In-Plane Anomalous Hall Effect

• In-plane anomalous Hall effect is a transverse Hall response arising from non-traditional Berry curvature contributions modulated by crystal symmetry and magnetic texture.
• Experimental studies on thin films, antiferromagnets, and topological semimetals show that mechanisms like skew scattering and side-jump processes can generate measurable in-plane signals.
• Tunability via gate voltage, strain, and annealing enables precise control of the Berry-curvature landscape, providing actionable insights for designing novel Hall-effect devices.

In-plane anomalous Hall effect (AHE) denotes a transverse Hall response that is generated in geometries where the decisive magnetic control variable lies in the Hall deflection plane, or more broadly where the Hall response is governed by off-diagonal conductivity components that are not reducible to the canonical out-of-plane Lorentz-force picture. In the modern formulation, the relevant response is still the anomalous Hall conductivity produced by Berry curvature, but the allowed tensor component, its field-angle dependence, and even the existence of the signal are set by crystal symmetry, magnetic point group, and the way spin-orbit coupling converts magnetic order, spin canting, orbital magnetization, or interfacial texture into a finite Hall vector (Sürgers et al., 2016, Nishihaya et al., 14 Feb 2025, Nakamura et al., 29 Jul 2025, Nishihaya et al., 6 Mar 2025).

1. Definitions and experimental scope

The ordinary Hall effect and the anomalous Hall effect are commonly written as

$\rho_{xy} = R_o\,\mu_0 H + R_s\,\mu_0 M,$

with $R_o$ the ordinary Hall coefficient and $R_s$ the anomalous Hall coefficient. In conductivity form, the intrinsic contribution is expressed through Berry curvature,

$$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$

so the anomalous velocity and the integrated Berry curvature, rather than the Lorentz force alone, control the transverse response (Sürgers et al., 2016).

In thin-film ferromagnets, a standard Hall-bar experiment already measures an in-plane Hall response in the literal geometric sense: current flows in the film plane, the Hall voltage is transverse but still in-plane, and the magnetic field is applied perpendicular to the film plane. This is the geometry used for Co$_2$FeSi and Co$_2$FeAl, where the reported $\rho_{xy}$ and $\rho_{\mathrm{ahe}}$ are in-plane Hall resistivities for a perpendicular magnetization component (Imort et al., 2011). The same planar current–voltage layout is used in heavy-metal/antiferromagnetic-insulator heterostructures such as Pt/NiO, where electrons flow in the heavy-metal plane and the Hall voltage is measured across the same plane, even though the emergent field that governs the response points out of plane (Liang et al., 2023).

A more restrictive modern usage refers to Hall responses induced by an in-plane magnetic field or by in-plane magnetization. In this sense, the defining feature is not simply that current and voltage electrodes lie in a plane, but that a finite Hall signal survives when the magnetic field or magnetization also lies in the Hall deflection plane. SrRuO$_3$ films on the (111) plane, EuCd$_2$Sb$R_o$0 thin films on the trigonal (001) plane, Cd$R_o$1As$R_o$2 (112) films under in-plane field rotation, and TaIrTe$R_o$3/Cr$R_o$4Ge$R_o$5Te$R_o$6 heterostructures all realize this more specific regime (Nishihaya et al., 14 Feb 2025, Nakamura et al., 29 Jul 2025, Nishihaya et al., 6 Mar 2025, Kao et al., 11 May 2025).

A further extension appears in low-symmetry antiferromagnets such as Mn$R_o$7Si$R_o$8, where “unusual” Hall configurations can produce sizable transverse voltages even when current is parallel to field or when the measured voltage is parallel to field. There the Hall tensor is sufficiently anisotropic that several off-diagonal components become experimentally accessible and are often grouped under the broader heading of in-plane Hall responses (Sürgers et al., 2016).

