Magnetic-field control of magnetoelastic coupling in the rare-earth pyrochlore ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

In the rare-earth pyrochlore ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, there are strong interactions between crystal field and phonon excitations resulting in the hybridization of crystal field excitations with both acoustic and optical phonon excitations, which may be implicated in its evasion of the expected long-range ordered states. The magnetoelastic coupling mechanisms are thought to involve large quadrupolar matrix elements between the crystal field states that allow them to couple with intersecting phonons. The character of the hybridized modes can be determined by polarized neutron scattering, as is done here for the case of a crystal field-optical phonon coupling. The coupling mechanism can be further investigated by applying a magnetic field to modify the energies of the crystal field states relative to the phonon spectrum. For a weakly dispersive optical phonon and crystal field level, this has the effect of detuning the quasidegeneracy necessary for hybridization and suppressing the magnetoelastic signal. For a strongly dispersive acoustic phonon crossing a crystal field level, the magnetic field moves the crystal field level, changing the wave vector and energy at which the modes intersect. The field-driven modification of matrix elements for dipole and quadrupole operators involved in the formation of the coupled mode results in the suppression of the magnetoelastic coupling. As the crystal field states shift to higher energy and the magnetoelastic coupling is suppressed, the spin system is driven closer to classically anticipated ordered structures.

Figure 1

Spin structures in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$/spin ices with field applied along $\left[\overline{1}10\right]$: in an ice rule-obeying structure the “two-in–two-out” arrangement of the spins forms chains along tetrahedron edges that are parallel ($\alpha$ chains, red with black and red spins) or perpendicular ($\beta$ chains, blue with green and cyan spins) to the field (a), the $\beta$ chains can remain uncorrelated with each other. Flipping one spin (now colored yellow) per tetrahedron in the ice rule-obeying structure results in the fully ordered double-layer monopole structure (DL) (b). Each tetrahedron can now be seen to host a monopolar spin configuration (“three-in–1-out” or vice versa) and the $\beta$ chains have disappeared.

Figure 2

Typical examples of the line shape of CEF3 measured at $\vec{Q}=(2,2,0)$ and $\vec{Q}=(3,3,0)$ ($T=1.9\mathrm{K}, \vec{H}=0\mathrm{T}$) showing the three peaks and their assignment (a). At $\vec{Q}=(3,3,0)$, the modes are entirely magnetic, as shown by the absence of any signal in the NSF channel (b).

Figure 3

Calculated phonon energy and intensity along the $(h,h,0)$ propagation directions for $\vec{Q}=(2,2,0)$ (a) and $\vec{Q}=(0,0,8)$ (b) zone centers (red lines show the phonon dispersion relations, the color maps show the predicted neutron scattering intensity), and the predicted intensities at $\vec{Q}=(2,2,0), \vec{Q}=(3,3,0)$, and $\vec{Q}=(1,1,8)$ (c) (from Ref. [40]).

Figure 4

(a) Energy scans at $\vec{Q}=(3,3,0)$ and $\vec{Q}=(1,1,8)$ in both the SF and NSF channels at $T=1.8\mathrm{K}$. The lines are fits to multiple Lorentzian peak shapes, as described in the text. The CEF2 excitation around 10.2 meV is included in the scan at $\vec{Q}=(1,1,8)$ because the broader instrumental resolution of this setting causes it to overlap with the CEF3 envelope. At both positions, most of the signal is magnetic and appears in the SF channel. (b) The NSF channel at $\vec{Q}=(1,1,8)$ does contain a signal, albeit much weaker than the magnetic signal. At $T=200\mathrm{K}$, the theoretical phonon line shape from Fig. 3 can be scaled well onto the experimental data. At 1.8 K, the NSF signal is even weaker and may be tentatively ascribed to two separate peaks. (c) The temperature dependence of the scattering intensity at $\vec{Q}=(1,1,8)$ and $E=14\mathrm{meV}$ in the NSF (purple diamonds) and SF (blue triangles) channels. The intensity in the NSF channel increases approximately as expected for the Bose statistics of a phonon at this energy. The SF channel is described by the Boltzmann population factors of the MEOMs and CEF3, as in Ref. [17], but also incorporating a contribution from CEF2 due to the broad energy resolution in this instrument setting.

Figure 5

Cuts through the CEF3 envelope at 0, 3, 6, and 10 T ($T=1.9\mathrm{K}$). The same three peaks were assigned and fitted as in Fig. 2. Modes at higher fields are shaded in darker tones–the darker the tone, the higher the field. As the field increases, the intensity of the two lower energy modes (red and blue) decrease, while the highest energy (green) mode retains its 0 T magnitude. The yellow peak, which appears only at 10 T, is spurious.

Figure 6

The locations of peaks in the CEF3 envelope, fitted by three Lorentzians and plotted as a function of magnetic field at $\vec{Q}=(2,2,0)$ (a) and $\vec{Q}=(3,3,0)$ (b). Also shown is the CEF3 eigenenergy calculated using the parameters of Ref. [17] (thin black line). The signal from the experiment using the 9 T cryomagnet ($T=1.6\mathrm{K}$) is shown as the colormap. In the data taken in this experiment (circle markers), above 6 T the MEOMs have disappeared and the only remaining excitation is the CEF3 singlet. However, in the data taken in the 10 T cryomagnet ($T=1.9\mathrm{K}$) (diamond markers), which has a lower background, including the MEOM excitations improves the accuracy of the fit even up to 10 T [e.g., (c)].

