Origin of $A$-type antiferromagnetism and chiral split magnons in altermagnetic $\alpha$-MnTe

The origin of the $A$-type antiferromagnetic ordering, characterized by ferromagnetic layers coupling antiferromagnetically, in the prototype semiconductor altermagnet $\alpha$-MnTe has been a topic of ongoing debate. Experimentally, $\alpha$-MnTe exhibits an in-plane ferromagnetic exchange interaction, whereas previous \emph{ab initio} calculations predicted an antiferromagnetic interaction. In this paper, we resolve this discrepancy by considering an expanded set of magnetic configurations, which reveals a ferromagnetic in-plane exchange interaction in agreement with experimental findings. Additionally, we demonstrate that the 10th nearest-neighbor exchange interaction is directionally dependent, inducing a nonrelativistic chiral splitting in the magnon bands, as recently observed experimentally. We further show that applying a compressive strain may significantly enhance both nonrelativistic spin and chiral magnon splittings. The strain can also change the sign of the in-plane exchange interaction. Computing magnetic susceptibility, we show that strain enhances the N{\'e}el temperature, significantly. Our results highlight the critical importance of convergence in the number of magnetic configurations for spin interactions in antiferromagnetic materials.