order charge response, but point-group symmetry determines which tensor components survive. A single mirror plane in bilayer WTe$2\omega$4 constrains the Berry curvature dipole to lie along the crystallographic $2\omega$5-axis, which is why the observed nonlinear Hall voltage appears only for specific current directions (Ma et al., 2018). More generally, the review literature emphasizes that NLHE is exceptionally sensitive to discrete and crystal symmetries, and that this sensitivity is one of its main spectroscopic uses (Du et al., 2021).

The Berry-curvature-dipole picture is not exhaustive. A full quantum diagrammatic theory showed that nearly all relevant diagrams in nonlinear Hall transport account for disorder effects, and identified intrinsic, side-jump, intrinsic skew-scattering, and extrinsic skew-scattering contributions within a unified framework (Du et al., 2020). That work also established symmetry classes in which intrinsic BCD contributions are forbidden but a pure disorder-induced NLHE remains allowed, including $2\omega$6, $2\omega$7, $2\omega$8, $2\omega$9, and $_2$0 in 2D, and $_2$1, $_2$2, $_2$3, and $_2$4 in 3D (Du et al., 2020).

This is directly relevant for systems with threefold symmetry. In KTaO$_2$5 (111) interfaces, the threefold symmetry forbids a nonzero BCD, so the observed NLHE was attributed to extrinsic skew scattering and side jump, with the skew term dominating over the full tunable range (Zhang et al., 19 Jul 2025). In Bi(111) thin films, the Berry curvature dipole is likewise symmetry-suppressed, while a finite Berry curvature triple activates side jumps and skew scatterings that generate nonlinear transverse currents (Makushko et al., 2023). In practice, conductivity-scaling analyses have become a standard diagnostic: in Sb-doped MnBi$_2$6Te$_2$7, a linear relation $_2$8 with opposite slopes below and above the magnetic transition was taken as evidence for skew scattering (Wang et al., 2024), whereas in the oxide 2DEG a cubic transport-time scaling isolated a dominant skew-scattering contribution (Zhang et al., 19 Jul 2025).

Taken together, the modern microscopic picture is plural. NLHE can be intrinsic and BCD-driven, disorder-dominated, controlled by higher Berry-curvature multipoles such as the Berry curvature triple, or mixed. Identifying the operative mechanism is therefore a symmetry-and-scaling problem rather than a purely phenomenological one.

3. Materials platforms and representative realizations

A broad materials landscape has now emerged.

Platform	Representative feature	Source
Bilayer WTe$_2$9	First clear transport observation of NLHE under time-reversal-symmetric conditions; Hall angle about $j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$0	(Ma et al., 2018)
TaIrTe$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$1	Room-temperature NLHE and wireless RF rectification with zero external bias and magnetic field	(Kumar et al., 2020)
BaMnSb$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$2	Strong bulk room-temperature NLHE from a spin-valley locked Dirac state	(Min et al., 2022)
Bi thin films	Room-temperature surface NLHE and geometric enhancement in curved devices	(Makushko et al., 2023)
CaZrO$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$3/KTaO$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$4 (111) 2DEG	Light-induced nearly five orders of magnitude enhancement to $j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$5 with sign reversal	(Zhang et al., 19 Jul 2025)
Ta/Pt/[Ir/Fe/Co/Pt]$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$6/Pt	Robust NLHE in sputtered polycrystalline magnetic multilayers from $j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$7 K to room temperature	(Kamal et al., 6 Jul 2026)

These realizations cover distinct physical regimes. Bilayer WTe$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$8 established the time-reversal-symmetric, Berry-curvature-dipole paradigm (Ma et al., 2018). TaIrTe$j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\cdots,$9 demonstrated that a type-II Weyl semimetal can host a room-temperature NLHE and support zero-bias wireless RF rectification (Kumar et al., 2020). BaMnSb$\sigma_{ij}$0 extended the phenomenon to a strong room-temperature bulk response in a spin-valley locked Dirac material, together with wireless microwave detection and frequency doubling (Min et al., 2022). Bi thin films showed that technologically relevant polycrystalline elemental films can display a strong surface NLHE at room temperature, and that geometry can amplify the response (Makushko et al., 2023). Oxide 2DEGs added a strongly tunable, sign-reversible, optically controlled platform (Zhang et al., 19 Jul 2025). Most recently, polycrystalline heavy-metal/ferromagnet multilayers established a scalable thin-film spintronic architecture for BCD-driven nonlinear Hall transport (Kamal et al., 6 Jul 2026).