Figure 7

The energy difference between the center of CEF3 and the mean position of the two MEOMs plotted against field at $\vec{Q}=(2,2,0)$ and $\vec{Q}=(3,3,0)$. The intensity of the MEOMs are shown by the color scale, and they indicate that as the energy gap between the modes increases, the intensity of the MEOMs decreases. Dashed lines are guides to the eye, meant to demonstrate that the change in energy with field occurs beginning at $\approx 3\mathrm{T}$, at which point the intensity of the MEOMs begins to decrease.

Figure 8

Dispersion of the CEF1 doublet and spin wave excitations in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ under applied magnetic field parallel to the $\left[1\overline{1}0\right]$ axis. (a) Measurement on the RITA-II spectrometer using sample EP2 with $H=4\mathrm{T}$. (b) Measurement on the TASP spectrometer using sample MH1 with $H=6\mathrm{T}$.

Figure 9

Fits of constant $\vec{Q}$ scans at $\vec{Q}=(1,1,0)$ across the CEF1 envelope. At zero field, the measurements on TASP (open circles) are compared to TOF data collected on IN5 (closed squares). A large quasielastic (QENS) component and the three exciton branches in the CEF1 envelope were identified (a). At $H=2\mathrm{T}$, the low-energy component is interpreted as a spin wave (SW) and one exciton branch disappears from the CEF1 envelope (b). The two remaining modes shift up in energy with increasing field strength $H$ [(c) and (d)].

Figure 10

Field dependence of the CEF1 envelope and spin wave excitations in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ under $\vec{H}\parallel \left[1\overline{1}0\right]$. The arrow denotes the onset of the AFM long range magnetic order, which is accompanied with the appearance of SW excitations. The exciton modes of the CEF1 doublet start to shift up only above $H\approx 2\mathrm{T}$.

Figure 11

Constant energy scans across the MEM in applied magnetic field. The hybrid mode with $\vec{k}\parallel (h,h,0)$ was measured at $5\mathrm{meV}$ on TASP and at $7\mathrm{meV}$ on EIGER (a and b). Similarly, the MEM propagating along the $(h,h,h)$ direction was measured at $E=7\mathrm{meV}$ (c). The intensity of the MEM decreases in large magnetic fields, while their dispersion remains approximately constant.

Figure 12

Peak amplitudes of the MEM measured in different BZs and energy transfers as function of applied magnetic field. Once the long range AFM order begins to develop, indicated by the dotted line at $H=2\mathrm{T}$, the intensity of the hybrid modes decreases. Data for $\hslash \omega =5\mathrm{meV}$ were measured at TASP where the field was limited to 6 T.

Figure 13

Linear correlation between the energy gap separating the CEF1 doublet from the electronic ground state, the development of AFM long range order (a) and the intensity of the MEM (b). Once the gap increases, the magnetoelastic coupling is weakened and long range AFM order can develop accordingly. The data points where the AFM Bragg peak has zero intensity arise from the intermediate field regime $0<H<2\mathrm{T}$, in which the second exciton mode strongly softens, see Fig. 12.

Figure 14

Calculated field dependence of the energies of the CEF3 level and matrix elements for ground state transitions under $\vec{H}\parallel \left[1\overline{1}0\right]$. The applied field is differently oriented at sublattices in the $\alpha$ and $\beta$ chains, which have a longitudinal field component and a purely transverse field respectively. Solid lines show the transition energies for spins in the $\alpha$ chains (cyan) and $\beta$ chains (yellow). The color map indicates the strength of matrix elements for transitions from the ground state to the excited states for various combinations of operators: $J_x+J_y+J_z$, all observable contributions to the neutron spectrum (a); the quadrupole operators implicated in the coupling mechanism $Q_{xz}+Q_{yz}$ (b, note that no geometric factor for neutron scattering is applied in this case). Broken lines are contours indicating some very weak intensities barely visible in the color map. At 1.9 K, the temperature used for the related measurements, the population of any of the excited states becomes negligible essentially as soon as the ground state splits, so no excited state contributions appear.

Figure 15

Calculated field dependence of the energies of the ground state and CEF1 doublets and matrix elements for ground state transitions under $\vec{H}\parallel \left[1\overline{1}0\right]$. The applied field is differently oriented at sublattices in the $\alpha$ and $\beta$ chains, which have a longitudinal field component and a purely transverse field respectively. Solid lines show the transition energies for spins in the $\alpha$ chains (cyan) and $\beta$ chains (yellow). The color map indicates the strength of matrix elements for transitions from the ground state to the excited states for various combinations of operators: $J_x+J_y+J_z$, all observable contributions to the neutron spectrum (a); the quadrupole operators implicated in the coupling mechanism $Q_{xz}+Q_{yz}$ (b, note that no geometric factor for neutron scattering is applied in this case). At 0.05 K, the temperature used for the related measurements, the population of any of the excited states becomes negligible essentially as soon as the ground state splits, so no excited state contributions appear.