Universal classes of disorder scatterings in in-plane anomalous Hall effect

The in-plane anomalous Hall effect (IPAHE) with planar Hall current and magnetization/magnetic fields in various quantum materials has received increasing attention. Most of the current efforts are devoted to the intrinsic part due to the Berry curvature of electronic bands, however, how disorder scattering affects the extrinsic part (the skew scattering and side jump) remains largely elusive. Here we theoretically investigate the three universal classes of disorder scattering (scalar, spin-conserving, and spin-flipping) for the IPAHE, based on the prototypical two-dimensional massive Dirac fermion model with warping term under generic Zeeman fields. We find that the different disorder scattering results in a distinct dependence of the anomalous Hall conductivity on disorder strength, and we recover previously known results within some limits. Remarkably, the spin-flipping scattering could give rise to nontrivial contributions featuring sinusoidal oscillations with periods of \textgreek{π} and 2\textgreek{π} to the extrinsic part, in contrast to the standard two-dimensional massive Dirac fermions. Our work unveils the rich features of anomalous transport in planar Hall geometry in the presence of disorder scattering and provides some useful insights into the magnetotransport phenomena.

💡 Research Summary

This paper provides a comprehensive theoretical analysis of the extrinsic contributions to the in‑plane anomalous Hall effect (IP‑AHE) in two‑dimensional massive Dirac fermion systems with a hexagonal warping term and generic Zeeman fields. The authors start from a minimal Hamiltonian
(H_{0}(\mathbf{k}) = v(k_{y}\sigma_{x} - k_{x}\sigma_{y}) + \lambda k,\sigma_{z} + \mathbf{M}\cdot\boldsymbol{\sigma}),
where (v) is the Dirac velocity, (\lambda) quantifies the warping, and (\mathbf{M}=(M_{\parallel},M_{z})) represents the in‑plane and out‑of‑plane components of the magnetization (or Zeeman field). The warping term breaks the three‑fold rotational symmetry (C({3v})) and, together with an in‑plane field, shifts the Dirac point away from (\mathbf{k}=0) and opens a small gap (\Delta{1}\propto \lambda k’^{3}\sin 3\theta). An additional out‑of‑plane field contributes a gap (\Delta_{2}=M_{z}); the total gap is (\Delta\approx\Delta_{1}+\Delta_{2}).

The intrinsic Hall conductivity is obtained by integrating the Berry curvature of the conduction band, yielding
(\sigma_{\text{in}}^{xy}= -\frac{e^{2}}{h}\frac{M_{z}^{2}}{E_{F}} + \frac{e^{2}}{h}\frac{\lambda M_{\parallel}^{3}\sin 3\theta}{v^{3}E_{F}}).
The first term is the familiar out‑of‑plane contribution, while the second originates from the warping‑induced magnetic octupole and carries a (\sin 3\theta) angular dependence.

To address the extrinsic part, the authors introduce three universal disorder classes, each described by a short‑range potential (V_{\text{dis}}(\mathbf{r})=\sum_{i}(V_{0}\sigma_{0}+ \mathbf{V}\cdot\boldsymbol{\sigma})\delta(\mathbf{r}-\mathbf{R}_{i})):

• Class A (scalar disorder) – non‑magnetic impurities, represented by (V_{0}\sigma_{0}).
• Class B (spin‑conserving disorder) – magnetic impurities that preserve the (z) component of spin, (V_{z}\sigma_{z}).
• Class C (spin‑flipping disorder) – in‑plane magnetic impurities or spin‑fluctuations, (V_{x}\sigma_{x}) and (V_{y}\sigma_{y}).

A Gaussian white‑noise model is employed, and the self‑energy is calculated in the first Born approximation. The vertex corrections are treated within the ladder (non‑crossing) approximation. Using the Kubo‑Středa formula (\sigma^{xy}= \sigma_{I}^{xy}+ \sigma_{II}^{xy}), the authors separate the Fermi‑surface contribution ((\sigma_{I})) from the Fermi‑sea term ((\sigma_{II})). The extrinsic Hall conductivity is further decomposed into side‑jump ((\sigma_{\text{sj}})) and skew‑scattering ((\sigma_{\text{sk}})) parts.

Side‑jump contributions are of order (n_{i}^{0}) (independent of impurity density) and consist of a second‑order term (\sigma_{\text{sj},2}) and a fourth‑order “intrinsic skew” term (\sigma_{\text{sj},4}). For classes A and B the side‑jump retains the three‑fold symmetry, giving expressions proportional to (M_{z}^{2}/(E_{F}^{2}+M_{z}^{2})(E_{F}+3M_{z}^{2})^{2}) and a warping‑induced term (\propto \lambda M_{\parallel}^{3}\sin 3\theta/(v^{3}E_{F})). Class C, however, breaks C({3v}) symmetry and yields linear‑in‑(M{\parallel}) terms (\propto \sin\theta) together with the warping term, reflecting the anisotropic scattering caused by spin‑flip processes.

Skew‑scattering contributions arise from the non‑Gaussian third‑order disorder correlator and scale as (1/n_{i}). For classes A and B the skew term contains the same C({3v})‑symmetric denominators as the side‑jump but with opposite signs for the warping part. Class C produces a markedly different result: (\sigma{\text{sk}}^{xy}\propto \lambda E_{F} M_{\parallel}^{2}\sin 2\theta/v^{3}), i.e. a (\pi)-periodic sinusoidal dependence on the in‑plane magnetization angle. This is a novel feature absent in conventional massive Dirac models and directly stems from the symmetry‑breaking nature of spin‑flip scattering.

Collecting all pieces, the total Hall conductivity for each class reads (schematically):

• Class A: (\sigma^{xy}= \sigma_{\text{in}}^{xy}+ \sigma_{\text{sj}}^{xy}(A)+ \sigma_{\text{sk}}^{xy}(A)), with explicit formulas involving (M_{z}), (\lambda M_{\parallel}^{3}\sin 3\theta) and a skew factor (\propto u_{1}^{3}/(n_{i}u_{0}^{4})).
• Class B: Similar to A but with denominators ((3E_{F}^{2}+M_{z}^{2})^{2}) and an extra skew term (-16\lambda M_{\parallel}^{3}\sin 3\theta M_{z}/E_{F}^{3}(3E_{F}^{2}+M_{z}^{2})^{3}).
• Class C: (\sigma^{xy}= (e^{2}/h)