Non-Lifshitz invariants corrections to Dzyaloshinskii-Moriya interaction energy

TL;DR

This paper investigates the corrections to Dzyaloshinskii-Moriya interaction energy in two-dimensional chiral magnets, revealing non-Lifshitz invariants and higher-order effects due to spin-orbit coupling, with implications for magnetic order stability.

Contribution

It provides a microscopic analysis of DMI tensors showing non-Lifshitz invariants and higher-order corrections in Rashba spin-orbit interactions, expanding understanding of DMI in chiral magnets.

Findings

Non-Lifshitz invariants appear at third order in spin-orbit coupling.

Higher-order terms can enhance DMI at strong spin-orbit coupling.

DMI free energy depends on symmetry groups and spin-orbit interactions.

Abstract

We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling. The resulting free-energy density of the system contains linear terms in the spatial gradient of the magnetic texture, which mark an instability of the system towards the formation of nontrivial magnetic orders such as skyrmions or chiral domain walls. We perform a microscopic analysis of DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants features. In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order in the small exchange coupling. The higher-order terms may even lead to an enhancement of DMI interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of the conduction band. Finally, we also study the DMI free energy generated from Rashba spin-orbit interaction in different symmetry groups.