Intrinsic Anomalous Hall Effect in Quantum Materials

• Intrinsic anomalous Hall effect is a dissipationless transverse transport phenomenon driven by Berry curvature from the electronic band structure in solids with broken time-reversal symmetry.
• It is quantified using first-principles DFT and model calculations that integrate Berry curvature over the Brillouin zone, independent of impurity scattering.
• The effect is tunable via Fermi-level engineering and manipulation of spin–orbit coupling, with applications in spintronics, quantum transport, and nonvolatile low-power devices.

The intrinsic anomalous Hall effect (AHE) is a fundamental dissipationless transverse transport phenomenon in solids with broken time-reversal symmetry, where the Hall response arises as a direct consequence of the electronic band structure and the associated Berry curvature of the occupied Bloch states. Unlike extrinsic mechanisms, which depend on impurity scattering, the intrinsic contribution originates solely from geometric properties of the wavefunctions in momentum space. This effect plays a pivotal role in topological materials, spintronics, and quantum transport, and is observed in ferromagnets, ferrimagnets, certain antiferromagnets, topological semimetals, and superconductors. Recent research elucidates the universality, tunability, and novel regimes of the intrinsic AHE, including nonlinear and quantized responses, across a wide spectrum of correlated electron systems.

1. Theoretical Framework of the Intrinsic Anomalous Hall Effect

The intrinsic AHE is rooted in the modern theory of geometric phases and Berry curvature in crystalline solids. In the clean limit and in the absence of extrinsic scattering, the anomalous Hall conductivity tensor for a multiband Bloch system is given by the Kubo–Berry-curvature formula: $$\sigma^{\text{int}}_{ij} = -\frac{e^2}{\hbar} \sum_{n} \int_{\text{BZ}} \frac{d^d k}{(2\pi)^d} f(\epsilon_{n\mathbf{k}}) \, \Omega_{n,ij}(\mathbf{k}),$$ where

• $$\Omega_{n,ij}(\mathbf{k}) = -2\,\text{Im} \langle \partial_{k_i}u_{n\mathbf{k}} | \partial_{k_j}u_{n\mathbf{k}} \rangle$$ is the Berry curvature of band $n$,
• $f(\epsilon)$ is the Fermi–Dirac function,
• The sum runs over all occupied bands.

This formalism directly connects the intrinsic AHE to the geometric properties of electronic wavefunctions, specifically the accumulation of Berry curvature in regions of the Brillouin zone near band degeneracies, anticrossings, or Weyl/Dirac points (Singh et al., 28 Mar 2024, Ohno et al., 2022, Das et al., 2020). The actual value of the Hall conductivity is determined by the distribution of Berry curvature below the Fermi energy and is independent of scattering time, distinguishing it unambiguously from extrinsic mechanisms such as skew-scattering and side-jump.

A critical requirement is the presence of both time-reversal symmetry breaking and spin–orbit coupling (SOC), with the latter generating avoided crossings and ‘hot spots’ of sharply peaked Berry curvature by lifting symmetry-protected line degeneracies (Singh et al., 28 Mar 2024). In crystals with noncollinear or compensated magnetic order (e.g., certain antiferromagnets, ferrimagnets, or altermagnets), the symmetry conditions for a nonzero intrinsic AHE can become more subtle and strongly depend on the allowed tensor components under the relevant magnetic space group (Attias et al., 19 Feb 2024, Ohgata et al., 11 Dec 2024).

2. Band Structure, Spin-Orbit Coupling, and Berry Curvature

The magnitude and sign of the intrinsic AHE are exquisitely sensitive to the electronic band structure proximate to the Fermi energy. SOC is essential in lifting band degeneracies—often producing nodal lines or Weyl points in its absence—that, once gapped, yield Berry curvature peaks (hot spots) with strength inversely proportional to the gap size (∼Δ⁻²). The overall Hall response is then set by the net flux of Berry curvature over all occupied states (Singh et al., 28 Mar 2024, Zhang et al., 2011).

