Single-crystal growth

Single crystals of Yb 2 Ti 2 O7, which were also used in the previous published work24, were prepared by the floating zone method. Stoichiometric quantities of Yb 2 O 3 and TiO 2 powder were mixed, pressed into rods and sintered at 1,150 °C for 24 h. Using these rods, single crystals were grown in air at a rate of 1.5 mm/h. The crystals had a typical diameter of ~6 mm and a length of 20 mm. Powder X-ray diffraction measurements on a pulverized part of the single crystal showed no appreciable amount of any impurity phase.

For comparison with other experimental work that reported the absence of long-range magnetic order, the different methods of sample preparation are worth mentioning. Only three reports are available in the literature. Gardner et al.25 synthesized polycrystalline samples of Yb 2 Ti 2 O 7 by firing stoichiometric amounts of Yb 2 O 3 and TiO 2 at 1,350 °C for several days, with the quality checked by X-ray diffraction. Ross et al.18 synthesized single crystals of Yb 2 Ti 2 O 7 using the same floating zone method as our work, but in 4 bars of oxygen and at a faster growth rate of 5 mm/h. Single crystals prepared by Yaouanc et al.35 were also grown by the floating zone method but at an even faster growth rate of 8 mm/h in air with and without a subsequent heat treatment for 24 h at 1,100 °C under oxygen flow.

Characterization of different single-crystal samples

Here, we discuss the controversy over the presence or absence of a first-order phase transition in Yb 2 Ti 2 O 7 . We then demonstrate from measurements of the specific heat and the EXAFS that the quality of the samples significantly alters the low-temperature behaviour of this material. In particular, we show that the long-range ferromagnetic order realized through the first-order phase transition in the single-crystal sample reported in the main text is easily removed by reducing the Yb content within the sample.

First, let us briefly summarize the previous experimental reports that catalogue the strong sample dependence of the physical properties of Yb 2 Ti 2 O 7 . The Curie–Weiss temperature, Θ CW , varies considerably from sample to sample. Θ CW ~0.53 K for our sample reported on in the main text and in ref. 24, 0.40 K in ref. 21, 0.59 K in ref. 36 and 0.75 K in ref. 22. A significant sample dependence is also observed in the low-temperature specific heat in both powder and single-crystal Yb 2 Ti 2 O 7 samples35,37. In addition, according to Gardner et al.25, a magnetic Bragg peak was observed at (111) with polarized neutron scattering measurements on polycrystalline samples, although neutron spin depolarization was not observed and static ferromagnetic order was ruled out.

In an attempt to rationalize these conflicting reports, we have studied three single-crystal samples of Yb 2 Ti 2 O 7 using specific heat and EXAFS measurements. Sample A was prepared using the method described above and was the sample used for both our polarized neutron scattering measurements and the previous unpolarized measurements24. Sample B was produced from a rod sintered at 1,350 °C rather than 1,150 °C, and was grown in air using a faster growth rate of 5 mm/h (cf. Ross et al.18 who also used a faster growth rate but grew in 4 bars oxygen). Sample C was grown from another rod, which was prepared in the same manner as sample A. However, the molten zone was less stable during the crystal growth.

Supplementary Figure S1 shows the temperature variation of the specific heat for these three samples. Sample A has a sharp singularity at 0.214 K. This is consistent with the observation made in our neutron scattering experiments of the first-order phase transition to a ferromagnetic state. Similar sharp anomalies were also observed for polycrystalline samples in both the early and recent measurements21,37. For sample B, the specific-heat anomaly is broadened significantly, signalling the disappearance of the first-order phase transition. The temperature dependence of the specific heat of sample B closely resembles those of the single crystals35,37 used for unpolarized neutron scattering experiments18,37,38. In sample C, the peak in the heat capacity disappears completely.

The crystal structure of these three samples has been examined by EXAFS at the Yb L 3 -edge. Supplementary Figure S2 shows the radial distribution functions of these samples extracted from the Fourier transform of EXAFS oscillations, shown in the upper inset. The oscillations become increasingly damped for sample A (black), B (red) and C (green), respectively, indicating either a higher level of disorder or the appearance of more vacancies for sample A through to sample C. The Fourier-transformed radial distribution functions favours the vacancy scenario. Near-edge spectra reveal only the Yb3+ charge state (see the middle inset). More oxygen vacancies, particularly at the O(2) site located at the centres of Yb 4 O tetrahedra, are created than Yb and Ti vacancies in samples B and C. These observations suggest vacancies of all three elements are created cooperatively to maintain charge neutrality. Comparing the EXAFS data with the specific heat curves, one can conclude that the higher the quality of the sample, that is the closer the sample is to stoichiometry, the sharper the magnetic transition. In particular, the low-temperature ferromagnetism discussed in the main text is intrinsic to the most ideal single crystals such as sample A.

Polarized neutron scattering experiments

Measurements were performed at the high-flux polarized diffuse neutron scattering spectrometer DNS, FRM II (Garching, Germany). A 3He/4He dilution refrigerator insert with an Oxford Instruments cryostat was used for the experimental temperatures from 0.04 to 1 K. A neutron wavelength of 4.74 Å was chosen for all the experiments. The [1–10] direction of the crystals was aligned perpendicular to the horizontal scattering plane so that the (h,h,l) reciprocal plane can be mapped out by rotating the sample. The neutron polarization at the sample position was aligned along the [1–10] direction of the sample, that is the Z-direction of the chosen experimental coordinate system (Z-direction polarized neutron scattering). Within this setup, the SF and NSF scattering cross-sections are given by

respectively, where are the components of the magnetic scattering cross-section in and out of the (h,h,l) scattering plane, respectively, with Y being perpendicular to the scattering wavevector Q. I SI is the total nuclear spin incoherent scattering cross-section, and N*N is the nuclear coherent scattering cross-section. In general, the magnetic scattering cross-sections are not identical. Therefore, they can provide information on the anisotropy of the magnetic correlations present in the system. To confirm the observed magnetic scattering, X-direction-polarized neutron scattering has also been carried out. In this polarized neutron scattering setup, the neutron polarization is parallel to the scattering vector Q, and can be used to separate the magnetic scattering contribution to be mapped in the SF channel, and nuclear contribution in the NSF channel;

For details, see refs 39, 40.