J Gaudet
email: gaudej@mcmaster.ca

D D Maharaj

G Sala

E Kermarrec

K A Ross

H A Dabkowska

A I Kolesnikov

G E Granroth

B D Gaulin

Neutron Spectroscopic Study of Crystalline Electric Field Excitations in Stoichiometric and Lightly Stuffed Yb 2 Ti 2 O 7

Keywords: numbers: 75.25.-j, 75.10.Kt, 75.40.Gb, 71.70.Ch

Time-of-flight neutron spectroscopy has been used to determine the crystalline electric field (CEF) Hamiltonian, eigenvalues and eigenvectors appropriate to the J = 7/2 Yb 3+ ion in the candidate quantum spin ice pyrochlore magnet Yb2Ti2O7. The precise ground state (GS) of this exotic, geometrically-frustrated magnet is known to be sensitive to weak disorder associated with the growth of single crystals from the melt. Such materials display weak "stuffing" wherein a small proportion, ≈ 2%, of the non-magnetic Ti 4+ sites are occupied by excess Yb 3+ . We have carried out neutron spectroscopic measurements on a stoichiometric powder sample of Yb2Ti2O7, as well as a crushed single crystal with weak stuffing and an approximate composition of Yb2+xTi2-xO7+y with x = 0.046. All samples display three CEF transitions out of the GS, and the GS doublet itself is identified as primarily composed of mJ = ±1/2, as expected. However,"stuffing" at low temperatures in Yb2+xTi2-xO7+y induces a similar finite CEF lifetime as is induced in stoichiometric Yb2Ti2O7 by elevated temperature. We conclude that an extended strain field exists about each local "stuffed" site, which produces a distribution of random CEF environments in the lightly stuffed Yb2+xTi2-xO7+y, in addition to producing a small fraction of Yb-ions in defective environments with grossly different CEF eigenvalues and eigenvectors.

I. INTRODUCTION

Geometrically-frustrated magnetic materials are of great current interest due to the diversity of exotic ordered and disordered ground states (GSs) that they display 1 . In particular, cubic pyrochlore magnets with chemical composition A 2 B 2 O 7 have been a playground for geometric frustration as both the A 3+ and B 4+ sites, independently, reside on interpenetrating networks of corner-sharing tetrahedra (see Fig. 1), one of the canonical architectures for frustrated ground states in three dimensions 2 . The rare earth titanate pyrochlores have played a pivotal role in the development of the field, as Ti 4+ at the B site is non-magnetic, and the A site can be occupied by all trivalent rare earth ions from Sm 3+ to Lu 3+ . Many of these rare earth titanates therefore have a magnetic A sub-lattice, and the family as a whole gives rise to different combinations of magnetic anisotropies and interactions, which in turn are responsible for the diversity of exotic ground states 3 .

As an example, the classical spin ice state has been the focus of much attention [4][5][6] , and it results from a combination of local Ising anisotropy 7,8 and net ferromagnetic interactions 8 on the pyrochlore lattice, such that the Ising magnetic moments on each tetrahedron obey "ice rules" with two spins pointing into each tetrahedron and two spins pointing out, analogous to the water ice model proposed by Pauling 9 . This results in a sixfold degeneracy for a given tetrahedron, and a macroscopic degeneracy for the three dimensional network as a whole. Yb 2 Ti 2 O 7 has attracted much recent attention [10][11][12][13][14] as a candidate for a quantum spin ice ground state 15,16 , wherein effective S = 1/2 degrees of freedom decorate a pyrochlore lattice and interact via net ferromagnetic interactions 17 . The GS phase diagram and microscopic Hamiltonian appropriate to Yb 2 Ti 2 O 7 have been extensively studied 18,19 . The microscopic Hamiltonian itself has been determined by modelling spin wave dispersion and neutron intensities in the high magnetic field, low temperature state of Yb 2 Ti 2 O 7 20 . This work convincingly showed anisotropic exchange to be the relevant form of the interactions at low temperatures.

One of the most interesting features of the GS properties of Yb 2 Ti 2 O 7 is its apparent sensitivity to small levels of defects that are present in real materials [21][22][23] . Stoichiometric Yb 2 Ti 2 O 7 is known to display a large and sharp anomaly in its heat capacity, C p , near ∼ 265 mK. However, this anomaly has been observed to be sample dependent with samples displaying broader anomalies at lower temperatures depending on the exact stoichiometry of the material studied [24][25][26][27][28] . This phenomena is very unusual for a three dimensional magnet, as the defect levels involved are at the limits of detectability by conventional techniques. The variation in stoichiometry, where characterized, is on the order of 1% level, far removed from percolation thresholds in three dimensions. The pyrochlore lattice consists of interpenetrating networks of corner-sharing tetrahedra which are generated independently by the Yb 3+ (magnetic) and Ti 4+ (non-magnetic) cations. The pyrochlore lattice is said to be stuffed wherein a Yb 3+ ions, which are normally found on the 16d site (also called A-site), also occupy the Ti 4+ 16c site (B-site).

Polycrystalline samples of Yb

the sharpest and highest temperature C p anomalies, likely due to the lower temperatures required for their synthesis, leading to less TiO 2 volatization. Single crystals grown by floating zone image furnace techniques typically display broad low temperature C P anomalies, or no anomalies, and sometimes show multiple peaks in C P at low temperature. Ross et al. 21 undertook a detailed neutron crystallographic study of both powder and crushed single crystal Yb 2 Ti 2 O 7 samples, representative of those that displayed sharp and broad low temperature C P signatures, respectively, and showed that the crushed single crystal sample displayed weak "stuffing": a structural defect in which a slight excess of Yb occupies the Ti sub-lattice. Stuffing is illustrated schematically in Fig. 1, wherein Yb 3+ ions occupy both the 16d site of the F d 3m cubic space group, as well as act as impurities on the 16c site normally occupied by Ti 4+ . Weak stuffing was shown to occur at the 2.3% level in the crushed single crystals with composition Yb 2+x Ti 2-x O 7+y grown by floating zone image techniques. In contrast the powder sample grown by solid state synthesis was shown to be stoichiometric.

In this paper we report neutron spectroscopic measurements of the crystalline electric field (CEF) excitations in the two powder samples, the stoichiometric powder and the crushed single crystal with 2.3 % "stuffing" previously studied by Ross et al. These measurements allow us to accurately determine the eigenvalues and eigenvectors appropriate to the 4 doublets which make up the J = 7/2 CEF manifold for Yb 3+ in Yb 2 Ti 2 O 7 . Measurements on both stoichiometric and lightly "stuffed" samples allow us to investigate the role of "stuffing" on the CEF levels associated with Yb 3+ ions properly resid-ing on the A-site of the pyrochlore structure. With these benchmark measurements in hand, we calculated CEF eigenvalues and eigenvectors for the "stuffed" Yb 3+ ions residing on the B site, as well as for the A site Yb 3+ ions in the presence of oxygen vacancies 29 .

II.

CALCULATED CRYSTAL FIELD LEVELS FOR Yb 3+ AT THE A-SITE

Hund's rules enable the determination of the total angular momentum J of the Yb 3+ ion. The electronic configuration of Yb 3+ is 4f 13 , resulting in J = 7 2 which is 2J + 1 = 8-fold degenerate. Within the pyrochlore structure this degeneracy is lifted by the CEFs at the Yb 3+ site due largely to the presence of the eight neighbouring O 2-ions. As illustrated in Fig. 2 (left panel), the oxygen environment at the A-site consists of a scalenohedron, which is a cube distorted along one diagonal that forms the local [111] axis. Six oxygen ions, commonly referred to as O(2), are located on a plane perpendicular to this direction, which is a three-fold rotation axis. The other two oxygen ions, referred to as O(1), are located along the local [111] axis in the geometric centre of the tetrahedra defined by the A-site Yb 3+ ions. By contrast, the environment at the B-site is a trigonal anti prism made of six O(2) oxygen ions surrounding the transition metal, as shown in the right panel of Fig. 2.

