Effect of chemical pressure on the crystal electric field states of erbium pyrochlore magnets

We have carried out a systematic study of the crystal electric field excitations in the family of cubic pyrochlores ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7$ with $B=\mathrm{Ge}$, Ti, Pt, and Sn, using neutron spectroscopy. All members of this family are magnetic insulators based on $4f^{11}{\mathrm{Er}}^{3+}$ and nonmagnetic $B^{4+}$. At sufficiently low temperatures, long-range antiferromagnetic order is observed in each of these ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7$ pyrochlores. The different ionic sizes associated with different nonmagnetic $B^{4+}$ cations correspond to positive or negative chemical pressure, depending on the relative contraction or expansion of the crystal lattice, which gives rise to different local environments at the ${\mathrm{Er}}^{3+}$ site. Our results show that the $g$-tensor components are $XY$-like for all four members of the ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7$ series. However, the $XY$ anisotropy is much stronger for ${\mathrm{Er}}_2{\mathrm{Pt}}_2{\mathrm{O}}_7$ and ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7(g_{\perp }/g_z>25)$ than for ${\mathrm{Er}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ and ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7(g_{\perp }/g_z<4)$. The variation in the nature of the $XY$ anisotropy in these systems correlates strongly with their ground states as ${\mathrm{Er}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ and ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ order into ${\mathrm{\Gamma }}_5$ magnetic structures, whereas ${\mathrm{Er}}_2{\mathrm{Pt}}_2{\mathrm{O}}_7$ and ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ order in the ${\mathrm{\Gamma }}_7$ Palmer-Chalker structure.

Figure 1

Rietveld refinements of the time-of-flight neutron powder-diffraction patterns for each of the erbium pyrochlore magnets ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7,B=$ (a) Ge, (b) Ti, (c) Pt, and (d) Sn. The data sets were collected at 10 K using Bank 2 of the POWGEN diffractometer $\left({\lambda }_{\mathrm{center}}=1.066\AA \right)$. The data were refined against the $Fd\overline{3}m$ space group, and the goodness-of-fit parameters for each sample are given in Table 1. The insets show an expanded view of a high-$Q$ region from 12 to $16{\AA }^{-1}$.

Figure 2

Inelastic neutron-scattering measurements with an incident energy of $E_i=25\mathrm{meV}$ for the ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7$ pyrochlores, $B=$ (a) and (e) Ge, (b) and (f) Ti, (c) and (g) Pt, and (d) and (h) Sn. The top row shows the spectra at a base temperature (2 or 5 K) for each sample. In each case, three ground-state crystal electric field excitations are identified at the positions indicated by the arrows. The bottom row shows the spectra at 100 K, which is high enough to thermally populate transitions from the first and second-excited-state levels. The excited-state transitions at 100 K are indicated by the arrows. An empty can measurement has been subtracted from all data sets.

Figure 3

Inelastic neutron-scattering measurements with an incident energy of $E_i=150\mathrm{meV}$ for the ${\mathrm{Er}}_2{B}_2{\mathrm{O}}_7$ pyrochlores, $B=$ (a) and (e) Ge, (b) and (f) Ti, (c) and (g) Pt, and (d) and (h) Sn. The top row shows the spectra at a base temperature (2 or 5 K) for each sample. In each case, four higher-energy ground-state crystal electric field excitations are identified at the positions indicated by the arrows in addition to the transition at $\sim 20\mathrm{meV}$, which also was seen in the $E_i=25-\mathrm{meV}$ data set. The bottom row shows the spectra at 100 K, which is high enough to thermally populate transitions from the first- and second-excited-state levels. The excited-state transitions at 100 K are indicated by the arrows. An empty can measurement has been subtracted from all data sets.

Figure 4

(a) The $|\vec{Q}|$ dependence of the 15.7-meV CEF excitation of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is well captured by the single-ion magnetic form factor of ${\mathrm{Er}}^{3+}$. The same $|\vec{Q}|$ dependence is observed for all the CEF excitations resolved in this paper. (b) The neutron-scattering intensity as a function of energy transfer for ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, revealing the presence of three CEF excitations over the energy range shown. An example of a fit to the CEF spectra is shown. The fit captures the integrated intensity for all three CEF excitations and was used to optimize the CEF Hamiltonian. (c) The $|\vec{Q}|$ dependence associated with the ${\mathrm{Er}}^{3+}$ magnetic form factor can also be seen by performing energy cuts of the data with integrations over different $|\vec{Q}|$ ranges.

Figure 5

Comparison between the experimentally resolved CEF energy schemes of the four erbium pyrochlore magnets and those calculated using the CEF Hamiltonians given in Table 2. The calculated CEF spectra for ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ [39, 48] and ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ [29] derived from previous studies also are compared to our experimental data and are given by the gray dashed lines where asterisks mark the levels that were fit.

Figure 6

Inelastic neutron spectra for (a) ${\mathrm{Er}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$, (b) ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, (c) ${\mathrm{Er}}_2{\mathrm{Pt}}_2{\mathrm{O}}_7$, and (d) ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ showing the scattering intensity as a function of energy transfer at base temperature [(i) $E_i=25\mathrm{meV}$ and (ii) $E_i=150\mathrm{meV}]$ and 100 K [(iii) $E_i=25\mathrm{meV}$ and (iv) $E_i=150\mathrm{meV}]$. These data sets are extrated from the contour maps of Figs. 2 and 3 with the indicated integration in momentum transfer $|\vec{Q}|$. The calculated CEF spectra, using the CEF Hamiltonian parameters shown in Table 2, are plotted by the blue $(T=2$ and 5 K) and red $\left(T=100\mathrm{K}\right)$ lines.

Figure 7

An enhanced view of the lowest-energy crystal electric field levels in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ at (a) 5 K and (b) 100 K, revealing weak dispersion. (c) The maximum fitted intensity of these two CEF excitations as a function of $|\vec{Q}|$ at both 5 and 100 K.

Figure 8

Calculated CEF schemes for the rare-earth titanates $A_2{\mathrm{Ti}}_2{\mathrm{O}}_7(A=\mathrm{Tb}$, Dy, Ho, and Yb) obtained using scaling arguments from our fitted parameters for ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The calculated levels are indicated by the dashed lines where, for ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, the singlet levels are denoted by a lighter dashed line. The experimentally determined CEF energies, reproduced from Refs. [49, 54, 55], are given by the solid lines.