2. Symmetry conditions and tensor structure

The central symmetry statement is that in-plane AHE is not generic; it is admitted only when the magnetic point group allows the relevant off-diagonal Hall tensor component. In non-collinear antiferromagnets such as Mn$R_o$9Si$R_s$0, the non-collinearity and the loss of inversion symmetry remove the combined symmetries that would force Berry curvature to integrate to zero, so sizeable Hall conductivity can persist even though the net magnetization is tiny (Sürgers et al., 2016). In $R_s$1-symmetric antiferromagnets, the corresponding criterion is especially explicit: the in-plane anomalous Hall effect is allowed only if the magnetic point group lacks $R_s$2 operations in at least two directions, so that spin canting can lift the $R_s$3-enforced degeneracy and unmask a finite Berry curvature (Cao et al., 2022).

For purely in-plane magnetization, the decisive constraint is often mirror symmetry. In TaIrTe$R_s$4/Cr$R_s$5Ge$R_s$6Te$R_s$7, the interface lowers the point group from $R_s$8 to $R_s$9, leaving only one mirror. As long as magnetization has a finite component in the mirror plane, that last mirror symmetry is broken, and an anomalous Hall response proportional to an in-plane magnetization component becomes allowed; when $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$0, the mirror is preserved and the corresponding Hall response vanishes (Kao et al., 11 May 2025). The same logic appears in the prediction of an in-plane magnetization induced quantum anomalous Hall effect: a purely in-plane magnetization always preserves one reflection symmetry by itself, so all in-plane reflections must be broken by an additional ingredient such as hexagonal warping or shear strain before a Hall conductance can appear (Liu et al., 2013).

Trigonal systems provide another clean symmetry setting. EuCd$$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$1Sb$$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$2 thin films on the (001) principal plane have a threefold rotational axis $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$3 along $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$4 and lack any in-plane mirror plane, which is exactly the symmetry condition that allows an in-plane anomalous Hall effect on the principal plane of a trigonal crystal. In the paramagnetic phase, the leading in-plane Hall coupling is cubic in field, $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$5; in the forced ferromagnetic phase, a magneto-linear term dominates, $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$6 (Nakamura et al., 29 Jul 2025). In (111)-oriented SrRuO$$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$7, trigonal distortion likewise allows higher-order terms in the Hall conductivity, including terms such as $$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$8, which are absent in a perfect cubic treatment and permit a spontaneous in-plane AHE tied to in-plane spin magnetization and out-of-plane orbital ferromagnetism (Nishihaya et al., 14 Feb 2025).

A notable revision of the conventional symmetry picture comes from Fe and Ni. The observation of in-plane AHE in Fe(103) and Ni(111) is traced to an octupole of the anomalous Hall conductivity in magnetization space, represented by

$$\sigma_{ij} = -\frac{e^2}{\hbar}\sum_n \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\, f_n(\mathbf{k})\, \Omega^{n}_{ij}(\mathbf{k}),$$9

which in cubic crystals reduces to

$_2$0

The octupolar term misaligns $_2$1 from $_2$2 and thereby enables an in-plane Hall response even in common ferromagnets (Peng et al., 2024).

3. Microscopic mechanisms

The dominant microscopic mechanism in many in-plane AHE systems is intrinsic Berry curvature. In Mn$_2$3Si$_2$4, a non-collinear spin texture together with spin-orbit coupling produces substantial Berry curvature in momentum space, yielding $_2$5 at around 25 K despite $_2$6 (Sürgers et al., 2016). In the ferromagnetic Fe$_2$7Sn$_2$8 single crystal, the intrinsic anomalous Hall conductance is $_2$9, the fitted scaling exponent is $_2$0, and DFT+Wannier gives $_2$1, all of which identify a Berry-curvature-dominated in-plane AHE in a canonical in-plane Hall geometry (Li et al., 2020).

