Liquidlike correlations in single-crystalline Y${}_2$Mo${}_2$O${}_7$: An unconventional spin glass

The spin-glass behavior of Y${}_2$Mo${}_2$O${}_7$ has remained a puzzle for nearly three decades. Free of bulk disorder within the resolution of powder diffraction methods, it is thought that this material is a rare realization of a spin glass resulting from weak disorder such as bond disorder or local lattice distortions. Here we report on the single-crystal growth of Y${}_2$Mo${}_2$O${}_7$. Using neutron scattering, we present isotropic magnetic diffuse scattering occurring below the spin-glass transition. Our attempts to model the diffuse scattering using a computationally exhaustive search of a class of simple spin Hamiltonians show no agreement with the experimentally observed energy-integrated (diffuse) neutron scattering. This suggests that spin degrees of freedom are insufficient to describe this system. Indeed, a $T^2$ temperature dependence in the heat capacity and density functional theory calculations hint at the presence of a significant frozen degeneracy in both the spin and orbital degrees of freedom resulting from spin-orbital coupling (Kugel-Khomskii type) and random fluctuations in the Mo environment at the local level.

Figure 1

The crystal structure of Y${}_2$Mo${}_2$O${}_7$ (space group $Fd\overline{3}m$) can be thought of in a number of different ways. (a) A single layer of eightfold-coordinated Y-O and sixfold-coordinated Mo-O polyhedra share edges such that one can build the structure through ABCABC close packing of the area encircled in green along ⟨111⟩. (b) Two interpenetrating corner-sharing tetrahedral networks with cationic vertices and vacancy centers. In this figure, only the Mo-tetrahedral network is shown [The Y network is displaced by $k$ = (0.5, 0.5, 0.5)]. (c) The tetrahedral networks form a pseudo-close-packing network of kagome planes along ⟨111⟩.

Figure 2

Backscattered electron image showing a small cross section of single-crystal Y${}_2$Mo${}_2$O${}_7$ produced by wavelength-dispersive electron microprobe analysis. The gray area is stoichiometric Y${}_2$Mo${}_2$O${}_7$, while the white areas are small Mo metal inclusions as a result of a sputtering process used to probe the interior. These metal inclusions are a surface impurity only detected on this particular subsection of the crystal; thus they are not detected in any of our magnetic susceptibility, heat capacity, synchrotron x-ray, or neutron diffraction measurements.

Figure 3

Composite image of x-ray diffraction exposures taken using CCD imaging. The black spots are from the dominant Y${}_2$Mo${}_2$O${}_7$ phase refined to ICSD: 202522. It was determined that our 0.03-g single-crystal subsection contained a small collection of grains about 0.1 mm in size. The lattice constant was determined to be 10.28 Å, which is slightly higher than reported values of 10.230(1) Å [26]. While a larger lattice constant can be due to O deficiencies within the crystal, our annealing steps in the preparation, electron microprobe, and magnetic susceptibility suggest that this is not the case. The slightly larger lattice constant is instead due to the composite nature of the crystal at x-ray penetration depths.

Figure 4

Individual CCD exposures at 30 s (left) and 3 s (right). The Mo metal arcs have been labeled in red.

Figure 5

For all relevant panels, error bars are smaller than the symbol size. (a) dc susceptibility and inverse susceptibility of Y${}_2$Mo${}_2$O${}_7$ fit to the Curie-Weiss law using a field of 1 T applied along [111]. (b) The field dependence of $T_f$ as observed using dc susceptibility.

Figure 6

(a) The elastic ring observed using neutron scattering. Data from 100 K was subtracted from data at 1.5 K and binned over energy $E$ = [–0.12, 0.12] meV. The data has been smoothed in this figure to better show the broad features of the ring. The white points are calculated for constant Q = 0.44 Å⁻¹. (b) The ring is replicated at the center of the next Brillouin zone. Data has been binned over $L$ = [–0.1,0.1] r.l.u. (c) Inelastic data integrated over HH0 = [0,1] r.l.u. and 00$L$ = [–1,0] r.l.u. (Error bars are smaller than the symbol size.) (d) Raw data depicting the evolution of the rings. Peaks at 0.6 and 1.4 r.l.u. are spurious features of the instrument. Note that the data is shown on a logarithmic scale due to the enormous intensity difference between the (222) Bragg peak and the diffuse scattering. Error bars in the relevant figure panels represent one standard deviation.

Figure 7

Cuts along various directions through the ring. Data taken at 100 K was subtracted from data at 1.5 K. All cuts were made integrated over the elastic peak with step size 0.02 r.l.u. The data along [2$H$,2$H$,–$H$] appears to break trend due to a lack of detector coverage along that cut. Data along [$H$,$H$,$H$] is only shown up until the appearance of the (222) Bragg peak. (Inset) Powder-averaged cut of the same data.

