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

Pavel A. Andreev

TL;DR

The paper proposes a novel odd anisotropy of the symmetric Heisenberg exchange (OASEI) arising from ligand shifts, yielding a Keffer-like contribution and modifying the energy, torque, and polarization landscape in antiferromagnetic and multiferroic systems. Using a quantum hydrodynamic approach, it derives spin torques, force fields, and energy densities associated with this interaction, and links them to a generalized spin-current polarization mechanism. It analyzes XYZ-model antiferromagnets and a Dzyaloshinskii-Moriya term to balance cycloidal order, and derives spin-wave dispersion relations for collinear easy-axis, collinear easy-plane, and cycloidal states, highlighting how OASEI alters mode coupling and stability. The results suggest new pathways for magnetoelectric coupling and spintronic control in AFM and multiferroic materials, mediated by ligand-induced symmetry-breaking in exchange.

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.

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

TL;DR

The paper proposes a novel odd anisotropy of the symmetric Heisenberg exchange (OASEI) arising from ligand shifts, yielding a Keffer-like contribution and modifying the energy, torque, and polarization landscape in antiferromagnetic and multiferroic systems. Using a quantum hydrodynamic approach, it derives spin torques, force fields, and energy densities associated with this interaction, and links them to a generalized spin-current polarization mechanism. It analyzes XYZ-model antiferromagnets and a Dzyaloshinskii-Moriya term to balance cycloidal order, and derives spin-wave dispersion relations for collinear easy-axis, collinear easy-plane, and cycloidal states, highlighting how OASEI alters mode coupling and stability. The results suggest new pathways for magnetoelectric coupling and spintronic control in AFM and multiferroic materials, mediated by ligand-induced symmetry-breaking in exchange.

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.
Paper Structure (29 sections, 65 equations, 2 figures)

This paper contains 29 sections, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Keffer-like form of the exchange integral in symmetric Heisenberg Hamiltonian
  3. Heisenberg constant with the ligand shift parallel to the relative distance
  4. Heisenberg constant with combination of two ligand shifts
  5. Combined contribution of the ligand shifts
  6. Spin torque
  7. Force field in the Euler momentum balance equation
  8. Energy density
  9. Spin-current model
  10. Contribution of novel interaction in the polarization formation via the spin-current model
  11. Comparison with known form of polarization
  12. Energy density and spin torque for antiferromagnetic materials in XYZ model
  13. Energy density in XYZ model
  14. Spin torque in XYZ model
  15. Dzyaloshinskii-Moriya interaction
  16. ...and 14 more sections

Figures (2)

  • Figure 1: The figure shows a possible configuration of the magnetic ions (demonstrated as arrows) and ligands (oxygen ions demonstrated as black circles) in the antiferromagnetic material. Partial shifts of the ligand are demonstrated in the figure as $\hbox{\boldmath $\delta$}_{1}$ and $\hbox{\boldmath $\delta$}_{2,ij-AB}$.
  • Figure 2: This figure is similar to Fig. (\ref{['Fig 01']}), but we present the partial ligand shift $\hbox{\boldmath $\delta$}_{3,ij-AB}$ (which is parallel to the vector connecting magnetic ions $A$ and $B$$\textbf{r}_{AB}$) instead of the partial ligand shift $\hbox{\boldmath $\delta$}_{2,ij-AB}$ (which is perpendicular to the plane containing vectors $\textbf{r}_{AB}$ and $\hbox{\boldmath $\delta$}_{1}$).