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

## TL;DR

This paper investigates how strong spin-orbit coupling influences orbital and spin ordering in $d^1$ and $d^2$ double perovskites, explaining experimental puzzles and predicting complex magnetic phases through a microscopic model.

## Contribution

It introduces a microscopic mean-field model capturing orbital and spin interactions in spin-orbit coupled double perovskites, revealing novel magnetic and orbital phases.

## Key findings

- Orbital ordering occurs at higher temperatures than magnetic order.
- Distinct magnetic phases such as coplanar canted ferromagnetic and 4-sublattice antiferromagnetic states.
- Negative Curie-Weiss temperatures can arise in ferromagnetic materials due to orbital effects.

## Abstract

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.