Anomalous Hall effect from nonlinear magnetoelectric coupling

The anomalous Hall effect (AHE) is a topology-related transport phenomenon being of potential interest in spintronics, because this effect enables the efficient probe of magnetic orders (i.e., data readout in memory devices). It is well known that AHE spontaneously occurs in ferromagnets or antiferromagnets with magnetization. While recent studies reveal electric-field induced AHE (via linear magnetoelectric coupling), an AHE originating from {\it nonlinear} magnetoelectric coupling remains largely unexplored. Here, by symmetry analysis, we establish the phenomenological theory regarding the spontaneous and electric-field driven AHE in magnets. We show that a large variety of magnetic point groups host an AHE that is driven by uni-axial, bi-axial, or tri-axial electric field and that comes from nonlinear magnetoelectric coupling. Such electric-field driven anomalous Hall conductivities are reversible by reversing the magnetic orders. Furthermore, our first-principles calculations suggest Cr$_2$O$_3$ and CoF$_2$ as candidates hosting the aforementioned AHE. Our work emphasizes the important role of nonlinear magnetoelectric coupling in creating exotic transport phenomena, and offers alternative avenues for the probe of magnetic orders.

💡 Research Summary

The paper “Anomalous Hall effect from nonlinear magnetoelectric coupling” presents a comprehensive theoretical and computational study of an anomalous Hall effect (AHE) that is driven not by spontaneous magnetization alone but by electric fields acting through nonlinear magnetoelectric (ME) couplings. The authors begin by recalling that the AHE, a transverse voltage response to a longitudinal current, is intimately linked to the Berry curvature of occupied electronic states and traditionally appears in ferromagnets or antiferromagnets that possess a net magnetization. Recent experiments have shown that a linear ME coupling can induce an AHE under an applied electric field, yet the possibility of generating AHE via higher‑order (quadratic, cubic, quartic) ME interactions has not been systematically explored.

Using group‑theoretical analysis, the authors examine all 122 magnetic point groups (MPGs). They identify 90 MPGs (both type‑I and type‑II) that can support either a spontaneous Hall vector M (equivalent to the magnetization vector) or an electric‑field‑induced Hall vector through linear or nonlinear ME couplings. The Hall vector components (σzy, σxz, σyx) transform exactly like the three Cartesian components of magnetization, allowing a direct mapping between symmetry‑allowed magnetization directions and allowed Hall conductivities.

A central result is the formulation of an effective magnetic field Beff that can be generated by external electric fields. The authors write Beff as a series expansion:

• Zero‑order term λα (spontaneous magnetization),
• First‑order term λαβ,l El (linear ME coupling),
• Second‑order term λαβγ,lm El Em (quadratic ME coupling),
• Third‑order term λαβγδ,lmn El Em En (cubic ME coupling).

Each term is constrained by the symmetry of the specific MPG. For example, in the MPG (\bar{6}m2.1) the effective field components are: (B_x = λ_{xxz}E_xE_z + λ_{xyz}E_yE_z), (B_y = λ_{yxz}E_xE_z), (B_z = λ_{zx}E_x^3). Thus, σzy can be induced by a bi‑axial field (E_xE_z) (second order), σxz by (E_yE_z) (second order), and σyx by a uni‑axial field (E_x^3) (third order). The authors also discuss “quenching” conditions where specific ratios of electric‑field components (e.g., (|E_x|=|E_y|) in certain MPGs) cause the effective field to vanish, suppressing the AHE.

To validate the symmetry‑based predictions, the authors perform first‑principles density‑functional theory (DFT) calculations using VASP with PAW potentials, LDA+U (U = 3 eV for 3d electrons), and appropriate k‑point meshes. Structural relaxations are carried out without spin‑orbit coupling; subsequently, non‑collinear calculations with spin‑orbit coupling are performed on the relaxed structures under finite electric fields, employing the method of Refs.