Asymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet

TL;DR

This paper investigates the Dzyaloshinskii-Moriya interaction in a 2D Rashba ferromagnet, deriving a general expression for the DMI constant, and explores how band structure and temperature influence magnetic interactions and textures.

Contribution

It provides a microscopic derivation of the DMI constant in a generalized 2D Rashba ferromagnet and analyzes its dependence on band structure, temperature, and SOC strength beyond the conventional model.

Findings

The conventional relation between DMI and exchange stiffness does not hold beyond the Bychkov-Rashba model.

Both D and A vanish at zero temperature in the simple Rashba model with metallic bands.

Band structure and chemical potential can be tuned to control magnetic textures via gate voltage.

Abstract

Dzyaloshinskii-Moriya interaction (DMI) is investigated in a 2D ferromagnet (FM) with spin-orbit interaction of Rashba type at finite temperatures. The FM is described in the continuum limit by an effective $s$-$d$ model with arbitrary dependence of spin-orbit coupling (SOC) and kinetic energy of itinerant electrons on the absolute value of momentum. In the limit of weak SOC, we derive a general expression for the DMI constant $D$ from a microscopic analysis of the electronic grand potential. We compare $D$ with the exchange stiffness $A$ and show that, to the leading order in small SOC strength $\alpha_{\text{R}}$, the conventional relation $D=(4 m\alpha_{\text{R}}/\hbar)A$, in general, does not hold beyond the Bychkov-Rashba model. Moreover, in this model, both $A$ and $D$ vanish at zero temperature in the metal regime (i.e., when two spin sub-bands are partly occupied). For nonparabolic bands or nonlinear Rashba coupling, these coefficients are finite and acquire a nontrivial dependence on the chemical potential that demonstrates the possibility to control the size and chirality of magnetic textures by adjusting a gate voltage.