Quadrupole formation and coupling to magnetic and structural degrees of freedom in the $5d^1$ double perovskites Ba$_2$MgReO$_6$ and Ba$_2$NaOsO$_6$

Abstract

We investigate the interplay between charge, magnetic, and structural degrees of freedom in the isostructural and isoelectronic $5d^1$ double-perovskites Ba$_2$MgReO$_6$ and Ba$_2$NaOsO$_6$. Using first-principles-based electronic structure calculations, we show that both materials exhibit a tendency toward spontaneous quadrupolar order in the cubic paramagnetic phase, which is slightly weaker in Ba$_2$NaOsO$_6$ than in Ba$_2$MgReO$_6$. Our analysis further reveals an intimate coupling between the local magnetic moments and charge quadrupoles, mediated by the strong spin-orbit interaction, that leads to the unusual canted configuration of magnetic moments observed in these systems. When structural degrees of freedom are included, the two materials exhibit pronounced differences. In Ba$_2$MgReO$_6$ the strong coupling to Jahn-Teller distortions stabilizes the antiferroic $\mathcal{Q}_{x^2-y^2}$ order, yielding excellent agreement with available experimental data. In contrast, the Jahn-Teller coupling is significantly weaker in Ba$_2$NaOsO$_6$ and appears insufficient to stabilize the antiferroic quadrupolar order. While this is consistent with the absence of any measurable long-range structural distortion above the magnetic transition temperature, it contrasts with experimental results indicating a strong canting of the magnetic moments. Our analysis thus successfully describes the mechanisms shaping the properties of the Re-compound while a full quantitative description of the magnetic ground state of Ba$_2$NaOsO$_6$ is still elusive.