FIG. 2: A comparison between the A-site and B-site oxygen environment in the pyrochlore structure of Yb2Ti2O7. -The left panel of the figure shows the scalenohedron environment generated by the oxygen ions at the A-site where the Yb 3+ resides. The symmetry of this structure is similar to that at the B-site (right panel) where the Ti 4+ ions are located. Following Prather's convention 30 , the 3-fold axis should be placed along ẑ of the reference system in order to minimize the number of CEF parameters in the Hamiltonian. Therefore the resulting CEF Hamiltonian for Yb 3+ on the A-site can be written as:

Ô3 4 + B 0 6 Ô0 6 + B 3 6 Ô3 6 + B 6 6 Ô6 6 . (1)
Here we employ the Steven's operators Ôm n 31 and CEF parameters B m n to approximate the Coulomb potential generated by the crystalline electric field due to the neighbouring oxygen atoms.

The determination of the crystal field parameters B m n in Eq. 1 is well suited to inelastic neutron spectroscopy. The unpolarized neutron partial differential magnetic cross-section can be written within the dipole approximation as 32 :

d 2 σ dΩdE = C k f k i F (|Q|)S(|Q|, hω), ( 2
)
where Ω is the scattered solid angle, E the final neutron energy,

k f
ki the ratio of the scattered and incident momentum of the neutron, C a constant and F (|Q|) is the magnetic form factor. The scattering function S(|Q|, hω) gives the relative scattered intensity due to transitions between different CEF levels. At constant temperature and wave vector |Q|, we have:

S(|Q|, hω) = i,i ( α | i|J α |i | 2 )e -βEi j e -βEj L(∆E + hω), (3)
where α = x, y, z and L(∆E + hω) = L(E i -E i + hω) is a Lorentzian function which ensures energy conservation as the neutron induces transitions between the CEF levels i → i , which possess a finite energy width or lifetime.

The procedure for fitting our inelastic neutron scattering (INS) data assumes an initial set of CEF Hamiltonian parameters. The Hamiltonian is then diagonalized to find the corresponding CEF eigenfunctions and eigenvalues. S(|Q|, hω) is then computed and directly compared with the experimental results and, finally, the CEF parameters are tuned such that the χ 2 between the calculated and measured S(|Q|, hω) is minimized.

III. EXPERIMENTAL DETAILS

Now we turn our attention to the details of the INS experiment which was conducted on two samples of Yb 2 Ti 2 O 7 of different stoichiometry. We will refer to these two samples as the stoichiometric (x = 0) and stuffed powder (x = 0.046) samples hereafter. Specific details regarding their synthesis and characterization can be found in previous work by Ross et al. 21 . These samples were studied utilizing the SEQUOIA direct geometry time-of-flight spectrometer 33 , which is located at the Spallation Neutron Source at Oak Ridge National Laboratory. In an INS experiment the CEF excitations are manifested as dispersion-less features with the strongest scattering intensity expected at low |Q| positions as a result of their magnetic origin. SEQUOIA is the ideal instrument for the investigation of the CEFs since it provides low |Q| coverage and a large, dynamic (|Q|,E) range.

12g of each of the stoichiometric and stuffed powder samples were loaded in an aluminium flat plate with dimensions 50mm × 50mm × 1mm and sealed with indium in helium atmosphere. An empty can with the same dimensions was loaded with the two samples on a three sample changer in a closed-cycle refrigerator. Measurements have been performed over a range of temperatures from T = 5 K to 300 K, and utilizing neutrons with incident energy E i = 150meV giving an elastic energy resolution of ±2.8 meV. The energy resolution improves with increasing energy transfer and is ∼1.4meV and ∼1meV for energy transfers of 80 and 115meV, respectively. The corresponding chopper settings selected were T 0 = 120Hz and FC 1 = 600Hz. Similar measurements were conducted on the empty can for use as a background measurement.

IV. INELASTIC NEUTRON SCATTERING FROM CRYSTAL FIELD EXCITATIONS IN Yb2Ti2O7

The INS spectra taken on both the stoichiometric and stuffed powder samples are shown in Fig. 3 for energies up to E i = 150meV at T = 5 K. An empty can data set, taken as background at the same temperature, has been subtracted from both the sample data sets. As seen in Fig. 3, we observe a set of dispersion-less excitations arising from both the CEF transitions, as well as optical phonons. Below 60 meV, the intensity associated with the dispersion-less excitations increases with increasing |Q|, characteristic of inelastic scattering from phonons. In contrast, the excitations at 76.7 meV, 81.8 meV and 116.2 meV, highlighted with blue, grey and red arrows, respectively, in Fig. 3, show inelastic scattering which increases with decreasing |Q|, consistent with magnetic scattering. We therefore ascribe these three dispersionless excitations with CEF transitions from the GS doublet.

Yb 3+ possesses thirteen electrons in its almost filled 4f shell and, as a consequence of Kramers' theorem, its eight-fold degenerate CEF levels can be maximally split into four doublets. We associate the three magnetic excitations observed at T = 5 K in both the stoichiometric and the stuffed powder samples shown in Fig. 3, with the transitions between the CEF GS doublet and the CEF excited state doublets, as indicated in the inset of Fig. 4. These three transitions between CEF doublets account for all the CEF states within this J = 7/2 multiplet appropriate to Yb 3+ . As will be described in further detail, the CEF transitions observed at low temperatures in the stoichiometric sample are sharper in energy than those in the presence of light stuffing. We shall restrict our quantitative analysis of the CEF spectra to the case of the stoichiometric powder sample. Therefore we use the x = 0 data set in Fig. 3 a), and perform a |Q| integrated cut (|Q| = [4.5,5.25] Å-1 ), yielding the neutron scattering intensity as a function of energy. This data set is shown in the main panel of Fig. 4, wherein the intensity at the peak of the CEF transition at 81.8 meV has been normalized to unity. The relative intensities of the CEF transitions at 76.7 meV, 81.8 meV and 116.2 meV, as well as the energy of these transitions from the GS, constrain the CEF Hamiltonian.

The cut shown in Fig. 4 has been fitted to a model for the inelastic scattering 32 arising from dipole allowed transitions between the CEF GS doublet and the three excited states, using Eq. 3. The starting parameters in the CEF Hamiltonian for Yb 3+ were those determined by Bertin et al. 34 for Yb 2 Ti 2 O 7 within the point charge approximation. This calculation yielded CEF transitions at ∼60 meV, ∼70 meV and ∼90 meV, in contrast to those determined experimentally in Fig. 3. These starting parameters were then refined and a best fit (shown in solid red) to the |Q| integrated cut in Fig. 4 was obtained. All three CEF transitions were fit using the same resolution determined energy width. In addition, relatively weak Lorenzian lineshapes near 56 meV, 70 meV and 100 meV phenomenologically describe the inelastic scattering from the phonons which are in near proximity to the CEFs, and improved the fit when included. As can be seen, the overall description of the INS data from the stoichiometric sample in Fig. 4 is very good. The values which were obtained for the CEF Hamiltonian parameters from the fit relevant to Yb 3+ at the A-site are given in Table I and the resulting energy eigenvalues and eigenvectors are given in Table II. The GS doublet for Yb 3+ is comprised primarily of m J = ± 1/2 and the corresponding low temperature anisotropic gtensor components are given by g ⊥ = 3.62 ± 0. ent quantitative descriptions of the background phonons. Note that this determination of the g-tensor is performed at zero magnetic field, in contrast to, for example, the analysis of spin wave data in high magnetic fields 20 . The g-tensor is consistent with previous estimates for Yb 3+ in Yb 2 Ti 2 O 7 [35][36][37] , and the dominant m J = ± 1/2 character of the GS validates the effective S = 1/2 quantum description for the Yb 3+ moment. In order of ascending energy, the excited state doublets correspond primarily to m J = ± 7/2, m J = ± 3/2, and m J = ± 5/2.

