When an electric current passes through a conductor or magnet in a magnetic field, voltage is produced that is perpendicular to the directions of the electric and magnetic fields. This phenomenon is known as the “Hall effect” or “anomalous Hall effect.” A foundational statement in materials science, Onsager’s reciprocal theorem asserts that electron deflection direction is constant regardless of the direction of current in the plane perpendicular to the magnetic field or magnetization.

So far, no phenomenon has been found that defies this theorem. This paper reports the first observation of an anisotropic anomalous Hall effect in a thin layer of spinel oxide NiCo2O4 with conical magnetic anisotropy; this characteristic is dependent on the direction of the current.

Researchers examined the symmetry of the anisotropic anomalous Hall effect found in experiments from a phenomenological angle to comprehend this phenomenon. The results suggested that a magnetic structure known as a clustered magnetic toroidal quadrupole was involved.

As a result, they put forth a theoretical framework that satisfied Onsager’s reciprocal theorem while explaining the anisotropic anomalous Hall effect. Conical magnetic anisotropy allows this model to satisfactorily explain the coexistence of ferromagnetism and the magnetic toroidal quadrupole.