As demonstrated in tetragonal SmMn₂Ge₂, aligning the magnetization along different crystal axes can tune the Berry curvature landscape dramatically:

• For $\mathbf{M} \parallel c$, gapped nodal lines from Mn-d orbitals straddle $E_F$, resulting in a large ring of Berry curvature and a substantial $$\sigma_{xy}^{\text{int}} \sim 100\,\Omega^{-1}\,\mathrm{cm}^{-1}$$.
• For $\mathbf{M} \parallel a$, the nodal line shifts above $E_F$, rendering the Berry curvature contribution at $E_F$ negligible, and $\sigma_{xy}^{\text{int}} \to 0$ (Singh et al., 28 Mar 2024).

In kagome ferrimagnets such as Tb(Mn₁₋ₓCrₓ)₆Sn₆, substitutional doping adjusts $E_F$ to maximize overlap with Berry curvature arising from a dense mesh of three-dimensional anticrossings rather than from Dirac points, leading to substantial tunability of $\sigma_{xy}^{\text{int}}$ (from ∼200 S/cm up to over 1000 S/cm) while intrinsic contributions persist even at complete magnetic compensation (DeStefano et al., 6 Feb 2025).

For simple Weyl semimetals (e.g., EuCd₂Sb₂), the Berry curvature is concentrated near Weyl nodes, acting as monopole sources. Adjusting the Fermi level to align with these nodes maximizes the intrinsic σ_xy, confirmed both theoretically and experimentally (Ohno et al., 2022, Zyuzin et al., 2016). In type-II Weyl systems, tilting the cones introduces Fermi-surface contributions that can change the sign and magnitude of the intrinsic AHE, even allowing a nonzero value at coincident Weyl nodes if tilts differ (Zyuzin et al., 2016).

3. Material-Specific Manifestations and Tunability

Extensive experimental and computational work establishes the universality and tunability of the intrinsic AHE:

• Ferromagnetic Metals (e.g., Ni, FePt, CoPt, NiPt): In Ni, the intrinsic term dominates except in ultra-clean limits. Its strong temperature dependence (1100 Ω⁻¹ cm⁻¹ at 5 K to 500 Ω⁻¹ cm⁻¹ at 300 K) stems from the interplay between SOC and partially occupied t₂g hole pockets near the X-point (Ye et al., 2011, Fuh et al., 2011). Intrinsic AHE can be strongly anisotropic and even ‘quenched’ (σ_AHC→0) for certain magnetization directions in ordered alloys, with sign changes and “anti-ordinary” Hall effects occurring with magnetization rotation (Zhang et al., 2011).
• Compensated Ferrimagnets and Altermagnets: In Cr-doped kagome ferrimagnets, the intrinsic AHE remains robust at vanishing net magnetization, a consequence of Berry curvature from sublattice symmetry rather than net ferromagnetism. In altermagnets such as doped FeSb₂, the intrinsic AHE is symmetry-forbidden in the absence of SOC and emerges linearly with SOC strength as protected degeneracies are lifted, with Berry curvature sharply localized along nodal lines (Attias et al., 19 Feb 2024, Ohgata et al., 11 Dec 2024).
• Twisted Bilayer Graphene and Moiré Chern Insulators: In θ ≈ 1.15° twisted bilayer graphene, the intrinsic quantum anomalous Hall effect (QAH) is realized with perfect zero-field quantization (R_H = h/e² within 0.1%), a transport gap Δ/k_B ≈ 27 K that is much larger than the Curie temperature, and current-driven switching at nA-scale threshold, all arising from interaction-driven Chern insulator formation with fully spin- and valley-polarized minibands (Serlin et al., 2019).
• Superconductors and Topological Bosonic Systems: In multiband chiral superconductors, a nonzero optical Hall conductivity (and corresponding Kerr rotation) requires interorbital transitions with relative phase differences in the gap function. In bosonic chiral superfluids, the intrinsic AHE is linked to macroscopic loop current correlations and the Berry curvature at the condensation momentum, which can be probed via AC Hall response or circular dichroism (Taylor et al., 2011, Huang et al., 2022).