A. Temperature and weak stuffing dependence of the A-site CEF transitions

Inelastic neutron scattering measurements have also been carried out on both the stoichiometric and stuffed powder samples as a function of temperature, using E i = 150meV neutrons. Energy scans of the |Q| = [4.5,5.25] Å-1 integrated inelastic scattering for the stoichiometric powder sample are shown as a function of temperature in Fig. 5. The T = 5 K data set is the same x = 0 data set shown in Fig. 4, and the peak intensity associated with the 81.8 meV CEF transition at T = 5 K has been normalized to unity.

We note that the maximum temperature employed in these measurements, 300 K, corresponds ≈ 27 meV and consequently only the GS doublet of Yb 3+ is substantially occupied at any temperature. Qualitatively, the thermal fluctuations have three effects on the CEF neutron spectra: the CEF excitations broaden appreciably in energy; the maximum peak intensity diminishes; the energy of the CEF excitations softens slightly. This latter effect is somewhat subtle, but it can be seen in Fig. 5 by drawing a fiducial dashed line positioned at the centre of the CEF transitions at T = 5 K.

The energy width of the CEF excitations can be quantitatively examined by fitting the data sets as in Fig. 4, but now with a damped harmonic oscillator (DHO) lineshape for the three CEF transitions, each with the same temperature dependent energy width. At the energy transfers and temperatures of interest, the DHO can be approximated by a single Lorentzian for each mode.The form of this line-shape is given by: L(E) = 1 π

( Γ obs 2 ) (E -∆E) 2 + ( Γ obs 2 ) 2 , (4)
which is a Lorentzian function of energy with width Γ obs and centered on the energy of the CEF transition ∆E.

The falloff of the maximum peak intensity as a function of increasing temperature along with the concomitant broadening in the energy widths of the transitions ensures that the integrated spectral weight of this inelastic scattering is almost temperature independent, consistent with the CEF transitions being from the GS doublet of Yb 3+ at all temperatures considered here. The common energy width or inverse lifetime of the three CEF excitations extracted from this analysis is plotted as a function of temperature in Fig. 6, where we removed the resolution contribution to the widths using the following relation :

Γ 2 intrinsic (T) = Γ 2 obs (T ) -Γ 2 res . (5)
Figure 6 then shows pronounced growth in the intrinsic energy width of the CEF transitions from ∼ 3.5 meV at T =5 K to ∼ 9 meV at T =300 K. This growth mirrors the temperature dependence of the Yb 3+ mean squared displacements (MSDs) in stoichiometric Yb 2 Ti 2 O 7 as determined by powder neutron diffraction 21 , and this is also reproduced in Fig. 6 for direct comparison. It is worth noting that we plot one of the anisotropic components of the atomic displacement parameter, U 11 , which represents the anisotropic MSD of the Yb 3+ ions from their average positions. Although, for the specific case of the E(meV) - The CEF eigenvalues and eigenvectors for Yb 3+ at the A-site of Yb2Ti2O7. The first column displays the CEF eigenvalues of the system, while the corresponding eigenvectors are given in each row in terms of the mJ basis.

Yb 3+ ion in the stoichiometric powder, it can be shown that U 11 is equal to the other components of the atomic displacement, so that the MSD is effectively isotropic. The inset to Fig. 6 shows the slight softening of the CEF energies with increasing temperature. The softening is most pronounced for T < 125 K, and then it flattens out at higher temperature. The significance of 125 K = 11.3 meV is not completely clear, although it could be related to the temperature scale associated with the top of the acoustic phonon band in Yb 2 Ti 2 O 7 , which is typically 10 -15 meV for transition metal oxides. This effect is relatively small with a maximum observed softening of ∼ 0.8 meV at T = 300 K compared with T=5 K.

FIG. 6: Energy width and shift of the CEF transitions as a function of temperature-The intrinsic energy width of the CEF transitions as extracted using Eq. 5. The intrinsic energy width corresponding to the stuffed powder is higher at both 5 K and 300 K in comparison to the stoichiometric sample. The green dots shows the mean square displacement (MSD) of the Yb 3+ taken from ref 21 and scaled in such a way that the MSDs and the energy widths of the CEF excitations in the stoichiometric powder have the same value at T=100K. The inset shows the shift in energy of the CEF transitions as a function of temperature, relative to that at T =5 K.

The finite intrinsic energy widths of the CEF excitations in stoichiometric powder must originate from dy-namics in the lattice not captured by the static structure. The strong resemblance of the temperature dependence of the Yb 3+ MSDs to that of the CEF energy widths suggests that zero point fluctuations determine the low temperature energy widths, and the thermal population of phonons give rise to the larger widths at finite temperature. The ionic displacements associated with both zero point fluctuations and the phonons will distort the local environment at the Yb 3+ site, giving rise to a distribution of CEF transitions energies. Displacements of harmonic phonons time-average to zero, consequently the distribution of CEF transitions in the presence of the phonons is approximately centred on the zero temperature CEF transitions, and the transitions remain well defined at all temperatures.

Figure 7 shows a direct comparison of the T = 5 K, |Q| = [4.50,5.25] Å-1 integrated inelastic scattering for the stoichiometric and stuffed samples. This comparison clearly shows the CEF excitations in the lightly-stuffed sample have a considerably larger energy width even at low temperatures. The INS from the stuffed powder was also fit to the same DHO line-shape as was fit to the stoichiometric data, and the corresponding intrinsic energy widths at T = 5 K and 300 K are also plotted in Fig. 6. The intrinsic energy widths of the CEF excitations in the stoichiometric and lightly stuffed samples differ by only ∼ 5 % at T = 300 K; this is not surprising as the intrinsic energy widths in either sample are expected to be dominated by thermal fluctuations. However, at T = 5 K, the intrinsic energy width of the stuffed sample is ∼ 5.2 meV -approximately that displayed by the stoichiometric sample at ∼ 125 K.

The finite energy widths of the CEF excitations at low temperature in the lightly stuffed sample demonstrate that weak stuffing must induce a relatively large "volume of influence" about each "stuffed" Yb ion, that is, about each Yb ion residing on a B-site. The corresponding distortion field about each stuffed site gives rise to a distribution of A-site Yb 3+ environment, similar to those associated with phonons at finite temperature. Indeed, the stuffed B-site Yb ions would also give rise to CEF spectra completely distinct from those of Yb 3+ in A-site environments. However the concentration of such stuffed B-site Yb ions is known to be small, ≈ 2.3 %, in our lightly stuffed sample and we were unable to find convincing evidence for such excitations in this study.

V. CRYSTAL FIELD CALCULATIONS FROM FIRST PRINCIPLES WITHIN THE POINT CHARGE APPROXIMATION

As discussed in Sec. II, the CEF Hamiltonian parameters can be fitted using INS data. In principle, these parameters can also be calculated from first principles within a point charge model following the procedure outlined in Refs. [START_REF] Hutchings | Solid State Physics[END_REF][START_REF] Walter | [END_REF][40] The CEF interactions are treated as a perturbation to the spin-orbit coupling and the resulting eigenfunctions of the CEF Hamiltonian are expressed as a linear combination of the |J, m J states within the 2 F 7 2 manifold multiplets.

We have calculated the CEF Hamiltonian parameters in this manner and the ratio between these ab initio parameters and those extracted from fit of the INS data are shown in Table I. Note that the ionic positions were taken from the crystallographic refinement of stoichiometric Yb 2 Ti 2 O 7 21 . As can be seen, apart from the B 0 6 and B 3 6 terms which are relatively small, all the CEF parameters agree well with those extracted from fitting the INS data. This result is remarkable given the simplicity of the point charge approximation, and it gives us confidence that we can make reasonable predictions on the strength of the CEF Hamiltonian parameters determined in this way.

The CEF eigenvectors and eigenvalues determined from the fit to the INS in the stoichiometric powder at T = 5 K is shown in Table II. Tables III andIV show the CEF eigenvectors and eigenvalues for Yb ions in two different impurity sites: one "stuffed" on the B-site and one on the A-site but in the presence of one O(1) oxygen vacancy. Both of these impurities are expected to be present at some small concentration in the lightly stuffed sample. In principle O(2) vacancies can also occur, but these are higher energy defects than O(1) vacancies 29 . These CEF eigenvectors and eigenvalues have been calculated within the point charge approximation and then corrected with the same ratio of B m n parameters as were determined from the comparison between the INS fitted values and the calculated values in stoichiometric Yb 2 Ti 2 O 7 at low temperatures.

