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.

Figures (5)

• Figure 1: (a) Three-dimensional view of the double perovskite structure ($A_2BB'$O$_6$) showing the canted ferromagnetic configuration. $A$ ions are shown in gray and transition metal ions ($B'$) in light blue, with arrows indicating the local magnetic moments. The oxygen octahedra surrounding the $B$ and $B'$ ions are indicated in red and transparent gray. (b) The local magnetic moments are rotated away from the [110] direction (indicated by the black dashed lines) by an angle $\theta$, with $\theta>0$ indicating counterclockwise rotation around $z$. The thin gray line indicates the conventional cubic unit cell. Note that the Re atoms indicated with blue and red dipole moments are located in different adjacent [001] planes [see also subfigure (a)].
• Figure 2: Energy as function of the local quadrupole moment, $Q_t$, for FQ-$xy$ order (blue) and AFQ-$x^2-y^2$ order (purple) for structurally cubic Ba$_2$MgReO$_6$ (a) and Ba$_2$NaOsO$_6$ (b) in the paramagnetic case. Error bars indicate the standard deviation of the energies and local quadrupoles over the different paramagnetic configurations with the same value for the constraint $s_{t}$.
• Figure 3: Energy as function of the local quadrupole moments for Ba$_2$MgReO$_6$ (circles) and Ba$_2$NaOsO$_6$ (squares) for the ferromagnetic case with net magnetization along [110] and (a) FQ-$xy$ order or (b) AFQ-$x^2-y^2$ order.
• Figure 4: Evolution of canting angles of the local orbital (green) and spin (blue) moments as a function of the local $\mathcal{Q}_{x^2-y^2}$ component for an imposed AFQ-$x^2-y^2$ order, together with the angle $-\theta_{XY}$ (orange) indicating the rotation of the total local quadrupole moment.
• Figure 5: Evolution of the local quadrupole components $Q_{xy}$ and $Q_{x^2-y^2}$ (defined in the global coordinate system) on the two Re sites in the unit cell as a function of the canting angle $\theta_S$ between the spin moments (defined on Re 1) and the [110] direction in Ba$_2$MgReO$_6$. $\mathcal{Q}_{XY}(-\theta_S)$ represents the $xy$-type quadrupole (on Re 1) defined in a local coordinate system that is rotated in the opposite direction as the local spin moment, calculated according to eq:quad_rot.