Extrinsic Anomalous Hall Conductivity

• Extrinsic anomalous Hall conductivity is defined as the disorder-induced contribution to the AHE, primarily driven by skew scattering and side-jump mechanisms in systems with spin–orbit coupling.
• Quantum transport formalisms, including vertex corrections in approaches like Kubo–Bastin, enable the separation of extrinsic components from intrinsic Berry curvature effects.
• Engineering strategies such as impurity doping or suppression allow control over extrinsic AHC, optimizing device performance and revealing subtle electronic symmetry breakings.

Extrinsic anomalous Hall conductivity (AHC) designates the portion of the anomalous Hall effect (AHE) in conducting solids that arises from disorder-induced processes—specifically, the asymmetric scattering of charge carriers in the presence of spin–orbit coupling and broken time-reversal symmetry. Unlike the intrinsic contribution, which is governed by the momentum-space Berry curvature of the underlying band structure, extrinsic AHC emerges from microscopic mechanisms such as skew scattering and side-jump, and is typically operationally defined via the inclusion of vertex corrections in quantum transport formalisms. Quantitative separation and analysis of extrinsic AHC are central for the interpretation of Hall measurements in metals, alloys, magnetic semiconductors, topologically nontrivial conductors, and antiferromagnets.

1. Quantum Transport Formalism and Operational Definition

In the quantum Kubo–Středa or Kubo–Bastin approach, the total tensorial Hall conductivity, including all sources of the AHE, reads (for the off-diagonal component, e.g., $\sigma_{xy}$)

$\sigma_{xy} = \sigma_{xy}^{\text{VC}}\,,$

where $\sigma_{xy}^{\text{VC}}$ is computed with all disorder-induced vertex corrections included. The intrinsic contribution is obtained by omitting these vertex corrections,

$$\sigma_{xy}^{\text{intr}} \equiv \sigma_{xy}^{\text{no VC}},$$

and the extrinsic anomalous Hall conductivity is operationally defined as

$$\sigma_{xy}^{\text{ext}} = \sigma_{xy}^{\text{VC}} - \sigma_{xy}^{\text{no VC}}$$

(Lowitzer et al., 2010).

In fully relativistic $ab$ $initio$ implementations for alloys (e.g., Korringa-Kohn-Rostoker (KKR) method with Coherent Potential Approximation (CPA)), this procedure enables systematic and material-specific evaluation of all disorder and band-structure terms on equal footing. Experimentally, similar decomposition is deduced by fitting transport data using established scaling forms.

2. Physical Mechanisms: Skew Scattering and Side-Jump

Extrinsic AHC emerges from two distinct microscopic processes:

• Skew Scattering: Originates from an asymmetry in the scattering probability from spin–orbit-coupled impurity potentials, leading to a net transverse carrier deflection. Semiclassically, the lowest-order term yields a Hall conductivity proportional to the longitudinal conductivity,

$\sigma_{xy}^{\text{skew}} \propto \sigma_{xx}.$

The proportionality constant, sometimes called the skewness factor ($S$) or skew coefficient ($\alpha_{\text{sk}}$), depends on details of impurity potential strength, spin–orbit coupling, and impurity concentration (Lowitzer et al., 2010, Kotegawa et al., 6 Sep 2024).

• Side-Jump: Reflects a shift in the carrier’s trajectory at each impurity collision, caused by spin–orbit coupling, even for symmetric scatterers. It is independent of the scattering time (to leading order),

$\sigma_{xy}^{\text{sj}} \approx \text{const},$

and in resistivity form yields a term quadratic in $\rho_{xx}$ (Lowitzer et al., 2010, Kotegawa et al., 6 Sep 2024).

The total extrinsic AHC then takes the form

$$\sigma_{xy}^{\text{ext}} = S \sigma_{xx} + \sigma_{xy}^{\text{sj}},$$

which underpins the standard empirical fitting procedure in dilute magnetic alloys (Lowitzer et al., 2010).