4. Advanced: Nonlinear and Higher-Order Intrinsic Hall Effects

The geometric origin of the intrinsic Hall effect extends beyond linear response. Intrinsic second-order AHE (quadratic in electric field) is governed by the Berry-connection polarizability dipole, and appears in systems where the linear effect is symmetry-forbidden, as in PT-symmetric compensated antiferromagnets (e.g., Mn₂Au). The second-order intrinsic conductivity is odd under time reversal and independent of scattering time, distinguished both in scaling and symmetry from extrinsic nonlinear terms (Liu et al., 2021).

Third-order intrinsic AHE (cubic in E) arises from the second-order field-induced positional shift of wavepackets and requires an even stricter set of symmetry conditions (supported in 15 magnetic point groups). The magnitude can reach experimentally accessible levels in antiferromagnets with suitable band parameters, featuring distinct signatures such as J_H∝E³ scaling and angular dependencies tied to crystallographic symmetries (Xiang et al., 2022).

5. Methodologies: First-Principles and Model Calculations

Modern evaluation of the intrinsic AHE necessitates:

• Accurate DFT (GGA+U or hybrid) including SOC and correct magnetic ordering,
• Construction of maximally-localized Wannier functions (for k·p and tight-binding interpolation),
• Brillouin-zone integration of the Berry curvature on dense k-meshes, with adaptive refinement near degeneracies,
• Extraction of the current–current response via Kubo formula for metals or BdG formalism for superconductors/superfluids,
• Inclusion of model Hamiltonians to analyze symmetry constraints and locate band crossings/anticrossings responsible for Berry curvature hot spots (Singh et al., 28 Mar 2024, Fuh et al., 2011, Ohno et al., 2022, Ohgata et al., 11 Dec 2024, Xiang et al., 2022).

Experimental decomposition of the Hall resistivity into intrinsic versus extrinsic terms is commonly achieved via thickness- or residual-resistivity–tuning protocols in clean epitaxial films, combined with conductivity-scaling analyses (e.g., ρ_AH = a ρ_xx + b ρ_xx² fits) (Ye et al., 2011).

6. Physical Consequences, Design Principles, and Applications

The robust dissipationless nature and band-structure tunability of the intrinsic AHE have direct implications for low-power spintronic devices, nonvolatile Hall effect memories, and electrically switchable quantum devices. Key design principles emerging from current research include:

• Fermi-level engineering (doping, gating, or strain) to align $E_F$ with Berry curvature “hot spots”,
• Spin orientation control (thermal, field, or current-induced) to activate or deactivate intrinsic AHE channels,
• Exploiting compensated or topologically nontrivial magnetic order to achieve large AHE at zero net magnetization,
• Maximizing SOC and band anticrossings for enhanced Berry curvature.

Recent advances demonstrate: thermally reversible switching of intrinsic AHE (SmMn₂Ge₂, room T) (Singh et al., 28 Mar 2024); doping- and compensation-tuned giant AHE in kagome ferrimagnets with exceptional coercivity (>14 T) (DeStefano et al., 6 Feb 2025); maximized AHE by gating through Weyl points (Ohno et al., 2022); and quantized zero-field AHE with ultra-low-current rewriting in moiré materials (Serlin et al., 2019).

7. Outlook and Open Questions

Contemporary research on the intrinsic AHE reveals a vast landscape connecting band topology, symmetry, electron correlations, and transport anomalies. Open questions include:

• The role of electron-electron interactions and fluctuation effects beyond single-particle Berry curvature,
• Nonlinear and dynamical regimes (high-field, ultrafast), including frequency-dependent and higher-order Hall responses,
• Interplay with disorder, phonons, and quantum criticality,
• New realizations in bosonic, superconducting, and antiferromagnetic platforms with exotic symmetry constraints.

The unifying theme is the geometric structure of Bloch and Bogoliubov bands, encoded in their Berry curvature and related higher-order tensors, as the source of dissipationless, tunable Hall transport in a diverse array of quantum materials.