Keffer-like form of the symmetric Heisenberg exchange integral: Contribution to the Landau--Lifshitz--Gilbert equation and spin wave dispersion dependence

TL;DR

This paper introduces a new contribution to the symmetric Heisenberg exchange integral based on ligand shifts, affecting the Landau--Lifshitz--Gilbert equation and spin wave dispersion in magnetic materials.

Contribution

It proposes a novel term in the Heisenberg exchange constant arising from ligand shifts, influencing spin dynamics and energy density formulations.

Findings

Derived a new term in the Landau--Lifshitz--Gilbert equation.

Modified spin wave dispersion relations for antiferromagnetic materials.

Applied the effective spin current to multiferroic polarization models.

Abstract

The symmetric Heisenberg exchange interaction and antisymmetric Dzyaloshinskii-Moriya interaction are parts of the tensor potential describing effective spin-spin interaction caused by the superexchange interaction of magnetic ions via nonmagnetic ion. There is the Keffer form of the vector constant of the Dzyaloshinskii-Moriya interaction, which includes the shift of the nonmagnetic ion (ligand) from the line connecting two magnetic ions. It is suggested, in this paper, that the ligand shift can give contribution in the constant of the symmetric Heisenberg interaction in antiferromagnetic or ferrimagnetic materials. Hence, the constant of the Heisenberg interaction is composed minimum of two terms. One does not depend on the ligand shift an gives standard contribution in the energy density like term with no derivatives of the spin densities or term containing two spatial derivatives of the spin densities. It is demonstrated that additional term gives a term in the energy density containing one spatial derivative of the spin density. Corresponding contribution in the Landau--Lifshitz--Gilbert equation is found. Possibility of the noncollinear equilibrium order of spin under influence of new spin torque is discussed. Modification of the spin wave (normal modes) dispersion dependencies in the antiferromagnetic materials is found for the collinear order and for the cycloidal order of spins. Effective spin current is derived and applied for the spin-current model of the polarization origin in multiferroics.