On the generalized Keffer form of the Dzyaloshinskii constant: its consequences for the spin, momentum and polarization evolution

TL;DR

This theoretical work analyzes how microscopic Dzyaloshinskii–Moriya interaction contributions—including the Keffer form, ligand-shift parallel terms, and double-vector-product variants—propagate into macroscopic spin dynamics, momentum balance, and polarization in magnetic materials. By formulating a generalized Keffer form and deriving explicit spin-density evolution equations, energy densities, and polarization evolution laws for both ferromagnetic and antiferromagnetic regimes, the authors reveal distinct Lifshitz invariants and force densities that shape LL–G dynamics and magnetoelectric couplings. The study introduces novel DMI components aligned with ligand shifts and in double-vector-product configurations, deriving corresponding spin torques and Euler-force terms, and discusses AFM-specific contributions and potential analogs in symmetric Heisenberg exchange. These results provide a cohesive framework to analyze DMI-driven phenomena (e.g., spin waves, skyrmions, electromagnons) and motivate broader generalizations of Keffer-like forms for complex magnetic materials. The framework thus advances understanding of magnetoelectric effects and spin-orbit–driven dynamics in multiferroics and AFM systems, with implications for spintronics and functional materials.

Abstract

Different analytical features of the Dzyaloshinskii-Moriya interaction are related to different contribution to the Dzyaloshinskii constant in the microscopic Hamiltonian. Consequences appear in the macroscopic Landau--Lifshitz--Gilbert equation. It leads to various phenomena. Three contributions to the Dzyaloshinskii constant are reviewed and combined in the generalized Keffer form of the Dzyaloshinskii constant. Macroscopic consequences of these three mechanisms are well-known, but further possible generalizations of the Keffer form of the Dzyaloshinskii constant are suggested. Consequences for the spin evolution equations, the momentum balance equations, and polarization evolution equations are considered. Some analog of the Keffer form is suggested for the exchange integral in symmetric Heisenberg Hamiltonian demonstrating the nontrivial contribution of the ligands in this regime.

Figures (5)

• Figure 1: The figure shows two possibilities for the location of the ligand ions relatively magnetic ions and the possibilities of their shifts from the positions corresponding to the middle point between magnetic ions. Direction of the magnetic moments is fixed along "vertical" direction, but this is unnecessary choice, anisotropy axis is not presented in the figure, type of the magnetic (easy axis or easy plane) is not fixed either. Same picture can be made for the antiferromagnetic materials.
• Figure 2: The figure shows a possibility for the location of the ligand ions relatively magnetic ions in the antiferromagnetic materials, more complex in comparison with Fig. (\ref{['MFMextremumP Fig 01']}).
• Figure 3: The figure shows a possibility for the location of the ligand ions relatively magnetic ions in the antiferromagnetic materials, which is a modification of configuration shown in Fig. (\ref{['MFMextremumP Fig 02']}).
• Figure 4: The figure shows a possibility for the location of the ligand ions relatively magnetic ions in the antiferromagnetic materials, more complex in comparison with Fig. (\ref{['MFMextremumP Fig 01']}). It is seems possible to consider the regime given in this figure with the additional shifts of the ligands $\hbox{\boldmath $\delta$}_{2,ij-AB}$ and $\hbox{\boldmath $\delta$}_{3,ij-AB}$ demonstrated in Figs. (\ref{['MFMextremumP Fig 02']}) and (\ref{['MFMextremumP Fig 03']}).
• Figure 5: The figure illustrates configuration of spins with an nonzero angle between approximately antiparallel spins. Hence, both mechanisms of the polarization formation described in Sec. V can be realized.