Spin-orbit coupling effects in altermagnets: Interplay of weak spin and orbital ferromagnetism with relativistic splitting of electron states

The aim of the paper is to contribute to reaching a deeper understanding of the formation of relativistic effects in altermagnets. The focus of the paper is on the phenomena of weak ferromagnetism (WFM) and relativistic DFT calculations combined with the symmetry analysis on the basis of spin space groups. The consideration is performed on two different levels. On the first level, the atomistic magnetic structure of weak ferromagnetic state is calculated. Both spin and orbital atomic moments are taken into account. We study the dependence of the WFM moment on the strength of the SOC and obtain a peculiar nonmonotonous type of dependence. An interesting result is obtained in quasisymmetry (QS) calculation where only the component of the SOC collinear to the N\'eel vector is taken into account. In QS calculation the spin WFM is absent while the orbital WFM is present. This reveals a principal difference in the formation of the spin and orbital magnetic moments. On the second level, the study is focused on the properties of individual electron states. We introduce the notion of the magnetic structure of the electron state (MSES). It is shown how the collinear spin-MSESs of both metal and ligand atoms and compensated orbital-MSES of the ligand obtained in the nonrelativiatic calculation transform into complex noncollinear 3D MSESs of both spin and orbital nature. An important role in the formation of MSESs is played by the relativistic splitting of the accidental spin degeneracies at general {\bf k} points filling the volume of the Brillouin zone. The formation of the regions of avoided crossings in the relativistic band structure is related to the nonmonotonous behavior of the WFM moment. The importance of the metal-ligand hybridization in the formation of the AM properties is discussed. Most of the calculations are performed for MnTe.