Crystal-field parameters of the rare-earth pyrochlores $R_2{\text{Ti}}_2{\text{O}}_7$ ($R=\text{Tb}$, Dy, and Ho)

In this work, we present inelastic neutron scattering experiments which probe the single ion ground states of the rare-earth pyrochlores $R_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ ($R=\text{Tb}$, Dy, Ho). ${\text{Dy}}_2{\text{Ti}}_2{\text{O}}_7$ and ${\text{Ho}}_2{\text{Ti}}_2{\text{O}}_7$ are dipolar spin ices, now often described as hosts of emergent magnetic monopole excitations; the low-temperature state of ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ has features of both spin liquids and spin glasses, and strong magnetoelastic coupling. We measured the crystal-field excitations of all three compounds and obtained a unified set of crystal-field parameters. Additional measurements of a single crystal of ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ clarified the assignment of the crystal-field levels in this material and also revealed an example of a bound state between a crystal-field level and an optical phonon mode.

Figure 1

Inelastic neutron spectra of polycrystalline samples of ${\text{Ho}}_2{\text{Ti}}_2{\text{O}}_7$ (a), ${\text{Dy}}_2{\text{Ti}}_2{\text{O}}_7$ (b), and ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ (c) at $T=5$ K. The neutron intensities are represented in colormaps of arbitrary scale. Dispersionless excitations with highest intensity at lowest momentum transfers $|\vec{Q}|$ are magnetic excitations from the CEF ground state to excited CEF states. Intensities at large $|\vec{Q}|$ are due to scattering by phonons. The first two CEF excitations in ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ are particularly intense, and hence appear broad due to the cutoff of the intensity scale. Differences in the angular coverage of MERLIN (a,c) and 4SEASONS (b) are responsible for the different shapes of the projection of the data in the $\left(\right|\vec{Q}|,E)$ plane.

Figure 2

Details of the measured CEF spectrum in ${\text{Dy}}_2{\text{Ti}}_2{\text{O}}_7$. (a) $|\vec{Q}|$ dependence of the CEF excitations at $T=5$ K. The data points represent an integrated energy band centered at the indicated energy transfer with band widths of $\pm 1$ (9) meV for $E_i=55$ (150) meV (two of the six observed transitions are incorporated in the integration at 88 meV). The lines are fits to the dipolar form factor of ${\mathrm{Dy}}^{3+}$ ions, confirming the magnetic nature of the excitations. (b) Integration of raw data measured with $E_i=28$ meV at low and large momentum transfers ($\mathrm{\Delta }|\vec{Q}|=0.4 {\text{Å}}^{-1}$) to distinguish scattering of magnetic and phononic origins. Inelastic scattering at low-energy transfers stems dominantly from acoustic phonons, and the sharp feature at $E\approx 15.5$ meV is due to a nearly dispersionless transverse optic phonon (TOP) [67]. The first CEF excitation appears at $E\approx 21$ meV.

Figure 3

Comparison of the observed and calculated inelastic neutron spectra of polycrystalline ${\text{Dy}}_2{\text{Ti}}_2{\text{O}}_7$ at 5 and 200 K. The spectrum is concatenated from measurements in low-energy (LE, $E_i=55$ meV) and high-energy (HE, $E_i=150$ meV, multiplied by a factor of 6) settings. The theoretical spectra are calculated from the fitted CEF Hamiltonian and convoluted with the instrumental resolution. At $T=200$ K, the excitation peaks are broadened with a normalized Lorentzian of width 2 meV to account for thermal broadening observed in the experiment.

Figure 4

Summary of CEF excitations of the $R_2{\text{Ti}}_2{\text{O}}_7$ series. (a) The neutron intensity of dipolar transitions between the CEF states at $T=5$ K are presented in a color code. CEF doublet (singlet) states are illustrated by horizontal solid (broken) black lines. (b)–(d) Color-coded neutron intensity of transitions between thermally excited CEF states at $T=200$ K. Initial (final) CEF levels are located on the horizontal (vertical) axes.

Figure 5

Comparison of the $A_l^m$ parameters of the three compounds in $LS$ and intermediate (IM) coupling schemes. The parameters for ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ and ${\text{Dy}}_2{\text{Ti}}_2{\text{O}}_7$ are displayed relative to those for ${\text{Ho}}_2{\text{Ti}}_2{\text{O}}_7$, such that if they were exactly ion independent, they would all appear at zero. The colored band is a color map of a normal distribution of width 10%. Note that a parameter of the same magnitude but opposite sign appears at $-2$ (i.e., $A_6^3({\mathrm{Dy}}^{3+}$) is quite close to this situation).

Figure 6

Investigations of the bound state in ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$ by triple axis spectroscopy experiments on the CEF3 transition in a single crystal. (a) Color map representation of the temperature dependence of the inelastic neutron excitation spectrum measured at the Brillouin zone center $\vec{Q}=(2,2,0)$. The extracted excitation centers (widths) are presented by the points (errorbars). (b) High-resolution scan of the broad envelope including the CEF3 transition at 1.5 K for $\vec{Q}=(2,2,0)$ and $\vec{Q}=(3,3,0)$, where the latter is the Brillouin zone boundary. We find that the envelope consists of three modes, the CEF3 transition at highest energy, and two magnetoelastic optic modes (MEOM1 and MEOM2), which derive from a bound state between the CEF3 transition and the dispersionless optical phonon at $E\approx 15$ meV.

Figure 7

Wave vector and temperature dependencies of the components of the bound state in ${\text{Tb}}_2{\text{Ti}}_2{\text{O}}_7$. (a) Each of the three excitations follows the dipole form factor (solid lines) of the ${\mathrm{Tb}}^{3+}$ ion. (b) The temperature dependencies disentangle the nature of the three modes: the excitation at highest energy is evidently the CEF3 transition, as its intensity follows precisely the calculated temperature dependence (solid line). The sum of the integrated intensity of the two MEOMs is related to the population of the CEF3 state in a coupled four-level system including the ground-state and the bound state (dashed lines).