Frustrated spin-$\frac{1}{2}$ molecular magnetism in the mixed-valence antiferromagnets ${\mathrm{Ba}}_{3}M{\mathrm{Ru}}_2{\mathrm{O}}_9(M=\mathrm{In}$, Y, Lu)

We have performed magnetic susceptibility, heat capacity, muon spin relaxation, and neutron-scattering measurements on three members of the family ${\mathrm{Ba}}_{3}M{\mathrm{Ru}}_2{\mathrm{O}}_9$, where $M=\mathrm{In}$, Y, and Lu. These systems consist of mixed-valence Ru dimers on a triangular lattice with antiferromagnetic interdimer exchange. Although previous work has argued that charge order within the dimers or intradimer double exchange plays an important role in determining the magnetic properties, our results suggest that the dimers are better described as molecular units due to significant orbital hybridization, resulting in one spin-1/2 moment distributed equally over the two Ru sites. These molecular building blocks form a frustrated, quasi-two-dimensional triangular lattice. Our zero- and longitudinal-field $\mu \mathrm{SR}$ results indicate that the molecular moments develop a collective, static magnetic ground state, with oscillations of the zero-field muon spin polarization indicative of long-range magnetic order in the Lu sample. The static magnetism is much more disordered in the Y and In samples, but they do not appear to be conventional spin glasses.

Figure 1

A portion of the crystal structure of ${\mathrm{Ba}}_{3}M{\mathrm{Ru}}_2{\mathrm{O}}_9$, specifically using parameters for the $M=\mathrm{Y}$ sample, showing one plane of Ru-Ru dimers. (a) A view perpendicular to the $c$ axis showing the stacking of Ru ions to form dimers. (b) A view parallel to the $c$ axis, showing the triangular arrangement of Ru dimers.

Figure 2

Neutron powder diffraction measurements with a wavelength of 1.54 Å for (a) ${\mathrm{Ba}}_3{\mathrm{InRu}}_2{\mathrm{O}}_9$ (3.5 K), (b) ${\mathrm{Ba}}_3{\mathrm{YRu}}_2{\mathrm{O}}_9$ (3.5 K), and (c) ${\mathrm{Ba}}_3{\mathrm{LuRu}}_2{\mathrm{O}}_9$ (1.5 K). The corresponding structural refinements (black lines) are superimposed on the data points. The extra peaks in the In and Y patterns arise from the Al sample can.

Figure 3

(a) Energy-level and spin occupation diagram for a hybridized ${\mathrm{Ru}}^{4.5+}-{\mathrm{Ru}}^{4.5+}$ dimer with large bonding energy, which is likely to apply to the In, Y, and Lu samples [15]. (b) Energy-level diagram with lower bonding energy as expected to apply to the La sample. (c) Magnetic susceptibility of the In, Y, and Lu samples, with the fits using Eq. (1) superimposed on the data.

Figure 4

$f{\left(Q\right)}^2\mathrm{\Delta }{\chi }^{\prime \prime }=f{\left(Q\right)}^2{\chi }^{\prime \prime }$(5 K)-$f{\left(Q\right)}^2{\chi }^{\prime \prime }$(225 K) (arbitrary units) obtained with inelastic neutron scattering for (a) ${\mathrm{Ba}}_3{\mathrm{LuRu}}_2{\mathrm{O}}_9$, (b) ${\mathrm{Ba}}_3{\mathrm{InRu}}_2{\mathrm{O}}_9$, and (c) ${\mathrm{Ba}}_3{\mathrm{YRu}}_2{\mathrm{O}}_9$ as a function of wave vector and energy transfer. (d) Cuts of $f{\left(Q\right)}^2\mathrm{\Delta }{\chi }^{\prime \prime }$ integrated between $Q=2$ and 2.5 ${\AA }^{-1}$.

Figure 5

$f{\left(Q\right)}^2\mathrm{\Delta }{\chi }^{\prime \prime }=f{\left(Q\right)}^2{\chi }^{\prime \prime }$(${T)-f\left(Q\right)}^2{\chi }^{\prime \prime }$(225 K) (arbitrary units) obtained with inelastic neutron scattering as a function of wave vector and energy transfer, using a lower incident energy of $E_i=50$ meV for (a) ${\mathrm{Ba}}_3{\mathrm{LuRu}}_2{\mathrm{O}}_9$ at $T=1.5\mathrm{K}$, (b) ${\mathrm{Ba}}_3{\mathrm{InRu}}_2{\mathrm{O}}_9$ at $T=1.5\mathrm{K}$, and ${\mathrm{Ba}}_3{\mathrm{YRu}}_2{\mathrm{O}}_9$ at (c) $T=1.5\mathrm{K}$ and (d) $T=20\mathrm{K}$.

Figure 6

(a) Specific heat ($C$) of the samples measured here with dashed lines identifying the low-$T$ anomalies as $T_m$. (b) The paramagnetic fraction of the samples as a function of temperature, obtained by applying a transverse field and assessing the amplitude of the ${\mu }^{+}$ precession generated. (c) Fast-relaxation rates ${\lambda }_F$ for all three samples and the highest oscillation frequency in the Lu sample, ${\omega }_1=2\pi f_1$, as functions of temperature. (d) Slow-relaxation rate, or $1/T_1$, vs temperature for all three samples.

Figure 7

Elastic neutron-diffraction data from HB-1A for all three samples. The open circles are data taken at 1.5 K. Red points were taken above $T_m$, at 10 K (for Y and In) and 20 K (for Lu). The Lu data have been shifted downwards by 200 counts/minute for ease of view. No evidence of magnetic order is observed, possibly because the ordered moment is extremely small or spread out over an entire dimer.

Figure 8

Zero-field $\mu \mathrm{SR}$ asymmetry at various temperatures for (a) In, (b) Y, and (c) Lu, with the corresponding fits superimposed on the 2 K data only. Fourier transforms of the 2 K data, with the fits superimposed, are shown in (d) In, (e) Y, and (f) Lu. The Lu sample data show long-lived oscillations, whereas the data of the In and Y samples show only fast exponential relaxation (along with a slowly relaxing 1/3 tail). The narrow zero-frequency peak in the Fourier transforms comes from the $T_1$ time of the 1/3 tail, and is a measure of spin fluctuations rather than static internal fields.

Figure 9

Longitudinal-field $\mu \mathrm{SR}$ scans for the Y and In samples.

Figure 10

AC susceptibility (${\chi }^{\prime }$) of all three samples at two different frequencies, 333 and 9999 Hz. The high-frequency data has been normalized so that the peak susceptibilities are equal, since the frequency response of the PPMS system is not perfectly flat. Otherwise, the temperature dependence of the susceptibilities is very similar. In particular, the peak positions are independent of frequency.