Topological semimetals furnish a closely related mechanism. In ZrTe$_2$2, a sizable anomalous Hall effect persists when the magnetic field is rotated in-plane, where an ordinary Hall response from the Lorentz force should vanish. The anomalous Hall signal appears below $_2$3 K, together with negative longitudinal magnetoresistance and anomalous Nernst effect, and is attributed to Berry curvature generated by Weyl nodes (Liang et al., 2016). In Cd$_2$4As$_2$5 (112) films, the field-induced in-plane AHE is isolated by its three-fold angular component under in-plane field rotation, and the Hall angle reaches $_2$6 at 2 K in ultralow-electron-density films, consistent with a Berry-curvature response amplified near Dirac nodes (Nishihaya et al., 6 Mar 2025).

Real-space topology can also drive an in-plane Hall response. In Pt/NiO and related heavy-metal/antiferromagnetic-insulator heterostructures, a significant anomalous Hall resistivity up to $_2$7 appears only in a narrow temperature window around the AFM–PM transition of ultrathin NiO. Atomistic spin dynamics and continuum modeling attribute this to noncollinear AFM spin textures with net topological charge, stabilized by exchange, interfacial Dzyaloshinskii–Moriya interaction, thermal fluctuation, and field. The Hall signal is then detected electrically in the heavy metal as an in-plane anomalous Hall response (Liang et al., 2023).

Not all in-plane AHE is intrinsically Berry-curvature dominated. In Co$_2$8FeSi and Co$_2$9FeAl, the scaling

$\rho_{xy}$0

and the approximately linear $\rho_{xy}$1–$\rho_{xy}$2 relation show that skew scattering is the dominant mechanism in the measured in-plane Hall geometry, with $\rho_{xy}$3 and the temperature dependence attributed mainly to magnon scattering (Imort et al., 2011). In sputtered $\rho_{xy}$4 multilayers with in-plane magnetic anisotropy, the anomalous Hall coefficient follows $\rho_{xy}$5, which is interpreted as side-jump-dominated AHE, and both the saturation anomalous Hall resistance and the anomalous Hall sensitivity increase by about 24 times when $\rho_{xy}$6 decreases from 20 to 1 (Das et al., 15 Jan 2026).

The theoretical literature also includes explicitly in-plane-field-induced intrinsic Hall responses in nonmagnetic systems. In a two-dimensional hole gas grown along (113), the anomalous planar Hall effect is linear in the applied in-plane magnetic field $\rho_{xy}$7, arises from Berry-curvature monopoles of spin-3/2 holes, and has vanishing leading disorder contributions (Cullen et al., 2020). On the surface of a magnetic topological insulator, the anomalous Hall conductivity can be turned off in a system with in-plane magnetization by pushing the system into the fully metallic regime, because intrinsic, side-jump, and intrinsic-skew terms all vanish as $\rho_{xy}$8 in that limit (Sabzalipour et al., 2018).

4. Material platforms and representative phenomenology

Non-collinear antiferromagnets provide some of the clearest demonstrations that in-plane AHE need not scale with net magnetization. Mn$\rho_{xy}$9Si$\rho_{\mathrm{ahe}}$0 exhibits AF1, AF1′, and AF2 phases, and the Hall response is strongly anisotropic for $\rho_{\mathrm{ahe}}$1, $\rho_{\mathrm{ahe}}$2, and $\rho_{\mathrm{ahe}}$3. The AF1 and AF1′ phases both support nonzero AHE with different sign, whereas the high-field AF2-like collinear phase almost restores a vanishing Hall response (Sürgers et al., 2016). In $\rho_{\mathrm{ahe}}$4-symmetric antiferromagnets, strained CuMnAs and the VS$\rho_{\mathrm{ahe}}$5-VS heterodimensional superlattice were proposed as platforms where spin canting yields $\rho_{\mathrm{ahe}}$6, with $\rho_{\mathrm{ahe}}$7 and $\rho_{\mathrm{ahe}}$8 for strained CuMnAs, and $\rho_{\mathrm{ahe}}$9 and $_3$0 for VS$_3$1-VS (Cao et al., 2022).

