Multiphase magnetism in Yb2Ti2O7

Abstract

We use neutron scattering to show that ferromagnetism and antiferromagnetism coexist in the low T state of the pyrochlore quantum magnet [Formula: see text] While magnetic Bragg peaks evidence long-range static ferromagnetic order, inelastic scattering shows that short-range correlated antiferromagnetism is also present. Small-angle neutron scattering provides direct evidence for mesoscale magnetic structure that we associate with metastable antiferromagnetism. Classical Monte Carlo simulations based on exchange interactions inferred from [Formula: see text]-oriented high-field spin wave measurements confirm that antiferromagnetism is metastable within the otherwise ferromagnetic ground state. The apparent lack of coherent spin wave excitations and strong sensitivity to quenched disorder characterizing [Formula: see text] is a consequence of this multiphase magnetism.

Keywords: frustrated magnetism; neutron scattering; phase transitions; pyrochlore.

Fig. 1.

Refinement of ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ magnetic structure based on changes in integrated Bragg intensities upon cooling from 0.6 K to 80 mK in zero magnetic field. Inset shows the zero-field intensity of the (002) Bragg peak appears upon cooling from 300 to 80 mK, which rules out AIAO order. Note that the $S\left(Q\right)$ axis is quadratic.

Fig. 2.

(A–K) Inelastic magnetic neutron scattering from ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ along $\mathrm{\Gamma }\hspace{0.17em}\left(000\right)\to \mathrm{K}\hspace{0.17em}\left(2\overline{2}0\right)\to \text{UL}\hspace{0.17em}\left(2\overline{1}1\right)\to \mathrm{\Gamma }\hspace{0.17em}\left(000\right)$ in (A) the $B=0$ ZFC state, (B) the $B=0$ FC state, and (C) $B=1.5\hspace{0.17em}\mathrm{T}$, compared to linear spin wave theory predictions from (D and E) Ross et al.’s Hamiltonian (25), (F and G) Robert et al.’s Hamiltonian (26), (H and I) Thompson et al.’s Hamiltonian (20), and (J and K) this study. The vertical streaks in the zero-field data near $\mathrm{\Gamma }$ are due to saturated neutron detectors.

Fig. 3.

Fit to ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ spin wave spectrum. (A) Line through parameter space minimizing ${\chi }^2$ of the calculated vs. observed intensity data at 1.5 T. (B) Spin wave gap at (220) plotted against $J_2$, with the star indicating a gap of 0.11 meV. The gap goes to zero at the phase boundary between the FM and the AFM ${\mathrm{\Gamma }}_5$ phase. (C) ${\chi }^2$ along the “best-fit line” in A. D–K show the measured neutron-scattering data and the calculated neutron-scattering cross-section for the Hamiltonian fitted to the 1.5-T data. The experimental plots are averaged over ±0.038 Å⁻¹ in the direction perpendicular to the cut.

Fig. 4.

Calculated zero-field spin wave gap at (220) for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ as a function of $J_1$ and $J_2$ for four Hamiltonians: (A) Ross et al. (25), (B) Robert et al. (26), (C) Thompson et al. (20), and (D) this study. White stars indicate the best fit parameters for each model. ($J_3$ and $J_4$ are held fixed to the best-fit values for each respective Hamiltonian. White areas are where neither the FM nor ${\mathrm{\Gamma }}_5$ AFM are stabilized. The spin-configuration energy was minimized for each point.) The gap goes to zero upon entering the ${\mathrm{\Gamma }}_5$ AFM ordered phase, and this study as well as that by Robert et al. (26) places ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ within error of the AFM phase boundary.

Fig. 5.