3. Experimental Scaling and Quantitative Determination

Empirical identification and quantification of extrinsic AHC relies on characteristic scaling relations linking anomalous Hall and longitudinal transport coefficients. In resistivity form,

$$\rho_{xy}^{\text{AHE}} = a\,\rho_{xx} + b\,\rho_{xx}^2,$$

where $a$ parametrizes skew scattering and $b$ the side-jump (and/or intrinsic) contributions (Yang et al., 2019, Kotegawa et al., 6 Sep 2024, Lowitzer et al., 2010). In conductivity form,

$\sigma_{xy}^{\text{AHE}} = S \sigma_{xx} + C,$

or, in more refined treatments,

$$\sigma_{xy}^{\text{AHE}} = (\alpha/\sigma_{xx,0}) \sigma_{xx}^2 + b,$$

with $\alpha$ a skew constant, $\sigma_{xx,0}$ the residual longitudinal conductivity, and $b$ a disorder-independent offset (Yang et al., 2019).

Ab initio calculations in fcc-Fe$_x$Pd$_{1-x}$ and fcc-Ni$_x$Pd$_{1-x}$ alloys confirm this scaling: | System (Pd-rich) | $S$ ($10^{-3}$) | $\sigma_{xy}^{\text{sj}}$ [$(\mathrm{m}\Omega\,\mathrm{cm})^{-1}$] | |------------------|----------------|--------------------------------------| | Fe$_x$Pd$_{1-x}$ | –2.7 | ~0.1 | | Ni$_x$Pd$_{1-x}$ | 2.0 | ~0.4 | | Ni$_x$Pd$_{1-x}$ (Ni-rich) | –6.6 | ~4.8 | (Lowitzer et al., 2010).

Direct comparisons with experiment establish that in the clean limit (high $\sigma_{xx}$), $S \sigma_{xx}$ (skew scattering) dominates, while the side-jump term remains comparatively small and nearly concentration-independent.

4. Material Classes and Enhanced Extrinsic AHC

Extrinsic AHC plays a central or even dominant role in several material platforms:

• Nonmagnetic and Magnetic Weyl/Dirac Semimetals: KV$_3$Sb$_5$ exhibits giant extrinsic AHC ($$\sigma_{xy}^{\text{AHE}}\approx1.55\times10^4\,\Omega^{-1}$$cm$^{-1}$), entirely due to an anomalously enhanced and quadratic-in-$\sigma_{xx}$ skew-scattering mechanism. The Hall ratio ($\mathrm{AHR}\approx1.8\%$) and extrapolated Hall angles approach limits typically reserved for quantum anomalous Hall systems (Yang et al., 2019).
• $f$-Electron Antiferromagnets: Ce$_2$CuGe$_6$ displays an extrinsic AHC $~85\%$ of total, scaling as $\sigma_{xx}^2$, with a small intrinsic offset determined by point-group symmetry (Kotegawa et al., 6 Sep 2024).
• Chemically-Doped Topological Ferromagnets: In Co$_3$Sn$_2$S$_2$ and its Fe/In-doped variants, extrinsic AHC can be tuned to rival or exceed the intrinsic term via impurity choice and concentration as quantified by Tian-Ye-Jin (TYJ) scaling, with the anomalous Hall angle reaching up to $33\%$ in heavily skew-scattering-dominated regimes (Shen et al., 2020, Rathod et al., 2023).

5. Microscopic Theory and Model Calculations

Microscopic treatments employ diagrammatic perturbation theory or first-principles calculations within quantum transport frameworks to obtain extrinsic AHC.

• For two-band or Dirac models (e.g., quantum anomalous Hall systems (Lu et al., 2013)), side-jump and skew components can be obtained analytically in terms of impurity correlators ($V_0, V_m, V_3^0, V_3^z$), Berry-phase angles ($\theta_F$), and critical carrier densities:

$$\sigma_{xy}^{\text{sj}} = -\frac{e^2}{h} \cos\theta_F \left[ \frac{2\alpha}{1-\alpha} + \frac{3\alpha^2 \eta_B}{2(1-\alpha)^2} \right],$$

where $\alpha$ depends on the ratio of magnetic to nonmagnetic scattering.

