Nonlinear Planar Hall Effect Overview

• NPHE is a second-order Hall response producing a transverse, second-harmonic voltage in systems with inversion symmetry breaking.
• Experimental setups reveal distinct angular dependencies and scaling laws, distinguishing NPHE from conventional Hall effects.
• Mechanistic insights involve band geometry, Berry curvature dipole, and orbital effects, offering tunable responses in advanced materials.

Nonlinear planar Hall effect (NPHE) denotes a class of second-order transverse transport responses in which a Hall-like voltage or current appears in a planar field configuration and is typically detected through a second-harmonic signal. In the literature represented here, the effect is reported either as a second-harmonic Hall resistance obeying $R_{yx}^{(2\omega)} \propto E H \cos\varphi$ in $\mathrm{Bi_2Se_3}$ films or as an intrinsic Hall current of the form $j_a = \chi_{abcd} E_b E_c B_d$ in nonmagnetic polar and chiral crystals. Across these formulations, NPHE is distinguished from the ordinary Hall effect by its second-order character and by mechanisms rooted in symmetry breaking, band geometry, spin texture, orbital magnetic moment, or nonreciprocal vortex dynamics rather than a conventional out-of-plane Lorentz-force response (He et al., 2019, Huang et al., 2022, Du et al., 2021).

1. Definitions and phenomenology

In nonlinear Hall transport, the current is expanded as

$$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$

The experimentally relevant signature is a transverse voltage at double the driving frequency, measured as a second harmonic $V_{2\omega}$, which isolates the quadratic response. This general structure is shared by the broader nonlinear Hall effect and by its planar variants (Du et al., 2021).

Within planar geometries, the terminology is not completely uniform. In the $\mathrm{Bi_2Se_3}$ experiment, the nonlinear planar Hall resistance is introduced through

$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$

with $\varphi$ the angle between in-plane magnetic field and current. In the intrinsic NPHE literature, the same topic is formulated at the current-density level as a Hall current quadratic in electric field and linear in in-plane magnetic field, $j_H \propto E^2 B$ (He et al., 2019, Huang et al., 2022). This suggests that the label “NPHE” covers a closely related family of second-order transverse observables rather than a single universally normalized transport coefficient.

A recurrent phenomenological feature is the orthogonality between transverse and longitudinal nonlinear channels. In $\mathrm{Bi_2Se_3}$, the NPHE follows $\mathrm{Bi_2Se_3}$0, while the nonlinear longitudinal resistance associated with bilinear magnetoelectric resistance follows $\mathrm{Bi_2Se_3}$1, producing a $\mathrm{Bi_2Se_3}$2 angular offset (He et al., 2019). This differs from the $\mathrm{Bi_2Se_3}$3 offset that characterizes the conventional linear planar Hall effect and anisotropic magnetoresistance.

2. Symmetry structure and response tensors

The conventional linear Hall effect is forbidden by time-reversal symmetry unless time-reversal is explicitly broken. By contrast, the nonlinear Hall effect can survive time-reversal symmetry but requires inversion symmetry breaking. For planar nonlinear Hall responses in a magnetic field, time-reversal breaking is supplied by the field, while crystal symmetries determine which tensor components are allowed (Du et al., 2021, Wang et al., 2024).

The intrinsic NPHE is formulated through a fourth-rank conductivity tensor,

$\mathrm{Bi_2Se_3}$4

with

$\mathrm{Bi_2Se_3}$5

Here $\mathrm{Bi_2Se_3}$6 is the Berry-connection polarizability (BCP), $\mathrm{Bi_2Se_3}$7 its spin susceptibility, and $\mathrm{Bi_2Se_3}$8 the spin magnetic moment. The tensor is antisymmetric under $\mathrm{Bi_2Se_3}$9, which encodes the Hall character of the response (Huang et al., 2022).

The symmetry requirements are unusually permissive relative to other nonlinear Hall effects. The intrinsic NPHE requires broken inversion symmetry and broken horizontal mirror symmetry and is allowed in 16 out of 18 non-centrosymmetric 2D point groups, including $j_a = \chi_{abcd} E_b E_c B_d$0, $j_a = \chi_{abcd} E_b E_c B_d$1, and $j_a = \chi_{abcd} E_b E_c B_d$2 for $j_a = \chi_{abcd} E_b E_c B_d$3 (Huang et al., 2022). A later experimental-theory study further emphasized that many high-symmetry materials with $j_a = \chi_{abcd} E_b E_c B_d$4, $j_a = \chi_{abcd} E_b E_c B_d$5, $j_a = \chi_{abcd} E_b E_c B_d$6, and $j_a = \chi_{abcd} E_b E_c B_d$7 point groups, which forbid other nonlinear Hall effects, are still allowed to host NPHE (Jiang et al., 21 Jun 2025).

