Untangling the structural, magnetic dipole, and charge multipolar orders in ${\mathrm{Ba}}_2{\mathrm{MgReO}}_6$

We present a density functional theory study of the low-temperature structural, magnetic, and proposed charge-quadrupolar ordering in the double perovskite, ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$. ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$ is a spin-orbit-driven Mott insulator with a symmetry-lowering structural phase transition at 33 K and a canted antiferromagnetic ordering of $5d^1$ Re magnetic moments at 18 K. Our calculations confirm the existence of the proposed charge quadrupolar order and reveal an additional, previously hidden, ordered charge quadrupolar component. By separately isolating the structural distortions and the orientations of the magnetic dipoles, we determine the relationship between the charge quadrupolar, structural, and magnetic orders, finding that either a local structural distortion or a specific magnetic dipole orientation is required to lower the symmetry and enable the existence of charge quadrupoles. Our paper establishes the crystal structure–magnetic dipole–charge multipole relationship in ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$ and related $5d^1$ double perovskites, and illustrates a method for separating and analyzing the contributions and interactions of structural, magnetic, and charge orders beyond the usual dipole level.

Figure 1

Calculated band gap values of ${\mathrm{Ba}}_2\mathrm{Re}\mathrm{Mg}{\mathrm{O}}_6$ as a function of SOC scaling (that is the amount by which we scale the true ${\xi }_i$ values) and $U_{\mathrm{eff}}$.

Figure 2

Calculated lowest-energy structure of ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$. The space group is $P{4}_2/mnm$. The oxygen atoms at the corners of the $\mathrm{Re}{\mathrm{O}}_6$ octahedra are omitted for clarity. The spins are shown with yellow and red arrows for Re1 and Re2 atoms, respectively. The canting angle is indicated by $\varphi$.

Figure 3

(a) Canting angle as a function of $U_{\mathrm{eff}}$. (b) Canting angle as a function of ${\delta }_{\text{Re2-O}}^{\left[110\right]}$, defined as the ratio of Re2-O contraction to Re2-O expansion along [110]. ${\delta }_{\text{Re2-O}}^{\left[110\right]}=1$ means all Re-O bonds are equal. Distortion ${\delta }_{\text{Re2-O}}^{\left[110\right]}>1$ corresponds to the cases where the in-plane Re1-O and Re2-O bonds distort in the same direction as the optimized structure. ${\delta }_{\text{Re2-O}}^{\left[110\right]}<1$ occurs when the in-plane Re1-O and Re2-O bonds distort in the opposite direction from the optimized structure. In both cases the out-of-plane Re-O bonds are optimized.

Figure 4

Expectation values of the $Q_{xy}, Q_{x^2-y^2}$ and $Q_{z^2}$ charge quadrupole components in units of electron charges for Re1 (a, c) and Re2 (b, d) sites as a function of canting angle (a, b), and the scaling of SOC (c, d). The vertical dashed and solid lines in (a) and (b) represent our computationally determined and the previously experimentally estimated [18] values of the canting angle for ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$, respectively.

Figure 5

Evolution of the $Q_{xy}$ on the Re ions as the $Fm\overline{3}m$ parent structure is distorted to the low temperature $P{4}_2/mnm$ structure for (a) Re1 and (b) Re2 atoms. Values for four different magnetic orderings in canted antiferromagnetic configuration (canting angle = ${40}^{\circ }$) and ferromagnetic configurations along [001], [100], and [010] directions are shown. The unit of $Q_{xy}$ is in electron charges.

Figure 6

(a) Schematic of the paramagnetic supercell of ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$, constructed with the structure of the low-symmetry $P{4}_2/mnm$ phase. Magnetic dipole moments are shown by red arrows. Only $\mathrm{Re}{\mathrm{O}}_6$ octahedra are shown. (b) Schematic of the in-plane paramagnetic supercell of ${\mathrm{Ba}}_2\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_6$, again constructed with the structure of the low-symmetry $P{4}_2/mnm$ phase. Spins are shown by red arrows. Only $\mathrm{Re}{\mathrm{O}}_6$ octahedra are shown. (c) DOS (arbitrary units) of the canted antiferromagnetic unit cell (c-AFM), the paramagnetic supercell (PM), and the in-plane paramagnetic supercell (in-plane PM), all with $U_{\mathrm{eff}}=1.8$ eV. The Fermi level is indicated by the vertical dashed line at zero. Note the existence of a band gap in all three cases.

Figure 7

(a) $Q_{xy}, Q_{x^2-y^2}$, and $Q_{z^2}$ components of the charge quadrupoles in the unit of electron charges for (a) fully paramagnetic and (b) in-plane paramagnetic cases. The unfilled data points refer to the Re1 site and the filled points refer to the Re2 site.