The differences between the fitted CEF parameters and the calculated ones originate mainly from the overlap of the 4f orbital of the rare earth with the 2p orbitals of the ligands. The corresponding effects on the B m n terms are difficult to estimate from first principles and are not considered here. Instead we calculate the CEF parameters for the defective environments, and then correct these with the same B m n ratios which were determined in the treatment of CEFs for the stoichiometric sample. Finally, we verified that this correction did not dramatically affect the properties of the system; that is that the CEF properties within the defective environments did not qualitatively depend on this correction.

A. Crystal Field Calculation for Yb 3+ at the A-site in an Oxygen Depleted Environment

We first consider an A-site Yb 3+ in the presence of a single O(1) oxygen vacancy. This depleted environment is expected to break the symmetries of the CEF Hamiltonian. However if Prather's convention is satisfied, only the inversion operation is lost and the number of CEF parameters in our Hamiltonian is unchanged from the stoichiometric case. Note that in general, a broken symmetry can dramatically affect the CEF Hamiltonian, and it may be necessary to add terms to the CEF Hamiltonian to better approximate the defective Coulomb potential.

The eigenvalues and eigenvectors calculated for A-site Yb 3+ in the presence of a single O(1) oxygen vacancy are shown in Table III. As previously discussed, we have performed the point charge calculation and then scaled these results by the B m n ratios taken from the fitted and calculated CEF Hamiltonian terms determined for the stoichiometric powder.

Kramers' degeneracy still protects the Yb 3+ ion in this defective environment and all the CEF eigenvectors appear as doublets. However, the gap between the GS and the first excited state is much larger than in the stoichiometric case. Moreover an examination of the eigenvectors in Table III shows that the GS is now a pure linear combination of m J = ±7/2 states. These GS eigenvectors are similar to what is found in stoichiometric Dy 2 Ti 2 O 7 crystals, where the GS is comprised of a linear combination The CEF eigenvalues and eigenvectors calculated for Yb 3+ in a depleted oxygen environment at the A-site in Yb2Ti2O7. The first column displays the CEF eigenvalues of the system, while the corresponding eigenvectors are given in each row in terms of the mJ basis. The ratio correction was applied in order to arrive at the final CEF parameters.

E(meV) - 7 2 -5 2 -3 2 -1 2 1 2 3 2 5
of the maximal m J states, pure m J =±15/2. With the CEF eigenfunctions in hand, we can calculate the local magnetization of the A-site Yb 3+ in the presence of a single O(1) oxygen vacancy. This is shown in Fig. 8 where the magnetization anisotropy for Yb 3+ moment within the defective scalenohedron is displayed. In both Figs. 8 and9, coloured spheres represent the tips of magnetization vector centred at the Yb 3+ ion, in response to a given applied external field of H = 1 T with a particular [111]-component and which precesses around the plane normal to [111]. Many values of the magnetization are shown simultaneously for applied fields uniformly distributed on the unit sphere. The colour scale is chosen in order to match the strength of the magnetic moment from 0µ B (dark blue) to 4µ B (dark red). For the same [111] component of magnetization, the centre of the spheres are sufficiently close to each other, that the final impression is that of rings (this is seen more clearly in Fig. 9). The tightness of the rings about the [111] axis implies stronger Ising-like behaviour of the Yb 3+ moment. For the case of the A-site Yb 3+ in the presence of a single O(1) oxygen vacancy in Fig. 8, the calculated magnetic moment along the local [111] direction is 3.953µ B , close to the full moment of 4µ B . This suggests negligible precession around this easy axis and thus strong Ising-like behavior. The local magnetization anisotropy for the Asite Yb 3+ in the presence of a single O(1) oxygen vacancy is dramatically different from the stoichiometric case, where it was planar or XY-like 36,37,41 , and is similar to that displayed by the classical spin ice magnets Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 . This anisotropy is present even without the B m n ratio corrections and thus it depends only on the depleted environment. Neutron powder diffraction measurements on the stuffed powder samples, derived from crushed single crystals, show such Yb 2 Ti 2 O 7 samples to be lightly stuffed at the 2.3 % level. We therefore calculate the CEF eigenvectors and eigenvalues for "stuffed" Yb 3+ at the B-site. This is an interesting case to consider since the local environment at the B-site is very different from that considered to this point for A-site Yb3+. Nonetheless, the symmetry at the B-site is similar to the A-site, and we can simply rotate the trigonal anti-prism cage around the B-site so that the local [111] direction is aligned along ẑ and the C 2 axis along ŷ. Therefore the form of the B-site Yb 3+ CEF Hamiltonian is identical to Eq. 1. The calculated CEF eigenvectors and eigenvalues for B-site Yb 3+ are displayed in Table IV. Once again we have performed the point charge calculation and then scaled these results by the B m n ratios determined for the stoichiometric powder. Kramers' degeneracy is again invoked and all CEF levels are again doublets, now with a gap of 105 meV between the GS and the first excited state. The eigenvectors within the GS doublet are again almost pure m J =±7/2 spin states, and we expect the anisotropy to be Ising like. This is borne out by again applying a small rotating external magnetic field to probe the shape of the anisotropy of the GS magnetic moment as was done for Yb 3+ at the A-site in an oxygen depleted environment. Following the previous convention, Fig. 9 shows spheres centred along the direction in which the magnetic moment points. The colour scale matches the magnetic moment size in units of µ B . In contrast with the previous case, the spheres are not located exactly along the local [111] direction, thus we anticipate a small precession about the easy axis with the maximum value of the magnetic moment given by µ = 3.25µ B . This is similar to what was calculated for the defective scalenohedron associated with Yb 3+ at the A-site in an oxygen depleted environment, but the anisotropy is not as Ising-like as it was for that case.

It is interesting that the influence of oxygen vacancies on A-site Yb 3+ ions and that of stuffed B-site Yb 3+ ions have similar effects on the CEF eigenvalues and eigenvectors. Stuffing Yb 3+ onto B-sites nominally occupied by Ti 4+ will require oxygen vacancies to preserve charge neutrality, and we expect both phenomena to occur at some level in single crystals grown from the melt. The symmetry of the defective scalenohedron, especially when two O(1) are missing, is reasonably similar and to the trigonal anti-prism associated with stuffed Yb 3+ at the B-site, so it is perhaps not so surprising that these environments produce similar effects on the CEF eigenvalues and eigenvectors of the magnetic ions. On the other hand the similarity of the CEFs associated with the defective sites may complicate the analysis of experimental data from sufficiently defective non-stoichiometric Yb 2 Ti 2 O 7 , as the measured spectrum may show overlap originating from at least two distinct defective environments.

VI. CONCLUSIONS

Neutron scattering measurements on the stoichiometric and stuffed powder samples of the quantum spin ice candidate system Yb 2 Ti 2 O 7 have been carried out to probe the CEF eigenvalues and eigenvectors associated with the J = 7/2 Yb 3+ ion in these environments. Analysis of the INS from the stoichiometric powder show the GS doublet to correspond primarily to m J = ± 1/2, with anisotropic g-tensor components given by g ⊥ = 3.62 ± 0.15 and g z = 1.85 ± 0.10, which therefore imply local XY anisotropy as expected. The eigenvalues and eigenvectors of the three excited state CEF doublets were all identified above a lowest CEF gap of ∼ 76.7 meV at T = 5 K. The energy widths, or inverse lifetimes of the CEF states broaden with increasing temperature from ∼ 3.5 meV at low temperatures up to the highest measured temperature, 300 K. This broadening mirrors the temperarure dependence of the Yb 3+ MSDs, and we associate it with the effects of zero point motion and phonons on the CEF environment around the A-site Yb 3+ site. Interestingly, the corresponding measurements and analysis of the lightly stuffed powder sample (x = 0.046) of Yb 2 Ti 2 O 7 shows a similar set of CEF transitions, however the energy width or inverse lifetime of the transitions are intrinsically further broadened even at T = 5 K. This effect is ascribed to a relatively large strain field, or "volume-of-influence", associated with each stuffed Bsite Yb 3+ .