Figure 8

Single-crystal neutron diffraction ($E_i$ = 20 meV) integrated over $E$ = [–0.02, 0.02] meV and $H$-$H$0 = [–0.2, 0.2] r.l.u. taken on the CNCS at (a) 300 K and (b) 1.5 K. The Huang scattering and its temperature dependence are both observed using the C5 instrument (CNBC, Chalk River, inset) at 4 K, although the latter is not shown here. Unlike the pattern observed with x rays, temperature dependence is observed, which might indicate spin-orbital coupling. Aluminum powder lines have been artificially colored black for clarity.

Figure 9

Example of Huang scattering as observed with synchrotron x rays at 10 K. The intensity is on a logarithmic scale. In addition to the observed feature at (660), Huang scattering orders of magnitude more intense were also observed on the (440), (666), (008), and (880) peaks, the latter of which is visible on the right here. But in general, this Huang scattering is extremely weak and was not observed using a two-circle laboratory source equipped with 14.4-keV x rays. No obvious temperature dependence was observed; the scattering persisted at all temperatures between 10 and 300 K.

Figure 10

Neutron-scattering plots in the HHL plane for Y${}_2$Mo${}_2$O${}_7$ at $T$ = 150 K, with a range of next-nearest-neighbor terms $J_2,J_3$ (in Kelvin). Each plot covers $H$ $\in$ [–2, 2], $L$ $\in$ [–3, 3]; the arbitrary scale of each plot is normalized to the color bar on the right. The Curie-Weiss constraint $4\left(J_1+2J_2+2J_3\right)$ = −200 K is applied, meaning $J_1$ is ferromagnetic for $J_2+J_3<$ −25 K. This is marked in the upper diagram where $J_1$ is ferromagnetic above the line and antiferromagnetic below it.

Figure 11

(a) Large-$N$ Heisenberg model of the diffuse scattering $J_{nnn}$/$J_{nn}$ = −0.25 at $T$ = 0.6$J_{nn}$. (b) Mean-field theory Heisenberg model at $T$ = 1.2$T_c$ (same couplings), where $T_c$ is the mean-field model transition temperature. Note that this choice of couplings fails to fulfill ${\theta }_{CW}$ < 0 K. The red and black outlines in each figure represent the data coverage of the spectrometer used in Fig. 6.

Figure 12

Error bars for all measurements on single or crushed crystals of Y${}_2$Mo${}_2$O${}_7$ are smaller than the symbols. (a) Raw heat-capacity data of single-crystal Y${}_2$Mo${}_2$O${}_7$ (black square), crushed crystals of Y${}_2$Mo${}_2$O${}_7$ (blue triangle), crushed crystals annealed in O${}_{2\left(g\right)}$ for 24 hours at 300 °C (green diamond), and Y${}_2$Ti${}_2$O${}_7$ from Johnson et al. [60] (red circle). Differences between Y${}_2$Mo${}_2$O${}_7$ samples at higher temperatures are due to a mass errors (or surface effects for the annealed crushed crystals), while no clear difference is discernible at lower temperatures (inset). (b) The low-temperature heat capacity of single-crystal Y${}_2$Mo${}_2$O${}_7$ is fit to a strict $T^2$ law (red curve) and is compared to powder samples (Raju et al. [30]). (c) Field dependence of single-crystal Y${}_2$Mo${}_2$O${}_7$ lattice-subtracted heat-capacity data using our own sample of Y${}_2$Ti${}_2$O${}_7$ (found to be consistent with previous experiments [60]). The field was applied along the [111] direction. Integrating the peak should yield the entropy released by this system at low temperatures (inset). Only 14.7% of the theoretical maximum entropy (9.13 J/mol K) is released. (d) Heat-capacity data from Raju et al. [30] is quite noisy, probably due to instrumental effects in the subtraction. The low-temperature data can be fit to both a $T$ (red curve) and $T^2$ dependence (blue and green curves) within a reasonable margin of error dependent on the region of fit.

Figure 13

Y${}_2$Mo${}_2$O${}_7$ unit cell with four Mo atoms (space group $P\overline{1}$). Density isosurfaces of the valence electrons are shown in yellow; one clearly distinguishes the different occupations of Mo valence orbitals in the ferromagnetic spin-order-orbital order-1 (FM-OO1) and antiferromagnetic spin-order-orbital order-2 (AFM-OO2) solutions.

Figure 14

Legend for Table 1. Mo atoms are labeled by the blue shaded boxes.