Electronic structure and exchange interactions in altermagnetic MnGeP 2 in the quasiparticle self-consistent G W approach

The quasiparticle self-consistent $GW$ method is used to study the electronic band structure, optical dielectric function, and exchange interactions in chalcopyrite, $I\overline{4}2d$, structure ${\mathrm{MnGeP}}_{2}$. The material is found to be an antiferromagnetic semiconductor with a lowest direct gap of 2.44 eV at the $\mathrm{\ensuremath{\Gamma}}$ point and a lower indirect gap of 1.87 eV from $\mathrm{\ensuremath{\Gamma}}$ to $M$. The material is an altermagnet because the two magnetic atoms of opposite spin are related by a twofold rotation operation perpendicular to the main fourfold rotation inversion axis. The spin splittings along a low symmetry line like $PN$ is sizable while at k points on the diagonal mirror planes or on the twofold symmetry axes the spin splitting is zero. The exchange interactions are calculated using a linear response approach. The antiferromagnetic exchange interaction between nearest neighbors in the primitive unit cell is dominating and found to be slightly decreasing upon carrier doping but not sufficiently to change the interaction to become ferromagnetic. The bare (noninteracting) and interacting transverse spin susceptibilities, which provide interatomic site exchange interactions after averaging over the muffin-tin spheres, are calculated from the $GW$ band structure and wave functions. From these exchange interactions, the spin wave spectra are obtained along the high symmetry lines and the N\'eel temperature is calculated using the mean-field and Tyablikov estimations. The dielectric function and the optical absorption spectra are calculated including excitonic effects using the Bethe Salpeter equation. The exchange interactions around ${\mathrm{Mn}}_{\mathrm{Ge}}$ defect sites is also studied. While we find it can generate ferromagnetic interactions with neighboring spins, we did not find direct evidence of producing an overall ferromagnetic phase. First, if Mn antisites are introduced by exchanging Mn with a nearby Ge, the interactions stay largely antiferromagnetic. Second, when we add additional Mn, in other words in Mn-rich stoichiometry, the ${\mathrm{Mn}}_{\mathrm{Ge}}$ antisites produce a strong ferromagnetic interaction primarily with the Mn in the same basal plane but weaker ferromagnetic interaction with adjacent plane Mn. The interactions between regular lattice Mn stay antiferromagnetic as before and thus favor keeping the antiferromagnetic order along the [001] direction. Adding Mn antisites, however, does lead to a metallic band structure.