We have also calculated the CEF eigenvectors and eigenvalues associated with Yb 3+ in two defective envi- The CEF eigenvalues and eigenvectors calculated for Yb2Ti2O7 at B-site. The first column displays the Crystal Field spectrum of the system, while the corresponding eigenvectors are given in each row in terms of mJ basis. The ratio correction was applied in this case to the CEF parameters.

ronments of relevance for non-stoichiometric Yb 2 Ti 2 O 7 . These are for Yb 3+ at the A-site in an oxygen depleted environment, and for stuffed Yb 3+ at the B-site. These calculations were performed within the point charge approximation and make use of a comparison of the calculated and measured eigenvalues and eigenfunctions in the stoichiometric sample to benchmark these results. Both of these defective environments give rise to related effects wherein the GS doublet is now primarily made up of m J = ± 7/2, and the local anisotropy of the GS eigenfunctions are Ising-like, although with stronger Ising anisotropy for the oxygen depleted environment than for case of the stuffed Yb 3+ at the B-site.

These results put the nature of the CEF eigenfunctions and eigenvalues associated with Yb 3+ in the quantum spin ice candidate system on a much firmer footing and indicate how the nature of the GS moment and anisotropy in this system can be sensitive to small amounts of disorder, as is known to occur in single crystal samples grown from the melt using floating zone techniques.

2 FIG. 1 :

 21 FIG.1:The stuffed pyrochlore structure of Yb2Ti2O7. -The pyrochlore lattice consists of interpenetrating networks of corner-sharing tetrahedra which are generated independently by the Yb 3+ (magnetic) and Ti 4+ (non-magnetic) cations. The pyrochlore lattice is said to be stuffed wherein a Yb 3+ ions, which are normally found on the 16d site (also called A-site), also occupy the Ti 4+ 16c site (B-site).

FIG. 3 :

 3 FIG. 3: Inelastic neutron scattering spectra-S(|Q|, hω) obtained for the stoichiometric powder and the stuffed powder samples at T = 5 K are shown in panels (a) and (b) respectively, with the corresponding T = 5 K empty can subtracted from each data set. The three horizontal arrows in blue, gray and red highlight the three crystal field excitations which are found at 76.7 meV, 81.8 meV and 116.2 meV, respectively.

FIG. 4 :

 4 FIG. 4: |Q| integrated cut for the stoichiometric powder at T = 5 K. -A |Q| integrated cut (|Q| = [4.5,5.25] Å-1 ) obtained from the INS spectra in Fig.3(a) for the stoichiometric powder at T = 5 K is shown. The arrows indicate the corresponding positions of the CEFs. The inset outlines the corresponding CEF transitions from the GS doublet.

  15 and g z = 1.85 ± 0.10, where z corresponds to the local [111] axis. The error bars on the g values were estimated by exploring the sensitivity of the fitting procedure on the low temperature stoichiometric powder data, to differ-B m n (meV) Calculated Fitted Ratio B A comparison of the calculated crystal field parameters (B m n ) with those obtained by fitting INS data from the stoichiometric powder at T = 5 K.

FIG. 5 :

 5 FIG. 5: Energy cuts of the |Q| = [4.5,5.25] Å-1 integrated inelastic scattering for the stoichiometric powder sample as a function of temperature. -Energy cuts were taken with an appropriate background subtraction and all energy cuts for temperature above T = 5 K has been vertically translated for clarity.

FIG. 7 :

 7 FIG. 7:A direct comparison between the INS from CEF transitions in the stuffed powder (in orange) and the stoichiometric powder (in green) Yb2Ti2O7 samples. -Energy cuts taken at T = 5 K within a range |Q| = [4.50,5.25] Å-1 for the stuffed powder (orange) and stoichiometric powder (green) samples.

FIG. 8 :

 8 FIG. 8: Magnetization of the A-site Yb 3+ ion in presence of an O(1) oxygen vacancy. -A coloured sphere represents the tip of the magnetization vector centred at the Yb 3+ ion in response to an applied external field of H = 1 T which rotates within the plane normal to [111]. Many values of the magnetization are shown simultaneously for applied fields uniformly distributed on the unit sphere. For the same [111] component of magnetization, the centre of the spheres are sufficiently close to each other, that the final impression is that of rings. The tightness of the rings around the [111] axis implies stronger Ising-like behaviour of the Yb 3+ moment, in this case with a calculated magnetic moment of 3.953µB along the local [111] direction.

B

  . Crystal Field Calculation for "Stuffed" Yb 3+ at the B-site

FIG. 9 :

 9 FIG. 9: Magnetization of the "stuffed" Y b 3+ ion at the Bsite of Yb2Ti2O7. -Each sphere represents the tip of the magnetization vector centred at the Yb 3 + ion in response to an applied rotating external field of H = 1 T. Neighboring contiguous spheres form ring pattern. The ellipsoidal shape of the ring pattern suggests Ising-like behaviour of the rare earth moment along the local [111] direction, however the transverse extent of the ellipsoidal shape indicates a small precession around the Ising axis. The calculated magnetic moment is 3.25µB.

TABLE II :

 II

									7
									2
	0	0	0.0866	0	0	-0.9283	0	0	0.3616
	0	-0.3616	0	0	-0.9283	0	0	-0.0866	0
	76.706	0.9136	0	0	-0.3343	0	0	-0.2313	0
	76.706	0	-0.2313	0	0	0.3343	0	0	0.9136
	81.764	0	0	-1	0	0	0	0	0
	81.764	0	0	0	0	0	-1	0	0
	116.23	0	0.9690	0	0	0.1627	0	0	0.1858
	116.23	0.1858	0	0	-0.1627	0	0	0.9690	0

TABLE III :

 III

TABLE IV :

 IV

Acknowledgments

We would like to acknowledge helpful conversations with M. J. P. Gingras. We would also like to acknowledge T. E. Sherline for technical assistance with the measurements. The neutron scattering data were reduced using Mantid 42 and analyzed using the DAVE software package 43 . Research using ORNL's Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Work at McMaster University was funded by NSERC of Canada.


J Gaudet

K A Ross

⇤ E Kermarrec

N P Butch

G Ehlers

H A Dabkowska

B D Gaulin

Gapless quantum excitations from an Ice-like Splayed Ferromagnetic ground state in stoichiometric Yb 2 Ti 2 O 7

Keywords: numbers: 75.25.-j, 75.10.Kt, 75.40.Gb, 71.70.Ch

The ground state of the quantum spin ice candidate magnet Yb2Ti2O7 is known to be sensitive to weak disorder at the ⇠ 1 % level which occurs in single crystals grown from the melt. Powders produced by solid state synthesis tend to be stoichiometric and display large and sharp heat capacity anomalies at relatively high temperatures, TC ⇠ 0.26 K. We have carried out neutron elastic and inelastic measurements on well characterized and equilibrated stoichiometric powder samples of Yb2Ti2O7 which show resolution-limited Bragg peaks to appear at low temperatures, but whose onset correlates with temperatures much higher than TC . The corresponding magnetic structure is best described as an ice-like splayed ferromagnet. The spin dynamics in Yb2Ti2O7 are shown to be gapless on an energy scale < 0.09 meV at all temperatures, and organized into a continuum of scattering with vestiges of highly overdamped ferromagnetic spin waves present. These excitations di↵er greatly from conventional spin waves predicted for Yb2Ti2O7's mean field ordered state, but appear robust to weak disorder as they are largely consistent with those displayed by nonstoichiometric crushed single crystals and single crystals, as well as by powder samples of Yb2Ti2O7's sister quantum magnet Yb2Sn2O7.