Inelastic magnetic neutron-scattering data for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ compared to the calculated magnetic neutron-scattering cross-section for FM and AFM and mixed FM + AFM ground states. (A) Calculated magnetic neutron-scattering cross-section for the best-fit Hamiltonian FM ground state along high-symmetry directions $\mathrm{\Gamma }\hspace{0.17em}\left(000\right)\to \mathrm{K}\hspace{0.17em}\left(2\overline{2}0\right)\to \text{UL}\hspace{0.17em}\left(2\overline{1}1\right)\to \mathrm{\Gamma }\hspace{0.17em}\left(000\right)$. (B) Calculated spectrum for an AFM ${\psi }_3$ state just across the phase boundary. (C) Average of FM and AFM spectra. (D) Field-cooled 70 mK spectrum. (E) Constant 0.2-meV energy slice for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ (FC) at 70 mK scattering. (F and G) The 0.2-meV spin wave theory calculations for the FM and FM + AFM states, respectively. The overall scattering matches the AFM + FM result better, particularly near $\left(-\frac{4}{3},\frac{2}{3},-\frac{2}{3}\right)$ at 0.2 meV.

Fig. 6.

(A and B) MC + LSWT calculated neutron spectrum for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ (A) compared to the field-cooled 70 mK spectrum (B). (C–H) Constant energy (0.2 and 0.3 meV) slices for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ (FC) at 70 mK scattering compared to the FM and MC + LSWT spectra. The MC + LSWT simulation better matches the scattering pattern, particularly near $\left(-\frac{4}{3},\frac{2}{3},-\frac{2}{3}\right)$ at 0.2 meV.

Fig. 7.

(A) Spin wave gap at $\left(2\overline{2}0\right)$ for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ at 100 mK, plotted with the predicted gap from Robert et al. (26) and this study. (B) Zero-field spin wave spectrum around $\left(2\overline{2}0\right)$ (following the same cuts as Fig. 3) measured at higher resolution. (C) Spin wave spectrum plotted against distance from $\left(2\overline{2}0\right)$. (D and E) Calculated spin wave spectrum plotted against distance from $\left(2\overline{2}0\right)$ for (D) a ferromagnetic ground state and (E) the MC + LSWT spectrum. Neither one reproduces the low-lying flat feature above $\left(2\overline{2}0\right)$.

Fig. 8.

(A) SANS experimental configuration, with $\left[1\overline{1}0\right]$ vertical and [111] along a horizontal magnetic field. (B–D) SANS data for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ in (B) the zero-field–cooled state and (C) the field-cooled state after applying a 1-T field in the $⟨111⟩$ direction and subtracting (D) a 1.5-K background. (E) Dependence of the (111) and (201) scattering streaks on wave vector transfer $Q_{\perp }=\left(11\overline{2}\right)$ perpendicular to the plane of the detector, with 1.5-K data subtracted. Ranges of $Q$ integrated are $Q_{111}=0.015\pm 0.011$ Å⁻¹ and $Q_{201}=0.016\pm 0.011$ Å⁻¹ shown in Fig. 10. (F) Difference between B and C, revealing an enhancement of (111) in the scattering rod after cooling in a field applied along that direction.

Fig. 9.

Field- and temperature-dependent SANS data for ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$. (A–C) Temperature dependence of the (A) (111), (B) $\left(1\overline{1}0\right)$, and (C) (201) rods at 0, 0.1, and 0.2 T. (D–F) Field dependence of the (D) (111), (E) $\left(1\overline{1}0\right)$, and (F) (201) rods at 120 and 300 mK. The (111), $\left(1\overline{1}0\right)$, and (201) directions are defined in the insets to Fig. 10 A–C, Insets. (G–J) Colormap images of the ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ SANS pattern as a function of field at 100 mK.

Fig. 10.

Fits to SANS in ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$. (A–C) The $Q$-dependent scattering at various fields in the (A) (111), (B) $\left(1\overline{1}0\right)$, and (C) (201) directions. The regions defining these directions are shown in A–C, Insets. (D and E) The results of a Porod exponent fit $I=A{Q}^{-n}$.

Fig. 11.

Domain walls in ${\mathrm{Y}\mathrm{b}}_2{\text{Ti}}_2{\text{O}}_7$ rotating through a ${\mathrm{\Gamma }}_5$ antiferromagnetic phase. Top image shows a domain wall between $-z$ and $+z$ ferromagnetic domains, and Bottom image shows a domain wall between $+x$ and $+z$, rotating between two states in the ${\mathrm{\Gamma }}_5$ manifold.