Spin-Orbit Dimers and Noncollinear Phases in $d^1$ Cubic Double Perovskites

We formulate and study a spin-orbital model for a family of cubic double perovskites with $d^1$ ions occupying a frustrated fcc sublattice. A variational approach and a complementary analytical analysis reveal a rich variety of phases emerging from the interplay of Hund’s rule and spin-orbit coupling. The phase digram includes noncollinear ordered states, with or without a net moment, and, remarkably, a large window of a nonmagnetic disordered spin-orbit dimer phase. The present theory uncovers the physical origin of the unusual amorphous valence bond state experimentally suggested for ${\mathrm{Ba}}_{2}B\mathrm{Mo}{\mathrm{O}}_6$ ($B=\mathrm{Y}$, Lu) and predicts possible ordered patterns in ${\mathrm{Ba}}_{2}B\mathrm{Os}{\mathrm{O}}_6$ ($B=\mathrm{Na}$, Li) compounds.

Figure 1

(a) Density profile of $j=3/2$ quartet. The states $j^z=\pm \frac{1}{2}$ (bottom) have dominant $xy$-orbital character, while $j^z=\pm \frac{3}{2}$ components (top) are composed of $xz$ and $yz$ orbitals. Red and blue coloring denotes the up and down spin distribution, respectively. (b) Crystallographic unit cell containing four molybdenum ions (large circles). Oxygen positions are indicated by small circles. The nearest-neighbor bonds belonging to different cubic planes are distinguished by different colors. The $t_{2g}$ orbitals active along the corresponding bonds are also indicated.

Figure 2

Types of different interdimer bonds depicted as dashed lines. Dimers are represented with and are colored according to three cubic planes they belong to. Bonds ${\mathsf{b}}_1$ and ${\mathsf{b}}_2$ couple dimers perpendicular to these bonds. Bond ${\mathsf{b}}_3$ connects a dimer from an orthogonal plane with another one that is in the same plane as ${\mathsf{b}}_3$ itself.

Figure 3

Excitations of a spin-orbit dimer in the large $\lambda$ limit. Flipping locally, for example, the $j^z=-1/2$ component to $j^z=1/2$ or to $j^z=-3/2$ corresponds to flipping the pseudospin (top) or pseudo-orbital (bottom), respectively. Local excitations for small SOC correspond to changing the real spins or orbitals.

Figure 4

Phase diagram of the model Eq. (1) as the function of Hund’s coupling $\eta$ and the SOC $\lambda$ (in units of $J$). For small values of $\eta$ the dimer-singlet phase (see inset) is stable over the entire range of $\lambda$. With increasing Hund’s coupling, noncollinear coplanar phases with ordered moments in one of the cubic planes are stabilized. The ferro-type coplanar state, coplanar FM, has a finite net moment pointing along the [110] (or equivalent) direction. The illustrations show a tetrahedron of four molybdenum sites projected onto the plane of ordered $\overrightarrow{j}$ moments, depicted as arrows.