Nearly linear orbital molecules on a pyrochlore lattice

Abstract

The interplay of spin-orbit coupling with other relevant parameters gives rise to the rich phase competition in complex ruthenates featuring octahedrally coordinated Ru⁴⁺. While locally, spin-orbit coupling stabilizes a nonmagnetic J_eff = 0 state, intersite interactions resolve one of two distinct phases at low temperatures: an excitonic magnet stabilized by the magnetic exchange of upper-lying J_eff = 1 states or Ru₂ molecular orbital dimers driven by direct orbital overlap. Pyrochlore ruthenates A₂Ru₂O₇ (A = rare earth, Y) are candidate excitonic magnets with geometrical frustration. We synthesized In₂Ru₂O₇ with covalent In─O bonds. This pyrochlore ruthenate hosts a local J_eff = 0 state at high temperatures; however, at low temperatures, it forms a unique nonmagnetic ground state with nearly linear Ru─O─Ru molecules, in stark contrast to other A₂Ru₂O₇ compounds. The disproportionation of covalent In─O bonds drives Ru₂O molecule formation, quenching not only the local spin-orbit singlet but also geometrical frustration.

Fig. 1.. Structural changes of In₂Ru₂O₇ through multiple phase transitions.

(A) Top: Crystal structure of In₂Ru₂O₇ in the high-temperature cubic phase ( $\mathit{Fd}\overline{3}m$ ). The pink and yellow spheres represent two distinct oxygen sites, O1 and O2, respectively. Bottom: Ru pyrochlore network. (B) Powder x-ray diffraction patterns of In₂Ru₂O₇ in the cubic (493 K) and P4₁2₁2 tetragonal (263 K) phases. The peaks marked by asterisks are visible only in the P4₁2₁2 tetragonal phase. (C) Single-crystal XRD patterns at 500 K, 295 K, and 120 K of the $\mathit{Fd}\overline{3}m$ , P4₁2₁2, and $P\overline{4}{2}_{1}c$ fully identified phases, respectively. Example reflections not indexed by the $\mathit{Fd}\overline{3}m$ space group are circled in the middle. (D) Neutron powder diffraction patterns of In₂Ru₂O₇ showing the evolution of the 440 reflection when cooling from room temperature. Only the space groups and unit cell parameters have been identified for the two intermediate phases at 235 K and 225 K so far. The data shown here were used for the refinement shown in fig. S2. (E) The relationship between unit cells for the cubic ( $\mathit{Fd}\overline{3}m$ , a_c), orthorhombic (C222₁, a_o, b_o, and c_o), and three tetragonal structures [P4₁2₁2, a(c)_t and $P\overline{4}m2$ , a(c)_t′ and $P\overline{4}{2}_{1}c$ , and a(c)_t″]. $a_o,b_o\sim \sqrt{2}{a}_t$ , $a_{t\prime }\sim a_t/\sqrt{2}$ , and c_t″ ∼ 2c_t. The tetragonal phase around room temperature (P4₁2₁2) has a unit cell metric nearly identical to that of the cubic phase. a.u., arbitrary units.

Fig. 2.. Multiple phase transitions in In₂Ru₂O₇.

Temperature-dependent (A) resistivity ρ(T) and the derivative of the resistivity dρ/dT, (B) magnetic susceptibility χ(T), (C) heat flow determined with DSC, and (D) the lattice parameters of polycrystalline In₂Ru₂O₇, respectively. The data in the main panel were collected in the cooling process. The insets of (B) show the thermal hysteresis of two transitions, where the black arrows denote the zero-field-cooling (→) and field-cooling (←) curves. In (C), the entropy release identified in the three separate first-order phase transitions is indicated, and the inset shows both the heating (→) and cooling (←) DSC curves. In (D), the a, b, and c lattice parameters of the tetragonal and orthorhombic unit cells are normalized with respect to those of the $\mathit{Fd}\overline{3}m$ cubic structure.

Fig. 3.. Formation of Ru₂O orbital molecules on the pyrochlore lattice.

