Anomalous Hall Effect: Mechanisms & Applications

Updated 10 November 2025

Anomalous Hall effect is the generation of a transverse voltage from an applied electric field in systems with broken time-reversal symmetry and spin–orbit coupling.
It arises from intrinsic band-structure effects (Berry curvature) along with extrinsic mechanisms like skew scattering and side-jump processes.
This phenomenon serves as a probe of quantum geometry and is pivotal in designing spintronic devices and magnetic heterostructures.

The anomalous Hall effect (AHE) is the generation of a transverse voltage or current in response to an applied longitudinal electric field in systems with broken time-reversal symmetry, manifesting even in the absence of an external magnetic field. Its origin lies in the interplay of spin–orbit coupling, magnetization (or equivalent time-reversal-breaking order), and, in many cases, crystalline symmetry. The modern understanding of the AHE encompasses intrinsic band-structure effects—encoded in Berry curvature—and a hierarchy of extrinsic mechanisms arising from scattering off disorder, defects, or fluctuating magnetic textures. Investigations across metals, semiconductors, topological materials, and various magnetic and non-magnetic heterostructures reveal that the AHE is both a probe of underlying quantum geometry and a functional element in spintronic devices.

1. Fundamental Mechanisms and Theoretical Formulation

The total Hall resistivity in a broad class of conductors is conventionally decomposed as

$\rho_{xy} = R_0 B + R_s M_z,$

where $R_0$ is the ordinary Hall coefficient (OHE) proportional to the applied field $B$ , and $R_s$ is the anomalous Hall coefficient, proportional to the magnetization component $M_z$ (Culcer, 2022, Nagaosa et al., 2009). The AHE is fundamentally tied to spin–orbit coupling (SOC) and broken time-reversal symmetry, via either long-range magnetization, noncollinear or noncoplanar spin textures, or related time-reversal-breaking mechanisms.

1.1. Intrinsic (Berry Curvature) Contribution

The intrinsic AHE arises from the geometric properties of electronic Bloch bands. In the semiclassical picture, the velocity of a wavepacket centered at momentum $\mathbf{k}$ in band $n$ is (Culcer, 2022, Nagaosa et al., 2009): $\mathbf{v}_n (\mathbf{k}) = \frac{1}{\hbar} \nabla_{\mathbf{k}} \epsilon_n (\mathbf{k}) - \frac{e}{\hbar} \mathbf{E} \times \mathbf{\Omega}_n (\mathbf{k}),$ where $\mathbf{\Omega}_n (\mathbf{k}) = \nabla_{\mathbf{k}} \times \mathbf{A}_n (\mathbf{k})$ is the Berry curvature of band $n$ . The anomalous Hall conductivity is then given by: $\sigma_{xy}^{\mathrm{int}} = -\frac{e^2}{\hbar}\sum_{n} \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3} f_n(\mathbf{k}) \Omega_{n,z}(\mathbf{k})$ where $f_n$ is the Fermi–Dirac distribution (Železný et al., 2022, Nagaosa et al., 2009, Song et al., 13 Jan 2025). This term is independent of the quasiparticle scattering time and reflects the quantum geometry (topological monopoles, nodes, or Weyl points) in the band structure (Železný et al., 2022, Sürgers et al., 2016, Chen et al., 2013).

1.2. Extrinsic Contributions: Skew Scattering and Side-Jump

Extrinsic contributions arise from asymmetrical scattering of electrons by disorder or impurities due to SOC (Nagaosa et al., 2009, Culcer, 2022, Kovalev et al., 2010):

Skew scattering: Asymmetric impurity scattering appears at third order in the disorder potential, leading to $\sigma_{xy}^{\mathrm{skew}} \propto \sigma_{xx}$ (scaling linearly with the conductivity).
Side-jump: Each scattering event leads to a transverse coordinate shift independent of the net scattering time, giving $\sigma_{xy}^{\mathrm{sj}} \propto \tau^0$ (constant with respect to the scattering rate) and $\rho_{xy}^{\mathrm{sj}} \propto \rho_{xx}^2$ .

For disorder modeled by Gaussian potentials, the total AHE is rigorously decomposed as $\sigma_{xy} = \sigma_{xy}^{\mathrm{int}} + \sigma_{xy}^{\mathrm{sj}}$ , with both terms expressible via the clean-band electronic structure and its derivatives (Kovalev et al., 2010, Železný et al., 2022).

1.3. Quantum Kinetic and Kubo Formulations

The Kubo–Středa and Keldysh Green's function linear-response formalisms unify the theoretical treatment of the AHE. They establish the equivalence of the semiclassical Berry curvature intrinsic response to the quantum-mechanical expressions and systematically categorize all mechanisms, including side-jump and skew scattering, in a unified framework (Nagaosa et al., 2009, Culcer, 2022).