Ferromagnets and magnetic oxides show that the same response can be realized without antiferromagnetic compensation. Fe$_3$2Sn$_3$3 gives a robust intrinsic in-plane AHE from 5 to 350 K in a geometry with current along $_3$4, field normal to the $_3$5 plane, and Hall voltage along the remaining in-plane axis (Li et al., 2020). In (111)-oriented SrRuO$_3$6, the in-plane easy axes of spin magnetization support a spontaneous zero-field in-plane AHE whose sign depends on azimuthal angle and which persists after magnetization is removed, indicating an out-of-plane orbital ferromagnetic moment coupled to in-plane spin order (Nishihaya et al., 14 Feb 2025). Fe(103) and Ni(111) provide an even more elementary realization: the in-plane Hall conductivity in Fe(103) is about $_3$7, whereas the conventional out-of-plane anomalous Hall conductivity is about $_3$8, and the angular dependence follows $_3$9 in Fe(103) and $_2$0 in Ni(111), as predicted by the octupole model (Peng et al., 2024).

Nonmagnetic topological semimetals demonstrate that in-plane AHE does not require spontaneous magnetic order. ZrTe$_2$1 shows a large in-plane anomalous Hall signal for fields rotated in the $_2$2 plane, even though torque magnetometry detects no magnetic ordering (Liang et al., 2016). Cd$_2$3As$_2$4 films reveal a three-fold symmetric in-plane AHE component under in-plane field rotation, strongest in ultralow-electron-density samples (Nishihaya et al., 6 Mar 2025). Theoretical work further predicts that purely in-plane magnetization can induce a quantum anomalous Hall effect in Bi$_2$5Te$_2$6 thin films with magnetic doping and in HgMnTe quantum wells with shear strains when all reflection symmetries are broken (Liu et al., 2013).

Heterostructures expand the same physics into interfacial and electrically tunable settings. Pt/NiO, W/NiO, and related stacks show an unconventional high-temperature in-plane AHE tied to interfacial AFM topological textures (Liang et al., 2023). TaIrTe$_2$7/Cr$_2$8Ge$_2$9Te$R_o$00 realizes a gate-voltage-dependent AHE response proportional to both $R_o$01 and $R_o$02 through

$R_o$03

with the in-plane term disappearing when magnetization is aligned along the symmetry-preserving $R_o$04 axis (Kao et al., 11 May 2025). In oxide trilayers CaRuO$R_o$05/La$R_o$06Ca$R_o$07MnO$R_o$08/CaRuO$R_o$09 on NdGaO$R_o$10(110), the zero-field IP-AHE conductivity peaks at $R_o$11 for $R_o$12 u.c. and is tied directly to the monoclinic tilt angle $R_o$13 that quantifies mirror-symmetry breaking (Dai et al., 9 Jan 2026).

5. Switching, anisotropy, and tunability

A defining feature of in-plane AHE is its sensitivity to magnetic texture rather than to net magnetization alone. In Mn$R_o$14Si$R_o$15, the AF1 $R_o$16 AF1′ transition for $R_o$17 occurs at about $R_o$18–5 T and changes $R_o$19 by only about $R_o$20 at 25 K, yet the Hall resistivity undergoes a large jump and sign change comparable in magnitude to the zero-field switching. The higher-field AF1′ $R_o$21 AF2-like transition restores an almost vanishing $R_o$22 (Sürgers et al., 2016). This strongly anisotropic switching behavior extends to the unconventional configurations where current is parallel to field or voltage is parallel to field.

EuCd$R_o$23Sb$R_o$24 exhibits a different kind of tunability: in the paramagnetic and AFM phases around zero field the in-plane AHE is magneto-cubic,

$R_o$25

whereas in the forced ferromagnetic phase above $R_o$26, a magneto-linear dependence dominates and persists to at least 24 T. The octupolar coefficient scales approximately as $R_o$27, while the out-of-plane dipolar coefficient obeys $R_o$28 (Nakamura et al., 29 Jul 2025).

