Spin-orbit coupling, strong correlation, and insulator-metal transitions: The ${\mathrm{J}}_{\mathrm{eff}}=\frac{3}{2}$ ferromagnetic Dirac-Mott insulator ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$

The double perovskite ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ (BNOO), an exotic example of a very high oxidation state (heptavalent) osmium $d^1$ compound and also uncommon by being a ferromagnetic open $d$-shell (Mott) insulator without Jahn-Teller (JT) distortion, is modeled using a density functional theory based hybrid functional incorporating exact exchange for correlated electronic orbitals and including the large spin-orbit coupling (SOC). The experimentally observed narrow-gap ferromagnetic insulating ground state is obtained, but only when including spin-orbit coupling, making this a Dirac-Mott insulator. The calculated easy axis along [110] is in accord with experiment, providing additional support that this approach provides a realistic method for studying this system. The predicted spin density for [110] spin orientation is nearly cubic (unlike for other directions), providing an explanation for the absence of JT distortion. An orbital moment of $-0.4{\mu }_B$ strongly compensates the $+0.5{\mu }_B$ spin moment on Os, leaving a strongly compensated moment more in line with experiment. Remarkably, the net moment lies primarily on the oxygen ions. An insulator-metal transition, by rotating the magnetization direction with an external field under moderate pressure, is predicted as one consequence of strong SOC, and metallization under moderate pressure is predicted. Comparison is made with the isostructural, isovalent insulator ${\mathrm{Ba}}_2{\mathrm{LiOsO}}_6$, which, however, orders antiferromagnetically.

Figure 1

Orbital-projected density-of-states plot for BNOO using the oeeHyb functional (a) without SOC, and for three different spin-orbit coupling directions: (b) [001], (c) [110], and (d) [111]. The horizontal axis is in electronvolts relative to the Fermi level (a), or top of the gap (b,c,d). The vertical axis is in states/eV-atom/formula unit. The most relevant information is the relative occupations of $t_{2g}$ orbitals in the lower Hubbard band (–0.4–0.0 eV).

Figure 2

Band plots for ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ (a–c) and ${\mathrm{Ba}}_2{\mathrm{LiOsO}}_6$ (d–f) for the three directions of magnetization axis (indicated on the plots) and with a lattice parameter of 8.10 Å (experimental lattice parameter of ${\mathrm{Ba}}_2{\mathrm{LiOsO}}_6$). While the differences are quite small for the [001] magnetization direction, they become important along the $W\text{−}X\text{−}K$ lines for the other two spin directions. The chosen symmetry points are, in units of $\frac{2\pi }{a}$ and in order: $W=(1,\frac{1}{2},0),L=(\frac{1}{2},-\frac{1}{2},-\frac{1}{2}),\Gamma =(0,0,0),X=(1,0,0),W=(1,\frac{1}{2},0),K=(\frac{3}{4},\frac{3}{4},0)$.

Figure 3

Band plot along three different $W\text{−}K$ directions for ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ with [001] magnetization axis ($a=8.28$ Å), illustrating the large effect of symmetry lowering by the large spin-orbit interaction strength. The chosen points, in units of $\frac{2\pi }{a}$, are $\Gamma =(0,0,0),W_1=(-\frac{1}{2},0,1),W_2=(0,-1,\frac{1}{2}),W_3=(1,0,-\frac{1}{2}),K_1=(-\frac{3}{4},0,\frac{3}{4}),K_2=(0,-\frac{3}{4},\frac{3}{4}),K_3=(\frac{3}{4},0,-\frac{3}{4})$.

Figure 4

Spin-density contour plots [37] in three planes perpendicular to the cubic axes and cutting through the Os ion, for ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ using the oeeHyb functional. Panel (a) corresponds to the [001] spin direction, and panel (b) to the [111] spin direction. Os ions lie at the centers of the faces and at the corners, neighbored by four O ions in these planes. The blue contours in panel (a) reflect the negative spin polarization on the $2p_{\sigma }$ orbitals in the equatorial plane. In panel (b), isosurfaces encircling the Os ions are included to demonstrate that the spin density does not have cubic symmetry; the isocontours in the three planes would suggest incorrectly that cubic symmetry is present. See also Fig. 5.

Figure 5

Panel (a) Isocontour plot of the spin density in the limited range –0.4–0.0 eV, the occupied lower Hubbard band, for ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ at its experimental volume ($a=8.28$ Å) and [001] spin direction, obtained using the oeeHyb functional. This spin density is non-negative, in line with lack of any occupied minority spin (see Fig. 1). Lines connect Os to neighboring O and Na ions. Panel (b) Isocontour plot of the same density, showing more detail in planes perpendicular to the cubic axes and passing through Os ions. For example, the small spin density in the $x-y$ plane does not show up on the isocontour plot. The negative spin density on the equatorial O ions that is evident in Fig. 4 therefore arises due to the polarization of the O $2p$ bands (with no net moment).

Figure 6

Isosurface plot of the spin density of BNOO as in Fig. 5, but for [110] (top panel) and [111] (bottom panel) spin orientation. Note the near-cubic symmetry of the [110] density on both the Os ion and the neighboring O ions. The [111] density shows threefold symmetry around the chosen [111] axis for an $m_{\ell }=\pm 1$ orbital, as defined in Eq. (1).