2. Materials Realizations and Experimental Phenomenology

2.1. Conventional and Topological Ferromagnets

In transition metals (Fe, Co, Ni), the AHE transitions from skew-dominated ( $\sigma_{xy} \propto \sigma_{xx}$ ) in ultra-clean samples to intrinsic (Berry-curvature) and side-jump dominated ( $\sigma_{xy} \sim \mathrm{const}$ ) in good metals, and then to a sublinear scaling ( $\sigma_{xy} \sim \sigma_{xx}^n, n \approx 1.6$ ) in poorly conducting regimes (Nagaosa et al., 2009, Železný et al., 2022, Culcer, 2022). First-principles Berry-curvature calculations reliably reproduce experimental values within ~30%.

High-throughput database studies confirm that σ^AHE rarely exceeds that of elemental Ni (≈2400 S/cm) among 2871 surveyed ferromagnets. Materials with large AHE almost universally exhibit symmetry-protected nodal lines in their non-SOC band structure, with SOC lifting these degeneracies and generating Berry curvature "hotspots" (Železný et al., 2022).

2.2. Noncollinear and Noncoplanar Magnetism

Antiferromagnets with noncollinear (e.g. kagome lattice Mn₃Ir (Chen et al., 2013), Mn₅Si₃ (Sürgers et al., 2016)) or noncoplanar magnetism (skyrmion-hosting phases, pyrochlores) exhibit large intrinsic AHE despite vanishing or compensated magnetization. The necessary condition is the absence of sufficient symmetries (e.g., combined time-reversal and mirror or translation) that would otherwise enforce vanishing Berry curvature. In Mn₃Ir, σ_xy ≈ 200 Ω⁻¹ cm⁻¹ is realized in the absence of net M, matching conventional ferromagnets (Chen et al., 2013).

Metamagnetic transitions between noncollinear phases can switch the sign or magnitude of the AHE by reconstructing the Berry curvature distribution (as in Mn₅Si₃ (Sürgers et al., 2016)), enabling field- or strain-tunable AHE for spintronic applications.

2.3. AHE in Heterostructures and Nonmagnetic Systems

2.3.1. Spin Hall Anomalous Hall Effect and Nonlocal AHE

In bilayers of magnetic insulators and heavy metals (YIG|Pt), a nonlocal AHE originates from the combined action of the spin Hall effect in the metal and spin-dependent boundary scattering at the interface, without proximity-induced magnetization in the nonmagnetic layer (Zhang et al., 2015, Meyer et al., 2015). The transverse resistivity includes terms proportional to both the real and imaginary parts of the spin mixing conductance at the interface—sign changes in the imaginary part (G_i) lead to sign reversals in the AHE with thickness and temperature (Meyer et al., 2015). The effect scales linearly with the metal's spin Hall angle and is observed even for Pt thickness much less than the mean free path.

2.3.2. AHE from Magnetic Proximity and Berry Curvature Engineering

At interfaces of collinear antiferromagnets and heavy metals (Pt/Cr), a combination of proximity-induced spin polarization and interface-driven band topology (avoided crossings with large Berry curvature) enables an intrinsic AHE localized at the interface (Asa et al., 2019). This effect is quantified in both experiment and DFT-based Berry-curvature calculations and is robust to details of intermixing and roughness, with the potential for all-electrical readout of antiferromagnetic order.

2.3.3. Nonmagnetic Thin Films: AHE in Bismuth

Mechanically exfoliated thin-film Bi devices, with no net magnetization or magnetic doping, show robust saturating Hall signals at low field, with magnitudes and scaling indicative of an intrinsic mechanism (σ_xy ≈ 10³ Ω⁻¹ cm⁻¹) (Yu et al., 28 Feb 2024). Hidden time-reversal symmetry breaking at surfaces or interfaces (e.g., via structural strains, orbital magnetism, topologically nontrivial edge states) is implicated, and the lack of thickness dependence for the AHE amplitude points to a dominant role of surface/interface channels.

2.4. Heavy Electron Systems and Two-Fluid Phenomenology

In heavy-fermion metals, the AHE evolves from incoherent skew scattering off local moments at high T (R_s ∝ ρχ_l), to a coherent regime below the coherence temperature, where the itinerant “Kondo liquid” controls the Hall response. The empirical formula

$R_H(T) = R_0 + r_l\,\rho(T)\,\chi_l(T) + r_h\,\chi_h(T)$

unifies Fert–Levy and Kontani models, and accurately describes both T and material dependence across 4f and 5f systems (Yang, 2012).