The symmetry classification also distinguishes intrinsic and extrinsic contributions. The quantum diagrammatic theory of nonlinear Hall transport shows that disorder-induced terms obey a different tensor structure from Berry-curvature-driven intrinsic terms, and predicts pure disorder-induced nonlinear Hall effect in the 2D point groups $j_a = \chi_{abcd} E_b E_c B_d$8, $j_a = \chi_{abcd} E_b E_c B_d$9, $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$0, $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$1, and $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$2, where the Berry curvature dipole contribution vanishes (Du et al., 2020). In trigonal 2D crystals, linear anomalous planar Hall effect requires the magnetic field to break all mirror symmetries, but the nonlinear planar Hall response can survive even when one mirror symmetry remains because the Berry curvature dipole need not vanish (Battilomo et al., 2020).

3. Microscopic mechanisms

Several microscopic routes to NPHE have been established, and the mechanisms are not equivalent.

In topological-insulator surface states, the experimentally observed NPHE in $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$3 was attributed to the conversion of a nonlinear transverse spin current into a charge current by the concerted actions of spin-momentum locking and time-reversal symmetry breaking. The electric field produces a second-order distortion of the Fermi contour and generates a pure transverse spin current; an in-plane magnetic field lifts the balance between opposite spin contributions and partially converts that spin current into a transverse Hall charge current. Hexagonal warping and particle-hole asymmetry are critical ingredients of the theory (He et al., 2019).

A later analysis of strong-topological-insulator surface states rederived the nonlinear planar Hall current and argued that previous derivations were incomplete. In that treatment the effect is a magnetochiral-anisotropy-induced nonlinear planar Hall effect requiring broken time-reversal symmetry and broken particle-hole symmetry, the latter entering through the quadratic correction $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$4 to the Dirac Hamiltonian (Marinescu et al., 2023).

The intrinsic NPHE is instead a band-geometric response. Its natural objects are the BCP, the BCP dipole, and the BCP dipole susceptibility $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$5, defined through

$$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$6

In $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$7, first-principles calculations showed that the dominant contribution to the susceptibility tensor arises from coupling of the magnetic field to the orbital moment of Bloch electrons, and that in $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$8 the orbital contribution is an order of magnitude larger than the spin contribution (Jiang et al., 21 Jun 2025).

A third route is Berry-curvature-dipole-driven transport. In bilayer graphene under a steady in-plane magnetic field, time-reversal symmetry breaking induces a nonlinear planar Hall effect through an orbital effect in the complete absence of spin-orbit coupling. With inversion symmetry retained, the leading Berry-curvature-dipole components obey $$J_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \cdots .$$9 and $V_{2\omega}$0, and the response can be controlled by the planar magnetic field, gate voltage, and Fermi energy (Kheirabadi et al., 2022).

Disorder can be central rather than perturbative. The quantum transport theory based on Feynman diagrams identifies 69 diagrams for the nonlinear Hall effect, of which only 5 are intrinsic and 64 involve disorder. In the 2D tilted Dirac model, the total nonlinear Hall conductivity is enhanced after disorder corrections are included, and pure disorder-induced nonlinear Hall effect is symmetry-allowed in specific point groups (Du et al., 2020). A related disorder-based mechanism was developed for topological insulators with spin-momentum-locking inhomogeneity, where the total planar Hall conductivity contains a conventional linear term and an emergent bilinear term,

$V_{2\omega}$1

with the bilinear contribution changing sign under current or field reversal and potentially dominating the conventional planar Hall effect (Zarezad et al., 2022).

4. Experimental realization and measurement protocols

Experimentally, NPHE is most often isolated through harmonic transport. An ac current is driven through a Hall bar, and the second-harmonic transverse voltage is measured while the magnetic field, current amplitude, and field orientation are varied. This protocol suppresses linear Hall signals and directly tests quadratic scaling (He et al., 2019).

The first reported NPHE of this type was observed in high-quality $V_{2\omega}$2 films. The Hall resistance scaled linearly with both the applied electric and magnetic fields, $V_{2\omega}$3 at fixed field and $V_{2\omega}$4 at fixed current, and displayed the characteristic $V_{2\omega}$5 angular dependence, vanishing when field and current were perpendicular (He et al., 2019).

