Temperature scaling of the Dzyaloshinsky-Moriya interaction in the spin wave spectrum

The temperature scaling of the micromagnetic Dzyaloshinsky-Moriya exchange interaction is calculated from saturated to vanishing magnetization. We use Green's function theory to derive the finite-temperature spin wave spectrum of ferromagnetic systems described by a classical atomistic spin model Hamiltonian with temperature-independent parameters. Within this model, we find universal expressions for the temperature scaling not only of the Dzyaloshinsky-Moriya interaction but also of the Heisenberg exchange stiffness and the single-ion anisotropy. In the spirit of multiscale models, we establish a clear connection between the atomistic interactions and the temperature-dependent coefficients in the spin wave spectrum and in the micromagnetic free-energy functional. We demonstrate that the corrections to mean-field theory or the random phase approximation for the temperature scaling of Dzyaloshinsky-Moriya and Heisenberg exchange interactions have very similar forms. In the presence of thermal fluctuations and Dzyaloshinsky-Moriya interaction an anisotropylike term emerges in the spin wave spectrum which, at low temperature, increases with temperature, in contrast to the decreasing single-ion anisotropy. We evaluate the accuracy of the theoretical method by comparing it to the spin wave spectrum calculated from Monte Carlo simulations.