Spatial resolution has also entered the field. Real-space quantum-transport calculations in TaIrTe$\sigma_{ij}$1 nanoribbons showed that confinement modifies the NLHE, and that atomic-scale probe positioning can render nearly an order of Hall-voltage enhancement by exploiting local NLHE textures that are invisible in a purely momentum-space description (Luo et al., 2023).

4. Magnetic, layer-resolved, and three-dimensional extensions

Although the early literature emphasized inversion-broken, time-reversal-symmetric metals, later work established that magnetic systems can host qualitatively richer nonlinear Hall responses. In Sb-doped MnBi$\sigma_{ij}$2Te$\sigma_{ij}$3, the NLHE generation efficiency reached up to $\sigma_{ij}$4, survived up to $\sigma_{ij}$5 K, and exhibited a conductivity-scaling law with opposite slopes below and above the magnetic transition temperature, consistent with skew scattering as the dominant microscopic mechanism (Wang et al., 2024). This was important because it showed that a sizable NLHE can persist well above the temperature where long-range magnetic order disappears.

Layer-resolved variants have extended the topic into $\sigma_{ij}$6-symmetric antiferromagnets. In thin MnBi$\sigma_{ij}$7Te$\sigma_{ij}$8-type systems, the proposed nonlinear layer Hall effect originates from hidden Berry curvature dipoles that are opposite on $\sigma_{ij}$9-related layers. The resulting nonlinear Hall conductivity is even with respect to the antiferromagnetic order and odd with respect to the vertical electric field, and the hidden BCD and quantum metric dipole generate currents that flow in different directions (Chen et al., 28 Oct 2025). This establishes NLHE as a probe of hidden quantum geometry that is inaccessible to ordinary net Hall measurements.

Three-dimensional magnetic textures provide another extension. A microscopic theory for 3D magnetic systems showed that the leading NLHE can be proportional to the emergent toroidal moment $\chi_{ijk}$0, defined from the real-space emergent magnetic field of the texture; in that setting, the nonlinear Hall conductivity $\chi_{ijk}$1 is controlled by how the spin texture winds in three dimensions (Hou et al., 2024). This places real-space emergent electrodynamics on the same footing as momentum-space Berry-curvature-dipole physics.

A neighboring but conceptually distinct development is the magnetic nonlinear Hall effect in the altermagnet Mn$\chi_{ijk}$2Si$\chi_{ijk}$3. There the Hall conductivity takes the form

$\chi_{ijk}$4

so the nonlinear term is quadratic in magnetic field rather than electric field (Han et al., 7 Feb 2025). Because its nonlinearity is controlled by magnetic exchange, Haldane-like chiral flux phases, and field-driven reversal of hopping chirality, it is best regarded as an adjacent member of the Hall family rather than a conventional electric-field-driven NLHE.

5. Tuning strategies: gating, light, geometry, and thin-film engineering

A defining feature of NLHE research is the rapid emergence of external control knobs. In TaIrTe$\chi_{ijk}$5, an additional in-plane constant electric field can tune both the magnitude and the sign of the NLHE. With the AC current parallel to the $\chi_{ijk}$6-axis and an added constant electric field of $\chi_{ijk}$7 kV/cm along the $\chi_{ijk}$8-axis, the nonlinear Hall response strength was enhanced by $\chi_{ijk}$9 times at $j_y^{(2)}=\chi_{yxx}E_x^2$0 K relative to the intrinsic value, and scaling analysis suggested a combined effect of field-modified intrinsic BCD and disorder scattering (Yang et al., 9 Feb 2025).

