Anomalous Hall effect in rhombohedral graphene

TL;DR

This work addresses the anomalous Hall effect in rhombohedral (ABC-stacked) multilayer graphene by computing $\sigma_{xy}$ within the Kubo-Streda formalism, incorporating impurity scattering through Gaussian (weak-dense) and non-Gaussian (strong-sparse) disorder. The authors go beyond the non-crossing approximation by including crossed impurity-line diagrams (diffractive skew scattering) and, for strong sparsity, Mercedes-star diagrams, deriving analytic results for an isotropic model and semi-numerical warping corrections that capture trigonal warping effects. They show that the intrinsic contribution, side-jump, and diffractive skew scattering compete to determine $\sigma_{xy}$, with Gaussian skew scattering vanishing for $n>1$, and that warping introduces quantitative but not qualitative changes, especially near low energies. The framework provides a controlled description relevant to experiments, highlighting how disorder and lattice-induced warping shape anomalous transport in multiband graphene systems and offering a pathway to interpret AHE measurements in rhombohedral graphene.

Abstract

Motivated by recent experiments on rhombohedral stacked multilayer graphene and the observation of the anomalous Hall effect in a spontaneous spin-valley polarized quarter metal state, we calculate the anomalous Hall conductivity for this system in the presence of two types of impurities: weak and dense as well as sparse and strong. Our calculation of $σ_{xy}$ is based on the Kubo-Streda diagrammatic approach. In a model with Gaussian disorder applicable to weak dense impurities, this involves all non-crossing diagrams (intrinsic, side-jump and Gaussian skew-scattering contributions) and additionally diagrams with two intersecting impurities, X and $Ψ$, representing diffractive skew-scattering processes. A "Mercedes star" diagram (non-Gaussian skew scattering) is furthermore included to treat in the case of strong, sparse impurities. We supplement our asymptotically exact analytical solutions for an isotropic model without warping effects by semi-numerical calculations accounting perturbatively for warping, which plays a crucial role in the low-energy band structure.

Figures (7)

• Figure 2: Exemplary diagrams for the anomalous Hall conductivity: a) intrinsic contribution, b) side jump, c) Gaussian skew-scattering, d) diffractive skew-scattering, e) third moment (skewness).
• Figure 3: Example diagram of the side-jump contribution with a vertex correction (a), which is obtained by summing ladder diagrams (b).
• Figure 4: Intrinsic Eq. \ref{['eq: int']}, side-jump Eq. \ref{['eq: sj']}, skew-scattering components Eqs. \ref{['eq: xfull']}-\ref{['eq: psifull']} and the extrinsic part of the anomalous Hall conductivity for $n=4$. We plot separately the skew scattering parts in the non-crossing approximation (purple) and diffractive (gray). The vertical dashed line marks $\epsilon_F/m = 1$.
• Figure 5: Full anomalous Hall conductivity (one valley and one spin) for a) $n=4$ with warping parameter $\tilde{w}_4 = \frac{w}{m} \left(\frac{\epsilon_F ^2}{m^2}-1 \right)^{-1/2}$ and b) $n=3$ with $\tilde{w}_3 = \frac{w}{\upsilon^{1/4} m^{3/4}} \left(\frac{\epsilon_F ^2}{m^2}-1 \right)^{-3/8}$.
• Figure : (a)