Unraveling atomistic and electronic origins of multiaxial magnetic anisotropy

Multiaxial magnetic anisotropy (MA) refers to the phenomenon that multiple axes of the magnetic crystal correspond to different energy minima. Compared with the common uniaxial magnetic anisotropy, multiaxial MA facilitates novel forms of applications in spintronics. Here, by combining the first-principles-based spin Hamiltonian and tight-binding (TB) method, we reveal the microscopic origins, instead of the common phenomenological understanding, of biaxial MA and triaxial MA. In the example system of NiO, it is found that the multiple minima result from the fourth-order and the sixth-order single ion interactions, while the difference between [110] and \([1\bar{1}0]\) $$[1\overline{1}0]$$ directions originates from a second-order bond-dependent anisotropic pair interaction (i.e., the so-called Gamma interaction). Moreover, through the application of a newly developed general spin dependent TB approach, it is revealed that the triaxial MA arises from the special spin-orbital entangled Hund term, which is different from the orbital-independent Hund term in the usual Slater Koster TB method. Our work thus not only leads to a thorough understanding of the multiaxial MA in NiO, but also establishes a methodology that can be widely used to explore the microscopic origins of MA in different magnets.