Cu-Sb dumbbell arrangement in the spin-orbital liquid candidate ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$

The absence of both spin freezing and of a static Jahn-Teller effect have led to the proposition that ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$ is a quantum spin-orbital liquid. However, theoretical understanding of the microscopic origin of this behavior has been hampered by a lack of consensus on the lattice structure. Cu ions have been proposed to realize either a triangular lattice, a short-range ordered honeycomb lattice, or a disordered lattice with stripelike correlations. Here we analyze the stability of idealized versions of these arrangements using density functional theory. We find stripe order of Cu ions to be energetically favored, hinting towards stripelike local Cu-Cu arrangements, while long-range order is presumably hindered due to disorder effects. Furthermore, we find evidence of significant interlayer interactions between Cu-Sb dumbbells, which affects the out-of-plane arrangement. Analysis of the relaxed crystal structures, electronic properties, and tight-binding parameters provides clues as to the nature of the Jahn-Teller distortions.

Figure 1

The crystal structure of ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$, as revealed by x-ray- and neutron-diffraction experiments [15]. Cu-Sb dumbbells (red/blue) form a triangular lattice, but each dumbbell is in one of two possible orientations, with either the Cu or Sb ion on top. Experimentally, in the absence of long-range order, it is difficult to determine the local arrangement of the dumbbells.

Figure 2

The two honeycomb arrangements of Cu-Sb dumbbells taken as starting points for the density functional theory calculations. Cu ions are shown in red and Sb in blue. (a) Side and top views of the honey-$a$ structure, which has a net dipole moment. (b) Side and top views of the honey-$b$ structure, where the upper plane has been flipped with respect to the honey-$a$ structure. In the honey-$b$ structure the net dipole moment cancels between neighboring planes.

Figure 3

The two stripe arrangements of Cu-Sb dumbbells taken as starting points for the density functional theory calculations. Cu ions are shown in red and Sb in blue. (a) Side and top views of the stripy-$a$ structure. This configuration is chosen so as to minimize the energy of the electrostatic interaction between neighboring planes. (b) Side and top views of the stripy-$b$ structure, where the upper plane has been flipped with respect to the stripy-$a$ structure.

Figure 4

Total energy per formula unit of different dumbbell configurations, measured relative to that of the honey-$a$ structure [see Fig. 2]. Three different functionals are used: non-spin-polarized GGA, spin-polarized GGA, and $\text{GGA}+U$ in a ferromagnetic setup, and these give broadly consistent results. Additionally, we show the results for the full relaxation (fr), which is even enhancing the observed energy splittings. The lowest energy configuration is found to be the stripy-$b$ structure [see Fig. 3].

Figure 5

Local arrangement of the oxygen (grey) octahedra surrounding a Cu ion (red). The Jahn-Teller distortion as obtained in the spin-polarized GGA relaxation of the (a) stripy-$a$ and (b) stripy-$b$ structure (see Fig. 3). The Jahn-Teller distortion is more pronounced in the stripy-$b$ structure with one long bond of $2.57\text{Å}$, and five shorter bonds of between $2.01\text{and}2.15\text{Å}$. (c) The local coordinate system used to derive the orbital contribution to the bands.

Figure 6

Band structure and density of states of ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$, calculated using spin-polarized GGA. (a) Band structure associated with stripy-$a$ dumbbell ordering [see Fig. 3]. (b) Band structure associated with stripy-$b$ dumbbell ordering [see Fig. 3]. The color scheme assigns a predominant character to the bands associated with the Cu $d$ electrons. The density of states is shown on the righthand side.