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

Many Heusler compounds possess magnetic properties well-suited for applications as spintronic materials. The pseudo-binary Mn$_{0.5}$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 two magnetic phases, despite existing as a \textit{single} chemical phase. We have performed a first-principles study of the chemical and magnetic degrees of freedom in the Mn$_{1-x}$Fe$_{x}$Ru$_2$Sn pseudo-binary to determine the origin of the unique magnetic behavior responsible for exchange hardening within a single phase. We find a transition from antiferromagnetic (AFM) to ferromagnetic (FM) behavior upon replacement of Mn with Fe, consistent with experimental results. The lowest energy orderings in Mn$_{1-x}$Fe$_{x}$Ru$_2$Sn consist of chemically- and magnetically-uniform (111) planes, with Fe-rich regions preferring FM ordering and Mn-rich regions preferring AFM ordering, independent of the overall composition. Analysis of the electronic structure suggests that the magnetic behavior of this alloy arises from a competition between AFM-favoring Sn-mediated superexchange and FM-favoring RKKY exchange mediated by spin-polarized conduction electrons. Changes in valency upon replacement of Mn with Fe shifts the balance from superexchange-dominated interactions to RKKY-dominated interactions.