Staggered Dzyaloshinskii-Moriya interaction inducing weak ferromagnetism in centrosymmetric altermagnets and weak ferrimagnetism in noncentrosymmetric altermagnets

The Dzyaloshinskii-Moriya interaction (DMI) has explained successfully the weak ferromagnetism in some centrosymmetric antiferromagnets. However, in recent years, it was generally claimed that the DMI is not effective in such systems. We reconciled these views by separating the conventional antiferromagnets from altermagnets. Altermagnets are collinear magnets having zero magnetization preserved by crystal symmetries in the nonrelativistic limit. The spin-up and spin-down sublattices are connected by rotation (proper or improper and symmorphic or nonsymmorphic). Consequently, the system shows even-parity wave spin order in the k-space lifting the Kramer's degeneracy in the nonrelativistic band structure leading to unconventional magnetism. The staggered DMI is one of the mechanisms which can create weak ferromagnetism or weak ferrimagnetism in centrosymmetric and noncentrosymmetric altermagnets while it is not effective in conventional antiferromagnets. Once the spin-orbit coupling is included in an altermagnetic system (where the time-reversal symmetry is broken) with staggered DMI, the components of spin moments of the two sublattices along the N\'eel vector are antiparallel but the other two spin components orthogonal to the N\'eel vector can be null, parallel or antiparallel. In cases where we have different bands showing parallel and antiparallel spin components at the same time, the magnetic order results in weak ferrimagnetism. The altermagnetic compounds can host weak ferromagnetism, weak ferrimagnetism or zero magnetization. Restricted to the altermagnet with two atoms and staggered DMI, the Hall vector is orthogonal to the N\'eel vector in the case of weak ferromagnetism and weak ferrimagnetism with a magnetic component proportional to the DMI. When we keep the crystal symmetry, the NiAs phase and the wurtzite phase have the same weak ferromagnetism. We find a sign change of the magnetization, and possibly of the anomalous Hall effect, as a function of the band filling and N\'eel vector. We describe the dependence of the weak ferromagnetism on the charge doping.