Quenched crystal-field disorder and magnetic liquid ground states in ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$

Solid solutions of the “soft” quantum spin ice pyrochlore magnets ${\mathrm{Tb}}_2{\mathrm{B}}_2{\mathrm{O}}_7$ with B = Ti and Sn display a novel magnetic ground state in the presence of strong B-site disorder, characterized by a low susceptibility and strong spin fluctuations to temperatures below 0.1 K. These materials have been studied using ac susceptibility and $\mu \mathrm{SR}$ techniques to very low temperatures, and time-of-flight inelastic neutron scattering techniques to 1.5 K. Remarkably, neutron spectroscopy of the ${\mathrm{Tb}}^{3+}$ crystal-field levels appropriate to high B-site mixing ($0.5<x<1.5$ in ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$) reveal that the doublet ground and first excited states present as continua in energy, while transitions to singlet excited states at higher energies simply interpolate between those of the end members of the solid solution. The resulting ground state suggests an extreme version of a random-anisotropy magnet, with many local moments and anisotropies, depending on the precise local configuration of the six B sites neighboring each magnetic ${\mathrm{Tb}}^{3+}$ ion.

Figure 1

The measured $S\left(\right|Q|,\hslash \omega )$ for ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$ at $T=1.5$ K is shown. These low energy, high resolution data were taken with $E_i=11$ meV neutrons. All data have been corrected for empty can background and detector efficiency. Well defined transitions between the ground state crystal-field doublet and the lowest excited crystal-field doublet near 1.5 meV are observed for the end members ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7,x=0$, and ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7,x=2$, while intermediate concentrations see a much broader bandwidth to these low energy excitations.

Figure 2

Cuts of the inelastic neutron scattering data for ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$ samples at $T=1.5$ K are shown. (a) and (b) Inelastic scattering taken with $E_i=11$ and 45 meV neutrons, respectively. (c) Inelastic scattering taken with $E_i=120$ meV neutrons. All data have been corrected for empty can background and detector efficiency. The high energy crystal-field excitations in (c) were fit to extract their energies, and these are plotted in the inset to (c). Data sets in (a) and (b) have been integrated between $1<\left|Q\right|<1.4{\AA }^{-1}$, while that in (c) has been integrated between 2 and $3 {\AA }^{-1}$ in $\left|Q\right|$.

Figure 3

The real ${\chi }^{\prime }$ and imaginary ${\chi }^{\prime \prime }$ parts of the ac susceptibility are shown for ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$ at low temperatures $<1$ K. Measurements were performed in a zero dc magnetic field, at a frequency of 500 Hz.

Figure 4

The $\mu \mathrm{SR}$ relaxation rate measured with a small 50 Oe longitudinal field is shown as a function of temperature for selected ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$ samples. Both the relaxation rate and the temperature are plotted on logarithmic scales.

Figure 5

The low temperature muon relaxation rates for different members of the ${\mathrm{Tb}}_2{\mathrm{Sn}}_{2-x}{\mathrm{Ti}}_x{\mathrm{O}}_7$ series. Note that both axes are logarithmic. The inset shows differences of the relaxation rates measured in a 50 Oe field, on a linear scale, following protocols of a 800 Oe field cooling (FC) and a zero field cooling (ZFC) for samples with $x=0.1$, 0.2, and 1.0, compared with $x=0$ (from [40]). As with the main panel, the temperature axis is logarithmic. Lines are guides to the eye.

Figure 6

A minimal model for the effect of B-site disorder on the low energy CF levels in ${\mathrm{Tb}}_2{\mathrm{B}}_2{\mathrm{O}}_7$ is illustrated. B-site disorder can split the doublet CF states, as ${\mathrm{Tb}}^{3+}$ is a non-Kramers ion, but the singlets are simply shifted in energy. Disordered B-site samples possess a distribution of the seven variants (three ${\mathrm{Sn}}^{4+}$, three ${\mathrm{Ti}}^{4+}$ ions, four ${\mathrm{Sn}}^{4+}$, two ${\mathrm{Ti}}^{4+}$ ions, etc.) of local CF environment at the ${\mathrm{Tb}}^{3+}$ site. Transitions from the ground state to excited CF states are indicated with red arrows, while those terminating with a red dot indicate inelastic peak positions expected for high B-site disorder, due to averaging over local ${\mathrm{Tb}}^{3+}$ environments. The local environment at the ${\mathrm{Tb}}^{3+}$ site in the pyrochlore structure is indicated at the top. For clarity, the six ${\mathrm{O}}^{2-}$ ions around each ${\mathrm{Tb}}^{3+}$ ion are omitted.