Optical control has followed two distinct routes. A Floquet-theoretic proposal showed that off-resonant circularly polarized light can drive topological transitions and unlock large Berry curvature dipoles, yielding nonlinear Hall currents comparable to or larger than linear Hall contributions; a two-parameter quench protocol further generalized this control landscape (Qin et al., 2024). Experimentally, the CaZrO$j_y^{(2)}=\chi_{yxx}E_x^2$1/KTaO$j_y^{(2)}=\chi_{yxx}E_x^2$2 (111) interface exhibited a light-induced giant enhancement of NLHE: the second-order transverse conductivity increased by nearly five orders of magnitude, from $j_y^{(2)}=\chi_{yxx}E_x^2$3 in the dark to $j_y^{(2)}=\chi_{yxx}E_x^2$4 under illumination, together with a sign reversal traced to a sign change of the Berry curvature triple near a Ta $j_y^{(2)}=\chi_{yxx}E_x^2$5 band crossing (Zhang et al., 19 Jul 2025).

Device geometry itself can be an active tuning parameter. In Bi thin films, arc-shaped stripes boost the zero-field nonlinear transverse voltage through an extrinsic geometric classical counterpart of the NLHE, and this curvature-induced frequency doubling extends to optical second-harmonic generation in the THz range (Makushko et al., 2023). In TaIrTe$j_y^{(2)}=\chi_{yxx}E_x^2$6 nanoribbons, quantum confinement and atomic-scale probe placement modulate the local NLHE texture and can enhance the measured Hall voltage by nearly an order of magnitude (Luo et al., 2023).

Thin-film engineering has broadened the application space. Polycrystalline Ta/Pt/[Ir/Fe/Co/Pt]$j_y^{(2)}=\chi_{yxx}E_x^2$7/Pt multilayers display a frequency-independent second-harmonic Hall voltage from $j_y^{(2)}=\chi_{yxx}E_x^2$8 K to room temperature, and their conductivity scaling indicates a dominant conductivity-independent term consistent with an intrinsic BCD contribution (Kamal et al., 6 Jul 2026). This is significant because it moves NLHE from exfoliated or epitaxial single-crystal platforms into sputter-deposited, industry-compatible spintronic stacks.

6. Finite-frequency regime, insulators, and unresolved issues

A major conceptual extension is that NLHE is not restricted to metals. A finite-frequency theory for insulators showed that a nonvanishing nonlinear Hall conductivity can arise from frequency-dependent quantum geometric quantities built from the occupied bands, even when the Fermi-surface Berry-curvature-dipole picture is inapplicable (He et al., 2024). At resonance, the response represents an inter-band-transition-enabled nonlinear Hall current; near resonance, it becomes a nonlinear polarization transverse to the electric field. In that framework, the Hall component of the second-harmonic response is tied to the breakdown of the Kleinman conjecture, and biased Bernal bilayer graphene under uniaxial strain was proposed as a candidate system detectable by polarization-resolved second-harmonic microscopy (He et al., 2024).

This finite-frequency perspective clarifies another common misconception: NLHE is not synonymous with a low-frequency metallic Berry-curvature-dipole response. Occupied-band geometry, quantum metric dipoles, disorder, Berry curvature triples, and real-space emergent fields can all generate nonlinear Hall transport in different symmetry and frequency regimes (He et al., 2024, Chen et al., 28 Oct 2025, Hou et al., 2024). Experimental discrimination among these mechanisms remains central. For example, in polycrystalline magnetic multilayers, the vanishingly small third-harmonic transverse voltage was used to argue against a dominant quantum-metric contribution and in favor of a genuine second-order BCD response (Kamal et al., 6 Jul 2026).

The broader literature identifies several open directions. A review of the field emphasized the need for fuller quantum theories of nonlinear transport, nonlinear Onsager-type constraints, quantitative disentangling of intrinsic and extrinsic mechanisms, and systematic extensions to spin, thermal, gyrotropic, Magnus, hydrodynamic, and higher-order Hall effects (Du et al., 2021). Subsequent work has reinforced that agenda: disorder can decisively renormalize the intrinsic response (Du et al., 2020), magnetic order and hidden layer structure can reorganize the allowed tensor components (Chen et al., 28 Oct 2025), and finite-frequency interband processes can create nonlinear Hall transport even in insulating states (He et al., 2024).

Across these developments, the NLHE has become a unifying framework for probing quantum geometry beyond linear transport. Its modern form is not a single mechanism but a family of second-order transverse responses whose microscopic origin depends on symmetry, band filling, disorder, dimensionality, magnetic texture, and driving frequency.