I. INTRODUCTION

Pyrochlore magnets of the form A 2 B 2 O 7 have been of great topical interest as both the A and B sublattices independently form networks of corner-sharing tetrahedra, one of the canonical crystalline architectures supporting geometrical frustration in three dimensions [START_REF] Gardner | [END_REF]2 . The cubic rare earth titanates, of the form RE 2 Ti 2 O 7 have been especially relevant as many magnetic RE 3+ ions can occupy the A-site of the structure and where the non-magnetic Ti 4+ occupy the B-site. This pyrochlore family can also be relatively easily produced in both powder and single crystal form [3][4][START_REF] Prabhakaran | Conference on Crystal Growth (ICCG16)/The 14th International Conference on Vapor Growth and Epitaxy (ICVGE14)[END_REF] . One of the family members, Yb 2 Ti 2 O 7 , has been particularly topical as it has been proposed as a realization of quantum spin ice [START_REF] Chang | [END_REF][7][8][9][10][11][12][13][14][15][16] . It displays a net ferromagnetic Curie-Weiss constant of ⇠ 0.6 K 17,18 , and crystal field (CF) e↵ects give rise to a CF ground state doublet at the Yb 3+ site made up of primarily m J =± 1/2 eigenvectors and local XY anisotropy [19][20][21] . Yb 2 Ti 2 O 7 is therefore a good realization of quantum S e↵ = 1/2 spins decorating a network of corner-sharing tetrahedra -the pyrochlore lattice.

The microscopic spin Hamiltonian for Yb 2 Ti 2 O 7 has been estimated using neutron spectroscopic measurements of spin waves in its high field polarized state 12 . While the zero field phase of Yb 2 Ti 2 O 7 does not show well defined spin waves in single crystals, a magnetic field applied along the [1-10] direction pushes Yb 2 Ti 2 O 7 through a quantum phase transition or crossover near µ 0 H C ⇠0.5 T into a polarized phase that is characterized by resolution-limited spin waves 14 . Linear spin wave theory using anisotropic exchange produces an excellent description of the high field spin wave dispersion and intensities, and the resulting microscopic Hamiltonian has been used in high temperature series expansions which accurately describe the magnetization and heat capacity of Yb 2 Ti 2 O 7 in absolute units 22 . Yb 2 Ti 2 O 7 's spin Hamiltonian contains 4 symmetry-allowed nearneighbour anisotropic exchange terms and it has been proposed that the largest of these is J zz , which ferromagnetically couples together local z, or Ising components of spin 12 . These z-components are aligned directly into or out of the tetrahedra, and this combination of S e↵ = 1/2 spins, the spin Hamiltonian, and the pyrochlore lattice would then be responsible for the quantum spin ice phenomenology. Somewhat di↵erent phenomenology has also been discussed in which J zz is less dominant 23 .

While the general phase behaviour for an anisotropic exchange Hamiltonian of the form which describes Yb 2 Ti 2 O 7 in zero magnetic field has exotic quantum spin liquid and Coulomb ferromagnetic mean field ground states present within it 24,25 , the mean field ground state predicted on the basis of Yb 2 Ti 2 O 7 's spin Hamiltonian is a simple ferromagnet with a mean field phase transition of T MF ⇠ 3 K 12 . A broad "hump" in Yb 2 Ti 2 O 7 's experimentally-determined heat capacity (C P ) is observed at ⇠ 2 K, roughly co-incident with the calculated T MF , however there are no indications of order observed above a sharp C P anomaly which occurs near T C ⇠ 0.26 K in stoichiometric powders of Yb 2 Ti 2 O 7 11,26 . Taking the sharp C P anomaly at face value for an indication of the ferromagnetic phase transition predicted by mean field theory, this indicates that T MF is suppressed by a factor of ⇠ 12 by quantum fluctuations, geometrical frustration, or both. However, there are strong indications that Yb 2 Ti 2 O 7 's zero field phase below T C ⇠ 0.26 K is far removed from a conventional ferromagnet. To date there are experimental studies which support a relatively simple ferromagnetic ground state [START_REF] Chang | [END_REF]8 , but also studies which show extensive di↵use neutron scattering covering all of reciprocal space 10 , no changes in the spin relaxation observed when cooling below the sharp C P anomaly in µSR studies from stoichiometric samples 27 , and no evidence for the conventional spin waves that are expected as the normal modes of the magnetically-ordered state in any sample 13,14,23 .

An interesting feature of Yb 2 Ti 2 O 7 's exotic zero magnetic field ground state is its sensitivity to weak disorder 28,29 . The sharp C

P anomaly observed at T C ⇠0.26 K in stoichiometric powder samples, is observed to be broader and to occur at lower temperatures in all single crystal samples measured to date 29 . The C P anomalies can occur as low as ⇠ 0.15 K in single crystals grown from floating zone image furnace techniques, may not obviously occur at any temperature, or may appear as multiple peaks at lower temperatures.

In the interest of understanding the microscopic structure and defects responsible for the di↵erences between powder samples grown by solid state synthesis, and single crystals grown from the melt by floating zone techniques, neutron di↵raction studies were carried on both powder and crushed single crystal (CSC) samples, that were known to display di↵erent C P behaviour at low temperature 28 . The conclusions were that, while the powder sample was stoichiometric Yb 2 Ti 2 O 7 , the CSC was characterized as exhibiting weak "stu ng" wherein a small proportion of excess Yb 3+ resides on the Ti 4+ site, and the composition of the crushed single crystal was Yb 2+x Ti 2 x O 7+y with x = 0.046. Note that this defect level is close to the limit of detectability by conventional di↵raction techniques. This study was greatly aided by the fact that Ti displays a negative coherent neutron scattering length, hence there is significant neutron contrast for Yb, which has a positive coherent neutron scattering length, occupying the Ti site. In and of itself, sensitivity of the ground state to such weak disorder is a remarkable result, as conventional three dimensional ordered states are not sensitive to disorder at such a low level. For example, the phase transition to non-collinear 2 antiferromagnetic order in Er 2 Ti 2 O 7 is not sample dependent, and has been studied as a function of magnetic dilution and shown to be consistent with conventional three dimensional percolation theory [START_REF] Niven | Proceedings of the Royal Society of London A: Mathematical[END_REF] . A recent study of the Yb 3+ crystal field excitations in Yb 2 Ti 2 O 7 suggests that the anisotropy of Yb 3+ in defective environ-ments is Ising-like, rather than XY-like in stoichiometric Yb 2 Ti 2 O 7 , and this may be related to the e↵ectiveness of disorder at this low ⇠ 1 % level 21 .

It is therefore important to fully characterize powder samples of Yb 2 Ti 2 O 7 which are known to be stoichiometric and to display sharp C P anomalies at T C =0.26 K; that is to use this as a benchmark for understanding the zero field ground state of pristine Yb 2 Ti 2 O 7 . In this work, we report elastic and inelastic neutron scattering results from the same two powder samples previously studied by neutron di↵raction 28 We show that stoichiometric Yb 2 Ti 2 O 7 displays resolutionlimited magnetic Bragg scattering at low temperatures, however these persist to a much higher temperature scale than T C =0.26 K. Nonetheless, looking only at the low temperature Bragg intensities, we refine a static magnetic structure which is best described as an ice-like splayed ferromagnet with an ordered moment of 0.90(9)µ B . The spin dynamics at the lowest temperatures in stoichiometric Yb 2 Ti 2 O 7 in zero field are indeed far removed from conventional spin waves. They are gapless on an energy scale < 0.09 meV at all wave vectors, and characterized by a continuum of scattering with a bandwidth of ⇠ 1 meV. Vestiges of very overdamped ferromagnetic spin waves can be seen in the inelastic scattering, which itself is temperature independent below T MF ⇠ 3 K.

II. EXPERIMENTAL DETAILS

The powder samples employed in this study were the same samples previously studied by Ross et al 28 . 10 grams of stoichiometric Yb 2 Ti 2 O 7 and 8 grams of the CSC powder, Yb 2+x Ti 2 x O 7+y with x = 0.046, were separately loaded in aluminum sample cans with copper lids under 10 atms of helium exchange gas. This method of loading powder samples is known to provide good thermal contact to the cold finger of the dilution refrigerator, and maintains thermal conductivity below 1K by enabling enough superfluid Helium to coat the powder grains [START_REF] Ryan | [END_REF] . The 4 gram single crystal of Yb 2 Ti 2 O 7 (produced in a floating zone image furnace following similar procedures to those used to grow other single crystal titanate pyrochlores) was mounted on an aluminum holder.