Magnetic-insulator/topological-insulator heterostructures amplified the effect. In $V_{2\omega}$6 grown on TmIG and YIG, the NPHE was up to an order of magnitude larger than in $V_{2\omega}$7 grown on nonmagnetic GGG. TmIG/$V_{2\omega}$8 showed a linear-in-field nonlinear Hall resistance, whereas YIG/$V_{2\omega}$9 exhibited an extra hysteresis loop around zero field. The NPHE magnitude scaled inversely with carrier density and appeared only below $\mathrm{Bi_2Se_3}$0, consistent with a magnetic-proximity-effect interpretation (Wang et al., 2022).

The recent intrinsic NPHE experiment in $\mathrm{Bi_2Se_3}$1 established a different regime. There the Hall current is quadratic in electric field and linear in in-plane magnetic field, $\mathrm{Bi_2Se_3}$2, the signal persists up to $\mathrm{Bi_2Se_3}$3, and the NPHE conductivity obeys the scaling law

$\mathrm{Bi_2Se_3}$4

where the finite intercept $\mathrm{Bi_2Se_3}$5 identifies a relaxation-time-independent intrinsic contribution (Jiang et al., 21 Jun 2025).

A distinct room-temperature realization was reported in polycrystalline heavy-metal/ferromagnet multilayers. The second-harmonic transverse voltage satisfied

$\mathrm{Bi_2Se_3}$6

was independent of excitation frequency and applied out-of-plane magnetic field, and displayed a vanishingly small third-harmonic response. The conductivity scaling,

$\mathrm{Bi_2Se_3}$7

yielded a dominant conductivity-independent term $\mathrm{Bi_2Se_3}$8, identifying the Berry curvature dipole as the main driver (Kamal et al., 6 Jul 2026).

Platform	Reported signature	Mechanism emphasized
$\mathrm{Bi_2Se_3}$9 film	$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$0	spin-momentum locking and time reversal symmetry breaking
TmIG/YIG/$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$1	up to an order of magnitude larger than GGG; YIG shows hysteresis	magnetic proximity effect
$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$2	$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$3, robust up to $$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$4	BCP dipole susceptibility tensor
Polycrystalline magnetic multilayers	$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$5, dominant conductivity-independent term	intrinsic Berry curvature dipole
FeSe films	$$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$6 in vortex flow regime	thermally excited (anti)vortices

The FeSe case extends the experimental landscape beyond normal-state band transport. In the vortex-flow regime of 23-nm-thick FeSe films, a nonreciprocal transverse signal was observed when the in-plane magnetic field was parallel to the current, $$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$7. Because conventional Lorentz-force-driven vortex motion is not expected for $$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$8, the proposed explanation invokes thermally excited vortex-antivortex pairs in a two-dimensional superconductor with broken mirror symmetry at the film surfaces (Hashimoto et al., 8 Sep 2025).

5. Topology, orbital enhancement, and tunability

One of the most important developments has been the identification of topological and orbital amplification mechanisms. In $$V_y = R_{yx}^{(\omega)} I_x + R_{yx}^{(2\omega)} I_x^2, \qquad R_{yx}^{(2\omega)} \propto E H \cos\varphi ,$$9, first-principles calculations compared two nearby structural realizations: a topological-insulator state and an ideal Weyl semimetal state with 8 symmetry-related Weyl points exactly at the Fermi level. The NPHE in the Weyl state is much stronger than in the topological-insulator state and exhibits a resonance-like lineshape as the chemical potential passes through the Weyl-point energy. In the hole-doped regime, the peak value in the Weyl state is roughly one order of magnitude higher, $\varphi$0 versus $\varphi$1 for the topological-insulator state, with the enhancement dominated by orbital contribution amplified by Weyl points (Yang et al., 28 Apr 2025).

The orbital mechanism is especially significant because it weakens the common association of NPHE with strong spin-orbit coupling alone. In $\varphi$2, the dominant intrinsic contribution comes from the orbital moment of Bloch electrons, and the authors explicitly note that this mechanism does not require spin-orbit coupling (Jiang et al., 21 Jun 2025). In bilayer graphene, the nonlinear planar Hall effect was derived as an orbital effect of a steady in-plane magnetic field in the complete absence of spin-orbit coupling (Kheirabadi et al., 2022).

