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

Different analytical features of the Dzyaloshinskii-Moriya interaction are related to different contributions 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. The fourth possible form of the Dzyaloshinskii constant is suggested as well. 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.