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

Masamichi Sakai

doi:10.1063/1.4954808

Abstract

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.

Highlights

The combination of charge and spin transport is one of key techniques for realizing normally off state operation of logic and memory devices in computers
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
A summary of the results obtained in Section III is provided in Table I: (i) no anomalous Hall effect (AHE) is generated, and ordinary Hall effect (OHE) is solely observed in the dc case (ω = 0); (ii) the Hall coefficient of the ac OHE is modified by second harmonic that is generated by the simultaneous presence of the skin effect and the carrier spin polarization; (iii) when the magnetostatic field is absent, the simultaneous presence of dc and ac longitudinal currents generates an ac AHE with an additional second harmonic

Summary

INTRODUCTION

The combination of charge and spin transport is one of key techniques for realizing normally off state operation of logic and memory devices in computers. There is another feature specific to the ac case; the apparent Lorentz force given by Eq (1) is augmented by −(1/c2)(∂/∂t)(m⊥ × E⊥).[21] This feature is expressed in terms of ASOI as −(∂/∂t)ASOI; thereby, it is analogous to an ac electric field applied to charged particles, where the magnitude of −(∂/∂t)ASOI is given by (mω/c2)E⊥. The complete expression of the electromagnetic forces exerted on a moving CMP is given by Eq (13) This approximate expression is the similar to that obtained by Kholmetskii et al, who used Lorentz transformations for the magnetic and electric dipoles through their derivation.[23] In an electrostatic case, i.e., (∂/∂t)E = 0, Eq (13) is reduced to the expression obtained by Hnizdo, who used the Lagrangian formalism.[24] By substitutinf B = 0 and (∂/∂t)E = 0 into Eq (13), we obtain an expression similar to Chudnovsky,[1] who employed the nonrelativistic limit of the Dirac Hamiltonian for a spin–1/2 particle

Equation of motion for the Hall electric field

Solution under linear approximation

Perturbation approach for studying the magnetic moment effect

DISCUSSION

CONCLUSIONS