Orbital Longitudinal Magnetoelectricity in Quasi-2D Parity-Violating Antiferromagnets: Interplay of Berry Phase and Stacking Order

Abstract

The magnetoelectric (ME) effect traditionally arises from complex spin-orbital interactions in multiferroic materials. In this work, we propose a distinct, intrinsic mechanism for the magnetoelectric effect in quasi-2D magnetic systems that lack spatial inversion symmetry. We demonstrate that in such parity-violating magnets, the stacking order, which id coupled with Berry curvature and orbital magnetic moment, generates a stacking Berry curvature dipole (SBCD) and a stacking orbital magnetic moment dipole (SOMD). The SBCD and SOMD act as fundamental ingredients of the ME response. As concrete examples, we apply our framework to antiferromagnets such as monolayer Ca(CoN)$_2$ and multilayer MnBi$_2$Te$_4$ with stacking magnetic orders. Our results reveal the microscopic orbital origin of the ME effect in antiferromagnetic systems with vanishing net Berry curvature and orbital magnetization, governed by the interplay between layer stacking, Berry curvature, and magnetic order. The SBCD is also identified as the origin of electric-field-induced Hall effects.

Figures (5)

• Figure 1: (a) Schematic illustration of the longitudinal ME effect in a bilayer system. The interlayer distance is $d$. Under the applied magnetic (B) field and electric (E) field, the electric polarization and magnetization can be induced. (b) The stacking Berry curvature dipole (SBCD) and stacking orbital magnetic moment dipole (SOMD) are centralized near the band edge, which give rise to the orbital ME effect.
• Figure 2: (a) Schematic of the low-energy band structure with $C_{4z}\mathcal{PT}$ symmetry, illustrating the spin-layer locking. The $X$ and $Y$ valleys correspond to spin-up and spin-down states, respectively. The side panel shows magnetic moments localized primarily on the top and bottom Co atoms, aligned in opposite directions. (b) Momentum-space distribution of the Berry curvature component $\mathcal{B}_{v\bm{k}}$. (c) Calculated magnetoelectric coefficient $\chi{\text{me}}$ as a function of Fermi energy $\mu$. (d) Response of the electric Hall conductivity, $d\sigma_{H}/d\Delta$. Parameters for the model: $(v_1, v_2) = (0.6, 0.4)\mathrm{eV}\cdot \AA$ and $\Delta = 0.3$ eV.
• Figure 3: (a) Schematic plot of the AFM bilayers wth $\mathcal{PT}$ symmetry and interlayer AFM order. (b) The band structure for $N_l=2$. (c) and (d) The $k$-space SBCD $B_{n\bm{k}}$ for the band 1 and 2 in (b). (e) The ME coeifficient $\chi_{me}$ as a function if $\mu$ in the low-energy regime. Both Fermi-surface and Fermi-sea terms are shown. (f) The electric Hall effect $d\sigma_H/d\Delta$ as a function if $\mu$. (g) The band structure of multilayer AFM with $N_l=20$. The gapped surface state is labeled by orange lines. (h) The $\chi_{me}$ approaches $e^2/2h$ as the (even) number of layer goes up. Parameters: $A=1.4,B=1,M_0=1,m_z=0.3$. The temperature is set to be $k_B T=0.02$.
• Figure 4: (a) Schematic for an axion insulator with gapped surface state. The Dirac mass of top and bottom layers are opposite. (b) The FM bilayer with the quantum anomalous Hall state with the same Dirac mass for two layers. (c) $\chi_{me}$ as a function of $\kappa$. When $\kappa=1$, the interlayer coupling vanishes.
• Figure 5: summary of the origin of orbital magnetoelectricity in two representative systems: in chiral-stacked multilayer graphene, the effect is governed by the orbital ME moment $\alpha_{n\bm{k}}$, while in parity-violating antiferromagnets, it is dominated by the SBCD and SOMD, which is the primary focus of this work.