Tunability is likewise a central theme. In bilayer graphene, the magnitude and direction of the Berry-curvature dipole can be controlled by the in-plane magnetic-field orientation, gate voltage, and Fermi energy (Kheirabadi et al., 2022). In $\varphi$3, small changes in lattice parameters, specifically $\varphi$4 in the $\varphi$5 ratio, can drive the transition between topological-insulator and Weyl-semimetal states, thereby switching the NPHE amplitude through the phase boundary (Yang et al., 28 Apr 2025). In Janus monolayer MoSSe, first-principles calculations yielded an intrinsic nonlinear Hall conductivity $\varphi$6 for $\varphi$7, with the angular dependence reducing in polar $\varphi$8 crystals to a cosine law,

$\varphi$9

which makes the nonlinear output directly controllable by field orientation (Huang et al., 2022).

Not all field-tuned nonlinear Hall responses are dominated by the same geometric object. In the isotropic $j_H \propto E^2 B$0-cubed Rashba model, an out-of-plane magnetic field or magnetization is necessary to induce a nonzero Berry curvature, while an in-plane magnetic field tunes the Berry curvature and generates the Berry curvature dipole. However, the second-order Hall correction is generally dominated by the component independent of the Berry curvature dipole rather than by the dipole term itself (Krzyzewska et al., 2024). This is a useful corrective to overly narrow BCD-only interpretations.

A related extension into explicitly time-reversal-breaking systems is the magneto-nonlinear Hall effect driven by an in-plane electric field and a vertical magnetic field. In that case,

$j_H \propto E^2 B$1

where $j_H \propto E^2 B$2 is a generalized Berry curvature–velocity dipole. The effect vanishes in time-reversal-symmetric systems and was illustrated in a massive Dirac model relevant to a $j_H \propto E^2 B$3 quantum well (Zhang et al., 2023). This is not the canonical coplanar NPHE, but it broadens the geometric taxonomy of nonlinear Hall responses in planar-type measurement configurations.

6. Distinctions, adjacent effects, and unresolved issues

NPHE should not be conflated with the conventional planar Hall effect or with anisotropic magnetoresistance. In the conventional case, the planar Hall voltage is even under magnetic-field reversal, shows $j_H \propto E^2 B$4-periodic angular dependence, and vanishes when electric and magnetic fields are aligned. In the nonlinear $j_H \propto E^2 B$5 case, the transverse and longitudinal nonlinear responses are offset by $j_H \propto E^2 B$6, not $j_H \propto E^2 B$7, and the transverse signal is explicitly second harmonic (He et al., 2019). In 2D trigonal crystals, the anomalous planar Hall effect can even be odd in magnetic field and independent of the relative angle between $j_H \propto E^2 B$8 and $j_H \propto E^2 B$9, while the nonlinear version yields both dc and $\mathrm{Bi_2Se_3}$0 components through a Zeeman-induced Berry curvature dipole (Battilomo et al., 2020).

Another common misconception is that NPHE is synonymous with Berry-curvature-dipole transport. The literature summarized here contains at least six distinct mechanisms: nonlinear spin-to-charge conversion in topological surface states, intrinsic BCP-dipole susceptibility, orbital-moment-enhanced intrinsic response near Weyl points, disorder-induced nonlinear Hall transport, bilinear response from spin-momentum-locking inhomogeneity, and nonreciprocal vortex-flow transport in superconductors (He et al., 2019, Jiang et al., 21 Jun 2025, Du et al., 2020, Zarezad et al., 2022, Hashimoto et al., 8 Sep 2025). The general reviews therefore emphasize mechanism disentangling through symmetry analysis, conductivity scaling, and harmonic diagnostics rather than by terminology alone (Du et al., 2021, Wang et al., 2024).

Open problems follow directly from this multiplicity. One is the quantitative separation of intrinsic and extrinsic channels in realistic materials. Another is the extension of theory beyond semiclassics toward fully quantum, disordered, strongly correlated, or hydrodynamic regimes. A further direction is systematic exploitation of NPHE as a probe of topological phase transitions, as suggested by the resonance-like response near Weyl points in $\mathrm{Bi_2Se_3}$1 (Du et al., 2021, Yang et al., 28 Apr 2025).

The conceptual framework is already extending beyond charge transport. The intrinsic nonlinear planar thermal Hall effect introduces a dissipationless heat-current response proportional to $\mathrm{Bi_2Se_3}$2, governed by a thermal Berry connection polarizability tensor and permitted only in noncentrosymmetric crystal point groups lacking horizontal mirror symmetry (Barman, 3 Nov 2025). A plausible implication is that NPHE belongs to a broader family of nonlinear transverse responses controlled by field-induced corrections to band geometry, with charge, spin, and heat transport as parallel realizations of the same geometric logic.