Elastic neutron scattering measurements were carried out on the SPINS triple axis instrument at the NIST Center for Neutron Research. SPINS used pyrolytic graphite as both monochromator and analyzer, and 80' collimators before and after the sample, producing a 5.0 meV elastically scattered beam with an energy resolution of 0.25 meV. Cooled beryllium filters placed before and after the sample helped to eliminate higher harmonic contamination. The sample was first cooled to T =8K where scans of the (111), (002), ( 220), ( 113) and (222) Bragg peaks were collected. Afterwards, the sample was cooled down to 100mK and sat at this temperature for 2 hours before collecting measurements at the same Bragg positions as above. Finally, the sample was warmed up to 700mK where another scan of the same Bragg peaks was also collected.

Inelastic neutron scattering measurements were also performed on both the stoichiometric powder as well as the CSC, stu↵ed powder using the time-of-flight disk chopper spectrometer (DCS) at NIST 32 . For these measurements, monochromatic incident neutrons of wavelength of 5 Å were employed, giving an energy resolution at the elastic position of 0.09 meV. Empty can measurement have been used as a background for these data. Inelastic neutron scattering measurements on the single crystal of Yb 2 Ti 2 O 7 were performed using the cold neutron chopper spectrometer (CNCS) 33 at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. For this time-of-flight experiment, the single crystal was mounted in a dilution refrigerator and aligned with the [HHL] plane in the horizontal plane. An incident neutron energy of 3.3 meV was employed which gave an energy resolution of 0.1 meV at the elastic position. Sample rotation methods were employed wherein the single crystal sample was rotated 360 about the vertical direction in 1 steps. The background is approximated using a measurement done under the same conditions but without the sample. Magnetic and structural refinement were performed using SARAh Refine 34 and FullProf [START_REF] Rodriguez-Carvajal | satellite meeting on powder di↵raction of the XV congress of the IUCr[END_REF] .

III. RESULTS AND DISCUSSION

A. Elastic neutron scattering and magnetic structure determination:

Fig. 1 shows elastic neutron scattering data taken on the stoichiometric powder sample of Yb 2 Ti 2 O 7 using SPINS. No magnetic Bragg peaks were observed at nuclear-disallowed positions upon cooling below T C , and consequently we focused our attention on nuclear-allowed Bragg peaks typical of Q=0 magnetic structures. Fig. 1 a) and b) show the (111) Bragg peak at three temperatures: T = 0.1 K (below T C ), T=0.7 K (above T C ), and T=8 K, which is above both the T MF calculated for Yb 2 Ti 2 O 7 on the basis of its microscopic spin Hamiltonian, as well above the higher temperature "hump" in its C P near ⇠ 2 K. Fig. 1 c) shows the corresponding elastic scattering at the (113) Bragg position at T=0.1 K and T=8 K. Elastic scattering at both wave vectors is dominated by nuclear contributions, but a weak temperature dependent magnetic contribution is identified as can be seen in Fig. 1 b) and d) which show T=0.1 K T=0.7 K and T=0.7 K T=8 K data sets for (111) and a T=0.1 8 K data set for (113), respectively. Similar elastic scattering measurements were also carried out at the (220) Bragg position. The magnetic intensities extracted from these di↵erences are listed in Table I, where we see that for all of the (111), ( 113) and (222) Bragg positions, the increase in scattering, relative to T=8 K, is 3.2(4) %, 3.3(5) %, and 3.6(6) %, respectively. Relative to T=0.7 K, the magnetic intensity at T=0.1 K is increased by only ⇠ 1.3 % at the (111) Bragg position. We also note that the di↵erences in the elastic scattering have the same Qlineshape as the original Bragg scattering, implying that the corresponding magnetic order is long range, with a correlation length exceeding 80 Å.

Ignoring for the time being the fact that the magnetic order parameter does not go to zero above T C =0.26 K, we can nonetheless refine a magnetic structure based on the relative intensities of the di↵erences in Bragg scattering between T=0.1 K and 8 K. This di↵erence is plotted as a function of |Q| in Fig. 2 a), along with the best fit to a model for a canted or splayed ferromagnet of the form depicted in the inset of Fig. 2 b). This generalized structure has the Yb moments along a [100] direction, but also allows for a canting angle, ✓, which can be either towards (+) or away from (-) the local [111] direction, that is the direction pointing into or out of the tetrahedron. A collinear [100] ferromagnet would correspond to a canting angle of zero, while the ordered spin ice ground state, in which all moments point directly into or out of the tetrahedra, would correspond to ✓=54.7 . A negative canting angle gives an XY splayed ferromagnet. The best fit, shown as the solid line in Fig. 2 a), corresponds to a splayed ice-like ferromagnetic ground state with a positive canting angle of 14 ± 5 , and an ordered moment (relative to 8 K) of 0.90(9) µ B . The saturated moment determined by DC magnetization is 1.75 µ B which compared to the order moment we obtained gives a spin polarization of 51 %. Note that, as shown in Fig. 1 b), most of the decrease in the Bragg scattering on raising the temperature occurs above 0.7 K, more than a factor of two above T C =0.26 K. Our re-finements of these magnetic structures are summarized in Table I, where we have calculated the expected increases in Bragg scattering at each of (111), ( 113) and ( 222) for two di↵erent Q=0 magnetic structures. The splayed XY ferromagnet, with a negative canting angle, is known to describe Yb 2 Ti 2 O 7 's sister pyrochlore magnet Yb 2 Sn 2 O 7 (✓ ⇠ -10 ) [START_REF] Yaouanc | [END_REF] , while the splayed spin ice structure has been reported in some studies of single crystal Yb 2 Ti 2 O 7 but with a smaller splay angle (✓ ⇠ 1 ) 6,8 . Fig. 2 b) shows how the goodness of fit parameter ( 2) vary as a function of splay angle for this model of splayed or canted ferromagnets. Fig. 3 shows the temperature dependence of the (111) Bragg intensity from the stoichiometric powder sample of Yb 2 Ti 2 O 7 , relative to T=0.7 K. This data set was collected while warming from 100mK to 700mK with a warming rate of 1 mK/min. For reference, we overploted the elastic scattering perviously measured near the (111) position in single crystal Yb 2 Ti 2 O 7 (note: both curves are scaled to their intensities at 100 mK, after subtracting their intensities at 700 mK). It is clear that these temperature dependencies mirror each other. We also included C P as measured on a powder sample of Yb 2 Ti 2 O 7 with T C =0.26 K. One can see that these magnetic intensities do not resemble a conventional order parameter with T C =0.26 K. The elastic intensities are approximately constant up to ⇠ 0.35 K, and then begin to decrease. The inelastic spectral weight shown in Fig. 4 is known to be magnetic in origin from its previously determined field dependence, and from its temperature dependence to be discussed shortly. We see that S(| Q|, E) in stoichiometric Yb 2 Ti 2 O 7 at T=0.1 K is largely characterized by a continuum of scattering below an upper band edge of ⇠ 1 meV, and it is gapless on a energy scale <0.09 meV at all wave vectors measured. The inelastic scattering has some weak structure to it, resembling overdamped ferromagnetic spin waves which disperse as 12 . This calculation successfully accounts for the spin wave spectrum in the high magnetic field, polarized state at all wavevectors, but does not resemble the inelastic neutron scattering seen in Fig. 4. In particular the measured inelastic magnetic scattering is gapless at all wavevectors, while the calculated S(| Q|, E), shown here, possesses a minimum gap of ⇠ 0.25 meV.

h! ⇠ Q 2 and whose intensity peaks at small Q. This continuum spin excitation spectrum is very di↵erent from that expected for spin waves within the ferromagnetically ordered state predicted by mean field theory on the basis of the anisotropic exchange Hamiltonian determined previously from the high field, polarized state. Fig. 5 shows the powder average of this theoretical spin wave spectrum broadened by the instrumental resolution, and one clearly expects sharp and dispersing excitations. The predicted spin wave spectrum is gapped at all wave vectors, with a minimum gap of ⇠ 0.25 meV at Q=0. Such a gap would be easily observed with our inelastic measurements using DCS.