Electrical control is now a major theme. In TaIrTe$R_o$29/Cr$R_o$30Ge$R_o$31Te$R_o$32, multiple devices reveal a gate-voltage-dependent AHE response, consistent with a tunable Berry-curvature landscape in a low-symmetry proximitized semimetal (Kao et al., 11 May 2025). In Cd$R_o$33As$R_o$34, lowering the electron density below $R_o$35 enhances the three-fold in-plane AHE component, implying that proximity to Dirac-node-related Berry-curvature hot spots is an effective tuning knob (Nishihaya et al., 6 Mar 2025). In CaRuO$R_o$36/LCMO/CaRuO$R_o$37, ionic liquid gating between $R_o$38 and $R_o$39 V protonates the CaRuO$R_o$40 layers, suppresses the monoclinic tilt transferred to LCMO, and switches the IP-AHE completely off at $R_o$41 V, with reversible ON/OFF cycling under opposite gate bias (Dai et al., 9 Jan 2026).

Structural tuning can be equally effective. In $R_o$42 multilayers, decreasing $R_o$43 from 20 to 1 enhances both $R_o$44 and the anomalous Hall sensitivity $R_o$45 by about 24 times, and for $R_o$46 yields $R_o$47 over $R_o$48 to $R_o$49 kOe (Das et al., 15 Jan 2026). In Co$R_o$50FeSi and Co$R_o$51FeAl, annealing controls crystal order, residual resistivity, and therefore the skew-scattering-dominated in-plane AHE magnitude (Imort et al., 2011). These examples suggest that interface density, chemical order, and carrier density are practical control parameters alongside magnetic field orientation.

6. Relation to neighboring Hall phenomena and conceptual significance

In-plane AHE is frequently discussed together with ordinary Hall, planar Hall, and topological Hall effects, but the distinctions are precise. Ordinary Hall transport requires a Lorentz force and therefore a field component that acts on carrier trajectories in the conventional way. This is why the large in-plane AHE of ZrTe$R_o$52 is described as “quite anomalous”: it appears for field orientations where the ordinary Hall response should vanish (Liang et al., 2016). The planar Hall effect, by contrast, is even in magnetic field and arises from anisotropic magnetoresistance; this is why Fe/Ni and Cd$R_o$53As$R_o$54 separate odd and even angular harmonics, and why SrRuO$R_o$55 uses antisymmetrization and threefold symmetry analysis to distinguish its spontaneous in-plane AHE from planar Hall backgrounds (Peng et al., 2024, Nishihaya et al., 6 Mar 2025, Nishihaya et al., 14 Feb 2025).

Another common misconception is that antiferromagnets cannot host large AHE because their uniform magnetization is nearly zero. Mn$R_o$56Si$R_o$57, strained CuMnAs, and VS$R_o$58-VS directly contradict that expectation: what matters is broken time-reversal symmetry, the absence of the spatial symmetries that cancel Berry curvature, and in some cases field-induced canting that lifts $R_o$59-protected degeneracies (Sürgers et al., 2016, Cao et al., 2022). A related older assumption was that cubic ferromagnets forbid in-plane AHE. The observation of in-plane AHE in Fe and Ni, traced to the octupole of anomalous Hall conductivity in magnetization space, shows that the linear “Hall vector collinear with magnetization” ansatz is incomplete even for common elemental ferromagnets (Peng et al., 2024).

Taken together, the literature establishes in-plane AHE as a symmetry-sensitive manifestation of Berry-curvature transport rather than a single material-specific anomaly. It can originate from non-collinear antiferromagnetism, interfacial topological textures, Weyl- or Dirac-node band structures, mirror-symmetry breaking in low-dimensional heterostructures, octupolar angular dependence in magnetization space, or, in some metallic ferromagnets, extrinsic skew or side-jump processes (Liang et al., 2023, Kao et al., 11 May 2025, Cullen et al., 2020, Imort et al., 2011). A plausible implication is that in-plane AHE is best understood as a tensorial Hall phenomenon whose experimentally visible form is selected by symmetry, magnetic texture, and measurement geometry rather than by a single universal alignment of current, magnetization, and Hall vector.