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We demonstrate how exchange hardening can arise in a chemically-disorderedsolid solution from a first-principles statistical mechanics approach. Ageneral mixed-basis chemical and magnetic cluster expansion has been developed,and applied to the Mn$_{1-x}$Fe$_x$Ru$_2$Sn Heusler alloy system; single-phasesolid solutions between antiferromagnetic \ch{MnRu2Sn} and ferromagnetic\ch{FeRu2Sn} with disorder on the Mn/Fe sublattice that exhibit unexpectedexchange hardening. Monte Carlo simulations applied to the cluster expansionare able to reproduce the experimentally measured magnetic transitiontemperatures and the bulk magnetization as a function of composition. Themagnetic ordering around a site is shown to be dependent not only on bulkcomposition, but also on the identity of the site and the local compositionaround that site. The simulations predict that local antiferromagneticorderings form inside a bulk ferromagnetic region at intermediate compositionsthat drives the exchange hardening. Furthermore, the antiferromagnetic regionsdisorder at a lower temperature than the ferromagnetic regions, providing anatomistic explanation for the experimentally-observed decrease in exchangehardening with increasing temperature. These effects occur on a length scaletoo small to be resolved with previously-used characterization techniques.

Modeling magnetic evolution and exchange hardening in disordered magnets: The example of Mn$_{1-x}$Fe$_x$Ru$_2$Sn Heusler alloys