Beyond the orbitally-resolved magnetic exchange in CrI$_{3}$ and NiI$_{2}$

Abstract

The pertinent need for microscopic understanding of magnetic exchange motivated us to go beyond the existing theories and develop a systematic method to quantify all possible mechanisms that contribute to magnetic exchange for an arbitrary pair of atoms in a given material. We apply it to the archetypal 2D magnetic monolayers CrI3 and NiI2, to reveal the previously underrated dx2-y2,dx2-y2 contribution as either the leading or the second largest contribution to the total magnetic exchange. We proceed to explore the microscopic mechanisms behind all the non-zero orbital contributions in both CrI3 and NiI2, and generalize the findings to other magnetic monolayers dominated by d8 and d3 electronic configurations of the magnetic atoms.