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Many Heusler compounds possess magnetic properties well-suited forapplications as spintronic materials. The pseudo-binaryMn$_{0.5}$Fe$_{0.5}$Ru$_2$Sn, formed as a solid solution of two full Heuslers,has recently been shown to exhibit exchange hardening suggestive of twomagnetic phases, despite existing as a \textit{single} chemical phase. We haveperformed a first-principles study of the chemical and magnetic degrees offreedom in the Mn$_{1-x}$Fe$_{x}$Ru$_2$Sn pseudo-binary to determine the originof the unique magnetic behavior responsible for exchange hardening within asingle phase. We find a transition from antiferromagnetic (AFM) toferromagnetic (FM) behavior upon replacement of Mn with Fe, consistent withexperimental results. The lowest energy orderings in Mn$_{1-x}$Fe$_{x}$Ru$_2$Snconsist of chemically- and magnetically-uniform (111) planes, with Fe-richregions preferring FM ordering and Mn-rich regions preferring AFM ordering,independent of the overall composition. Analysis of the electronic structuresuggests that the magnetic behavior of this alloy arises from a competitionbetween AFM-favoring Sn-mediated superexchange and FM-favoring RKKY exchangemediated by spin-polarized conduction electrons. Changes in valency uponreplacement of Mn with Fe shifts the balance from superexchange-dominatedinteractions to RKKY-dominated interactions.

First-principles investigation of competing magnetic interactions in (Mn,Fe)Ru$_2$Sn Heusler solid solutions