• For disordered ferromagnets with Gaussian disorder (Czaja et al., 2013), the full Kubo–Bastin formalism reveals that "side-jump" and "intrinsic skew-scattering" terms both survive in the clean limit, i.e., they are scattering-independent in this approximation but distinct from conventional skew-scattering (which diverges as $1/n_{\text{imp}}$).
• In Rashba ferromagnets within a DLM-CPA framework (Sakuma, 2017), the extrinsic Fermi-surface vertex correction grows with increasing temperature, reaching magnitudes comparable to the intrinsic Berry curvature term.

The sign and magnitude of the extrinsic contributions depend sensitively on the nature and concentration of the scatterers, impurity SOC strength, carrier density, and details of the band structure.

6. Engineering, Suppression, and Enhancement of Extrinsic AHC

Manipulation of the extrinsic AHC is central for both fundamental investigations and device-level optimization:

• Suppression Strategies: In Fe-rich kagome magnets (Fe$_3$Sn), increasing the Fe:Sn ratio eliminates high-SOC Sn impurity centers, suppressing extrinsic skew scattering. The effective $\alpha_{\text{sk}}$ is thus reduced by more than an order of magnitude, isolating the intrinsic AHC plateau at $\sim$555 S/cm over a broad temperature range (Liu et al., 1 Nov 2024).
• Enhancement Strategies: Selective introduction of high-SOC impurity centers (“alien-atom scattering”) in materials such as Co$_3$Sn$_2$S$_2$ or by chemical doping in Dirac/Weyl semimetals, can amplify the extrinsic AHC component up to several hundred percent, often achieving record anomalous Hall angles (Rathod et al., 2023, Shen et al., 2020).
• Crossover Regimes and Scaling Laws: Disorder-induced crossover between extrinsic (e.g., $\sigma_{xy}\propto\sigma_{xx}^{1.6}$ in the dirty limit) and intrinsic (plateau $\sigma_{xy}^c$) transport regimes can be directly observed by tuning sample purity and conductivity in systems such as Fe$_3$GaTe$_2$ (Lee et al., 31 Jul 2025). The exponent $n\approx1.6$ proves characteristic of mixed skew/side-jump behavior beyond the conventional dichotomy.

7. Broader Theoretical Considerations and Future Directions

Recent theoretical work demonstrates that:

• In altermagnets, extrinsic AHC can be comparable to the intrinsic term in certain spin Laue symmetry classes (those allowing Dzyaloshinskii–Moriya-type interaction), persisting even in the clean limit (Osin et al., 5 Nov 2025). In classes without the requisite symmetry, extrinsic AHC is negligible.
• The nonanalytic dependence of both intrinsic and extrinsic AHC on spin–orbit coupling arises from symmetry breaking on momentum-space nodal planes.
• The parameter space of impurity types, spin-orbit strength, and carrier densities continues to reveal new mechanisms, including cluster-based skew scattering with quadratic scaling behaviors (Yang et al., 2019).

A plausible implication is that extrinsic mechanisms may be engineered not only to maximize Hall response for applications, but also to probe subtle electronic symmetry breaking via their sensitivity to impurity and spin–orbit physics, with ongoing advances in both materials synthesis and $ab$ $initio$ methodologies.

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In summary, extrinsic anomalous Hall conductivity represents a quantitatively and conceptually distinct channel for transverse charge transport in magnetic and topological conductors, arising from impurity-driven processes whose magnitude, sign, and scaling behavior are tunable via crystallographic, chemical, and electronic degrees of freedom. Contemporary research provides unified formalisms for separating and controlling these contributions, enabling both detailed comparison to experiment and strategic materials design.