(A) Temperature-dependent Ru─O bond lengths (left) and Ru─O─Ru angles (right) obtained from the three identified crystal structures of In₂Ru₂O₇. The bond lengths and bond angles relevant to the Ru₂O molecule are shown as squares and are enclosed in the gray region. Black, turquoise, and pink markers denote $\mathit{Fd}\overline{3}m$ , P4₁2₁2, and $P\overline{4}{2}_{1}c$ phases, respectively. (B) Crystal structure in the vicinity of the Ru3─O4─Ru3 bond in the low-temperature tetragonal phase of In₂Ru₂O₇. Four (Ru3)O₆ octahedra, where the four Ru3 atoms (gray spheres) form a tetrahedron within the pyrochlore lattice, are shown. (C) Ru₂O molecules on the pyrochlore lattice. All Ru atoms are involved in the formation of Ru₂O molecules. The gray lines denote the underlying Ru pyrochlore network. Only the atoms involved in the Ru₂O units are shown, where the gray and pink spheres are Ru and O atoms, respectively. The Ru₂O molecules that reach beyond the selected fragment of the pyrochlore network are truncated after the central O atom. The Ru and O atoms involve four and three unique crystallographic sites, respectively. For details, see fig. S8.

Fig. 4.. In─O covalent bond disproportionation.

(A) Temperature-dependent In─O bond lengths obtained from the three identified crystal structures of In₂Ru₂O₇. Yellow and pink markers denote In─O2 or In─O2-derived and In─O1 or In─O1-derived bonds, respectively. (B) The change of In─O bonding geometry with decreasing temperature. The yellow and pink spheres denote the O2 or O2-derived and O1 or O1-derived oxygen atoms, respectively. In the $P\overline{4}{2}_{1}c$ phase at 10 K, the green oxygen atom (O4) is part of the short Ru3─O4─Ru3 bond shown in Fig. 3B. For more details, see tables S3 and S4.

Fig. 5.. Electronic structure of the Ru₂O molecule.

(A) ZF-μSR time spectrum of polycrystalline In₂Ru₂O₇ at various temperatures. The solid lines show a stretched exponential fit of the data in the form P_z(t) = Ae^−(λt)β, where P_z(t) is the time-dependent asymmetry, A is the initial asymmetry, λ is the relaxation rate, and β is the exponent (see fig. S11 for the fitted values). (B) The total density of states per formula unit (f. u.) of the high-temperature cubic ( $\mathit{Fd}\overline{3}m$ ), room temperature tetragonal (P4₁2₁2), and low-temperature tetragonal ( $P\overline{4}{2}_{1}c$ ) phases of In₂Ru₂O₇. (C) Density of states projected onto Ru3 4d states (left) and O 2p states of O4 atom coordinating Ru3 atom (right), obtained by a scalar-relativistic calculation. The x, y, and z axes point along the Ru─O bonds with the z axis parallel to the Ru─O4 bond, as illustrated in the right inset. The gray shaded sections correspond to the bands related to MOs noted on the right and shown in (D). (D) MO scheme of Ru₂O unit with 180° Ru─O─Ru geometry. π-type Ru t_2g–O p and σ-type Ru e_g–O p hybridization is considered. σ^∗ and π^∗ denote antibonding MO, while nb corresponds to nonbonding MO. The individual Ru d levels are split due to local tetragonal distortion before Ru─O hybridization. As each Ru⁴⁺ ion provides 4 d-electrons and an O²⁻ ion contributes 6 p-electrons, there are 14 electrons in total per Ru₂O molecule.

Fig. 6.. Local electronic structure of In₂Ru₂O₇.

(A) Ru L₃-edge RIXS spectra of polycrystalline In₂Ru₂O₇ with incident energy E_i = 2839.2 eV at 260 K and 30 K. (B) Fitting of RIXS spectrum at T = 260 K. The dashed line shows the fitted elastic peak, while the colored peaks correspond to the excitations which are schematically shown in (C). The gray line corresponds to an arctangent function representing the electronic continuum. All peaks were fitted with a pseudo-Voigt profile. (C) The energy level diagram of the $t_{2g}^4$ electronic configuration in an isolated RuO₆ octahedron (J_H >> λ_SO and 10Dq = ∞). The $t_{2g}^4$ are split by J_H, then by λ_SO, and lastly by Δ_tri. The dashed lines indicates the mixing of states. The ϕ₁, ϕ₂, ϕ₃, ϕ₄, and ϕ₅ colored states correspond to the labeled excitations in (B).