Neutron spectroscopic study of crystal field excitations in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$

We present time-of-flight inelastic neutron scattering measurements at low temperature on powder samples of the magnetic pyrochlore oxides ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. These two materials possess related, but different ground states, with ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ displaying “soft” spin ice order below $T_{\mathrm{N}}\sim 0.87$ K, while ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ enters a hybrid, glassy spin ice state below $T_{\mathrm{g}}\sim 0.2$ K. Our neutron measurements, performed at $T=1.5$ and 30 K, probe the crystal field states associated with the $J$ = 6 states of Tb${}^{3+}$ within the appropriate $Fd\overline{3}m$ pyrochlore environment. These crystal field states determine the size and anisotropy of the Tb${}^{3+}$ magnetic moment in each material's ground state, information that is an essential starting point for any description of the low-temperature phase behavior and spin dynamics in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. While these two materials have much in common, the cubic stanate lattice is expanded compared to the cubic titanate lattice. As our measurements show, this translates into a factor of $\sim$2 increase in the crystal field bandwidth of the $2J+1=13$ states in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ compared with ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. Our results are consistent with previous measurements on crystal field states in ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, wherein the ground-state doublet corresponds primarily to $m_J=|\pm 5\rangle$ and the first excited state doublet to $m_J=|\pm 4\rangle$. In contrast, our results on ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ differ markedly from earlier studies, showing that the ground-state doublet corresponds to a significant mixture of $m_J=|\pm 5\rangle$, $|\mp 4\rangle$, and $|\pm 2\rangle$, while the first excited state doublet corresponds to a mixture of $m_J=|\pm 4\rangle$, $|\mp 5\rangle$, and $|\pm 1\rangle$. We discuss these results in the context of proposed mechanisms for the failure of ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ to develop conventional long-range order down to 50 mK.

Figure 1

The crystal environment immediately around the Tb${}^{3+}$ ion (purple) within Tb${}_2$$M_2{\mathrm{O}}_7$ is shown. Each Tb${}^{3+}$ ions is surrounded by eight oxygens (red and beige at the vertices of the cube) which form a distorted cube. The two apical oxygens (red) along one diagonal of the cube (the local $\left[111\right]$ direction) are closer to the Tb${}^{3+}$ ion than are the other six oxygens (shown in beige).

Figure 2

Color contour maps of energy vs $\left|\mathbf{Q}\right|$ from the inelastic neutron scattering data for ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at $T=1.5$ K are shown. The top two plots show data at relatively low energies taken with $E_i=11$ meV neutrons. Well-defined transitions between the ground-state crystal field doublet and the lowest excited state crystal field doublet between 1–2 meV as well as weak dispersion of this excitation, are observed for both samples. The middle panels show data up to $\sim$43 meV, using $E_i=45$ meV neutrons. The bottom panels show data taken to the highest energy transfers, employing $E_i=120$ meV neutrons. All data sets shown are for $T=1.5$ K. An empty can background was subtracted and the data were corrected for detector efficiency.

Figure 3

Cuts (blue) of the inelastic neutron scattering data shown in Fig. 2 are compared with theoretical calculations (red) of the magnetic neutron scattering cross section for crystal field excitations from (a) ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (b) ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, at $T=1.5$ K. Spectra over the three different energy ranges shown in Fig. 3 were taken using the three different $E_i$ shown in Fig. 2, and used an integration in $\left|\mathbf{Q}\right|$ as indicated in the figure.

Figure 4

Comparison between the low-energy inelastic scattering measurements ($<$6 meV) and calculations for the magnetic neutron scattering between crystal field states for (a)–(c) ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (d)–(f) ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. This shows the measured and calculated spectra at both $T=1.5$ and 30 K. The calculations shown in (b) and (e) use our newly determined crystal field parameters shown Table 2. For comparison, (c) and (f) show the equivalent calculation using the crystal field parameters determined previously by Mirebeau et al. [18]. As can be seen, the temperature dependence of the low-energy spectra does not distinguish between these two model calculations.

Figure 5

Comparison between inelastic scattering measurements at moderate energies ($6\text{meV}<E<20$ meV) and calculations for the magnetic neutron scattering between crystal field states for (a)–(c) ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (d)–(f) ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. This shows the measured and calculated spectra at both $T=1.5$ and 30 K. The calculations shown in (b) and (e) use our newly determined crystal field parameters shown in Table 2. For comparison, (c) and (f) show the equivalent calculation using the crystal field parameters determined previously by Mirebeau et al. [18]. At low temperatures, $T=1.5$ K, this energy regime shows transitions from both the ground-state doublet to crystal field states beyond the first excited state doublet. At $T=30$ K, it shows transitions from both the ground-state doublet and first excited state doublet to crystal field states beyond the first excited state doublet. For that reason, and in contrast to data shown in Fig. 4, this inelastic data is very sensitive to the specific details of the spectral decomposition (eigenfunctions) of both the ground-state and first excited state doublets.

Figure 6

Crystal field energy scheme for all the $2J+1=13$ levels for Tb${}^{3+}$ in (a) ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (b) ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, as derived from our calculations of the magnetic neutron scattering cross section from transitions between crystal field states (not to scale). These calculations employed the best-fit crystal field parameter sets determined for ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Tb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ shown in Table $\text{II}$.