The thermodynamic properties of exchange stiffness

Abstract

The effective temperature dependence of micromagnetic parameters such as anisotropy (K), exchange stiffness (A) and the Dzyaloshinskii–Moriya interaction (DMI) are usually written as a power law of the magnetisation. For anisotropy this is well founded, based on Callen-Callen theory1. For exchange stiffness and DMI the power laws are based on the assumption that they behave the same as anisotropy and from classical numerical models. Recently there has been a lot of interest in the temperature dependence of these properties because their value affects spin textures like skyrmions.Inferring the DMI from experimental measurements often requires first deducing the exchange stiffness by fitting Bloch's law to measurements of magnetisation at finite temperature. The problem is that Bloch's law contains the zero temperature value of exchange stiffness, but is used to fit a finite temperature range over which the value of the exchange stiffness should change significantly. Another issue is that Bloch's law is derived using quantum mechanics with magnons which obey Bose statistics – numerical models typically use classical statistics for the thermal occupation of spin waves. Classical models cannot reproduce Bloch's T3/2 law for ferromagnets or the T2 dependence of sublattice magnetisation in antiferromagnets2. So, the classical thermodynamics of exchange is also a poor approximation for real materials in which the quantum distributions of magnons plays a large role.Here we discuss a quantum implementation of atomistic spin dynamics2, compare with classical results and present the temperature scaling of the micromagnetic exchange stiffness for both ferromagnets and antiferromagnets calculated from magnon spectra. We show antiferromagnets have similar scaling behaviour to ferromagnets and that the results using quantum statistics doesn’t follow a power law.Figure 1 presents the thermal behaviour of the exchange stiffness as a function of reduced sublattice magnetisation of the prototypical antiferromagnet NiO used in antiferromagnetic spintronics3. At low temperature (high magnetisation) the exchange stiffness is almost temperature independent when using quantum statistics in stark contrast to a power law.