2.5. Cluster Magnets and Unconventional Scaling

The kagome Dirac semimetal KV₃Sb₅ exhibits a giant, extrinsic AHE despite the absence of detected magnetic order (Yang et al., 2019). The Hall conductivity scales quadratically with the longitudinal conductivity ( $\sigma_{xy} \propto \sigma_{xx}^2$ ), deviating markedly from standard skew-scattering theory. This effect is attributed to enhanced skew scattering from spin clusters intrinsic to the frustrated lattice. Hall angles approaching 90° may become accessible in ultraclean metals by this route, previously thought exclusive to quantum anomalous Hall insulators.

3. Role of Topology, Real-Space Chirality, and Magnetic Textures

Noncoplanar real-space spin textures, such as skyrmions, chiral domain walls, or merons, generate effective gauge fields ("emergent magnetic fields") that contribute a real-space Berry-phase mechanism to the AHE (Song et al., 13 Jan 2025, Sürgers et al., 2016). The topological charge density,

$q_{\mathrm{topo}} = m \cdot (\partial_x m \times \partial_y m),$

produces a real-space emergent magnetic field that acts on carriers, leading to topological Hall effects in addition to classical Berry curvature contributions. In heterostructures such as Cr₅Te₆/Pt, direct imaging of skyrmions correlates with enhanced and switchable AHE, demonstrating the fusion of real- and momentum-space topology in transport (Song et al., 13 Jan 2025). Similarly, in heavy metal/AFM heterostructures, finite net topological charge emerges only in the vicinity of the Néel temperature through the interplay of Dzyaloshinskii–Moriya interaction, exchange, and thermal fluctuations, giving an AHE maximal near the AFM–PM phase boundary and vanishing at low T (Liang et al., 2023).

4. Measurement, Scaling Regimes, and Material Design Principles

Comprehensive studies (Nagaosa et al., 2009, Železný et al., 2022) establish distinct scaling regimes for the AHE:

High-conductivity (σ_xx ≳ 10⁶ Ω⁻¹ cm⁻¹): Skew scattering dominates, σ{xy} ∝ σ{xx}
Good metal (σ_xx ~ 10⁴–10⁶ Ω⁻¹ cm⁻¹): Intrinsic and side-jump (Berry curvature) dominate, σ_{xy} ≈ const.
Poor metals (σ_xx ≲ 10⁴ Ω⁻¹ cm⁻¹): Crossover to σ{xy} ∝ σ{xx}^n, n ≈ 1.6—empirically observed in thin and disordered films; a theoretical understanding remains incomplete.

Key materials-design implications include targeting systems with strong SOC and symmetry-protected nodal lines for large intrinsic AHE, controlling the magnetization direction and crystalline symmetry for anisotropic responses, and engineering interfaces to tune real- and k-space topological charge (Železný et al., 2022). Interfacial engineering, particularly control over G_i in spin mixing conductance or the emergence of nontrivial band topology at interfaces, enables switchable and enhanced AHE suited to device applications (Meyer et al., 2015, Song et al., 13 Jan 2025).

5. Extensions: Novel Regimes, Dynamical Effects, and Open Questions

AHE research now encompasses:

Ultraclean/hydrodynamic channels: The AHE in ballistic or hydrodynamic 2DEGs exhibits unique scaling behaviors and edge-dominated voltage profiles, with all conventional mechanisms (skew, side-jump, Berry) playing nontrivial roles (Grigoryan et al., 2023).
Dynamical textures/magnons: Chiral spin waves (dipolar magnons) in homogeneous ferromagnets induce a skew-scattering AHE even in the absence of static magnetization textures or electronic SOC. The resulting Hall conductivity is generally small and highly temperature dependent (Yamamoto et al., 2015).
Compensated system AHE: Cases such as α-MnTe show robust spontaneous AHE with zero net magnetization, due to crystallographically induced alternation of spin polarization—an "altermagnetic" phase—demonstrating the importance of local symmetry and band filling (Betancourt et al., 2021).
Contributions from crossed impurity diagrams: Recent diagrammatic studies (e.g., 2D Dirac models) highlight the necessity to include “crossed” impurity ladder diagrams to correctly capture skew scattering and accurately describe the Hall conductivity at high Fermi energy, often yielding a strong suppression or sign reversal of the skew-scattering contribution (Ado et al., 2015).

Outstanding Challenges and Future Directions

Quantitative separation and control of competing mechanisms in complex and low-dimensional systems.
Engineering robust, large AHE without magnetization (quantum anomalous Hall effect and related topological phases).
Exploiting multi-degree-of-freedom coupling (spin, charge, topology) in device architectures.
Extending AHE analogues to thermal (anomalous Nernst), spin, nonlinear, and optoelectronic domains.
Ab initio prediction of Berry curvature and AHE tensor anisotropy for arbitrary orientations and field/control conditions, informing materials-by-design approaches.

Overall, the anomalous Hall effect serves as a sensitive probe of quantum geometry and magnetism, a platform for exploring topological phenomena, and an enabler of emergent device functionalities rooted in band structure, symmetry, and interface design.