Orbital and spin order in spin-orbit coupled $d^1$ and $d^2$ double perovskites

We consider strongly spin-orbit coupled double perovskites A$_2$BB'O$_6$ with B' magnetic ions in either $d^1$ or $d^2$ electronic configuration and non-magnetic B ions. We provide insights into several experimental puzzles, such as the predominance of ferromagnetism in $d^1$ versus antiferromagnetism in $d^2$ systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials, and the size of effective magnetic moments. We develop and solve a microscopic model with both spin and orbital degrees of freedom within the Mott insulating regime at finite temperature using mean field theory. The interplay between anisotropic orbital degrees of freedom and spin-orbit coupling results in complex ground states in both $d^1$ and $d^2$ systems. We show that the ordering of orbital degrees of freedom in $d^1$ systems results in coplanar canted ferromagnetic and 4-sublattice antiferromagnetic structures. In $d^2$ systems we find additional colinear antiferromagnetic and ferromagnetic phases not appearing in $d^1$ systems. At finite temperatures, we find that orbital ordering driven by both superexchange and Coulomb interactions may occur at much higher temperatures compared to magnetic order and leads to distinct deviations from Curie-Weiss law.