Macroscopic kinematics of the Hall electric field under influence of carrier magnetic moments

The relativistic effect on electromagnetic forces yields two types of forces which depend on the velocity of the relevant particles: (i) the usual Lorentz force exerted on a moving charged particle and (ii) the apparent Lorentz force exerted on a moving magnetic moment. In sharp contrast with type (i), the type (ii) force originates due to the transverse field induced by the Hall effect (HE). This study incorporates both forces into a Drude-type equation with a fully spin-polarized condition to investigate the effects of self-consistency of the source and the resultant fields on the HE. We also examine the self-consistency of the carrier kinematics and electromagnetic dynamics by simultaneously considering the Drude type equation and Maxwell equations at low frequencies. Thus, our approach can predict both the dc and ac characteristics of the HE, demonstrating that the dc current condition solely yields the ordinary HE, while the ac current condition yields generation of both fundamental and second harmonic modes of the HE field. When the magnetostatic field is absent, the simultaneous presence of dc and ac longitudinal currents generates the ac HE that has both fundamental frequency and second harmonic.