While the spin excitation spectrum in stoichiometric Yb 2 Ti 2 O 7 is exotic, and quite distinct from the expectations of anisotropic spin wave theory, comparison to the measured spectrum in lightly stu↵ed CSC Yb 2+x Ti 2 x O 7+y with x=0.046, in Fig. 4 b), and to single crystal Yb 2 Ti 2 O 7 , in Fig. 4 c) which is also likely lightly stu↵ed, shows that is not particularly sensitive to weak disorder. The upper band edge of the continuum slightly softens in the lightly stu↵ed CSC sample and single crystal, and the vestiges of the overdamped ferromagnetic spin waves are not as clear compared with the stoichiometric Yb 2 Ti 2 O 7 in Fig. 4 a). However, the continuum nature of the magnetic scattering and its lack of a gap at any wave vector are common to all three samples. lished 37 , albeit using a high temperature subtraction, as opposed to the empty can subtraction we employ here. Yb 2 Sn 2 O 7 cannot be grown as a single crystal due to the volatility of the SnO 2 starting materials, which also likely implies some level of defects in its powders. As seen on Fig. 6 f) -g), the shoulders are considerably more rounded in the lightly stu↵ed CSC sample, indicating that the vestiges of these overdamped excitations are even further overdamped in the presence of weak disorder.

The energy cuts through S(| Q|, E) in Fig. 6, also shows the temperature evolution of this inelastic scattering from T=0.1 K, well below T C =0.26 K, to T=8 K, above T MF and the "hump" in C P near 2 K. This temperature dependence is very similar for both the stoichiometric Yb 2 Ti 2 O 7 and the lightly stu↵ed CSC powder sample, so we focus our discussion on the stoichiometric Yb 2 Ti 2 O 7 energy cuts of S(| Q|, E) shown on the top panels of Fig. 6. It is worth noting that the temperature evolution of S(| Q|, E) on the negative energy side, shows clear evolution of the scattering from T=0.8 to T=0.3 K, implying that the stoichiometric Yb 2 Ti 2 O 7 was equilibrated down to at least 0.3 K. While not easily visible in Fig. 6, there is a continued decrease in the inelastic scattering near -0.1 meV from T=0.3 K to 0.1 K, strongly suggesting that the system equilibrated at all temperatures measured.

The most striking feature of the temperature dependence of S(| Q|, E) in Yb 2 Ti 2 O 7 is that there is none for temperatures less than ⇠ 2.5 K. That is, the gapless, continuum form of S(| Q|, E) in stoichiometric Yb 2 Ti 2 O 7 is maintained through its large C P anomaly at T C =0.26 K, up to 10 times this temperature. Changes in S(| Q|, E) only appear in the temperature range from 2.5 K to 8 K, consistent with a temperature evolution on the scale of the calculated T MF ⇠ 3 K, or the high temperature "hump" observed in T C near 2 K. Very similar temperature dependence of S(| Q|, E) is observed for the lightly stu↵ed CSC sample, as shown in the bottom panels of Fig. 6.

IV. CONCLUSIONS

New neutron scattering measurements on a stoichiometric powder sample of Yb 2 Ti 2 O 7 reveal low temperature magnetic Bragg peaks, overlapping with nuclearallowed Bragg peaks. This elastic scattering is interpreted in terms of a long range ordered, splayed spin ice static structure with a correlation length exceeding 80 Å and an ordered moment, relative to 8 K, of 0.90(9) µ B . However, the temperature dependence of the elastic Bragg scattering does not correlate with the T C =0.26 K expected for such powder samples. Rather, this elastic scattering begins to dissipate above ⇠ 350 mK and shows continued decrease above 700 mK.

The zero field magnetic inelastic spectrum of stoichiometric Yb 2 Ti 2 O 7 is exotic. It shows a gapless, continuum form, at T=0.1 K, well below T C =0.26 K, with an upper band edge of ⇠ 1 meV. No spin gap is observed at any wave vector to an upper limit of 0.09 meV, in contrast to the expectations of anisotropic spin wave theory within the ferromagnetically ordered state predicted from the mean field model using the spin Hamiltonian derived from high field spin wave measurements. The continuum inelastic spectrum shows some weak structure, with vestiges of overdamped ferromagnetic spin waves present at small Q. The inelastic magnetic spectrum in stoichiometric Yb 2 Ti 2 O 7 shows little or no temperature dependence up to temperatures greater than 2.5 K, much larger than T C , and on the order of T MF ⇠ 3 K, and the temperature characterizing the high temperature "hump" in C P near 2 K. The form of the magnetic inelastic scattering, its continuum nature and temperature dependence, is only slightly influenced by weak stu ng, as is known to characterize the non-stoichiometric CSC powder sample and single crystal samples that were also measured.

FIG. 1 :

 1 FIG. 1: The temperature dependence of the (111) and (113) Bragg peaks.-Panels a) and c) show the elastically scattered neutron intensity as a function of the momentum transfer | Q| around the (111) and (113) Bragg peaks respectively. Panels b) and d) shows the subtraction between the di↵erent | Q| cuts at di↵erent temperatures. A clear, | Q|-resolution-limited increase of the elastic scattering at these two Bragg peaks is observed and is representative of the data used to model the magnetic structure of stoichiometric Yb2Ti2O7. Error bars represent one standard deviation.

FIG. 2 :

 2 FIG. 2: Intensity of elastic magnetic scattering vs. Q and the corresponding fit to the splayed Ice ferromagnet -a) the best fit of the model for a splayed Ice ferromagnet to the di↵erence of elastic neutron scattering intensity at 100 mK and 8 K (i.e., the magnetic elastic scattering) is shown. The model employed is a splayed ferromagnet (a Q=0 structure with a net moment along the cubic axes) on the pyrochlore lattice, with splay angle of ✓ = 14(5) and an ordered moment of 0.90(9)µB. Note: the error bars are not shown in this figure for clarity, but are taken into account as weights in the least squared fitting of the model. b) the goodness-of-fit parameter, 2 , as a function of splay angle is shown. The inset to b) shows an illustration of the best fit magnetic structure model on a pair of tetrahedra.

FIG. 3 :

 3 FIG. 3:The temperature dependence of the elastic scattered intensity at | Q|=(111).-The scattered elastic neutron intensity, relative to T=700mK at | Q|=(111) shows similar temperature dependence for both a previously measured single crystal (data taken from Ross et al.14 ) and for the stoichiometric powder sample of Yb2Ti2O7 (this work). With increasing temperature, the elastic magnetic scattering begins to fall around T = 0.35 K, and decreases approximately linearly above this temperature. The specific heat anomaly TC ⇠ 0.26 K for a representative powder sample of Yb2Ti2O7 (data taken from ref Ross et al.28 ) is shown for reference, and reveals the relative insensitivity of the temperature dependence of the elastic scattering to the specific heat anomaly.

FIG. 4 :

 4 FIG. 4: S(| Q|, E) as measured for stoichiometric and weakly stu↵ed Yb2Ti2O7 with comparison to Yb2Sn2O7.a) and b) show S(| Q|, E) measured on the stoichiometric Yb2Ti2O7 powder and the lightly stu↵ed CSC powder of Yb2+xTi2 xO7+y with x=0.046, respectively, both at T=0.1 K. c) shows the corresponding S(| Q|, E) measurement on single crystal Yb2Ti2O7, also at T=0.1 K. Panel d) shows S(| Q|, E) measured on powder Yb2Sn2O7 at T=0.1 K, by Dun et al. 37 . All four data sets have had an empty can background data set subtracted from them, and have had intensities scaled such that the scattering at | Q| = 1 Å 1 and E=0.5 meV saturates the colour scale.

FIG. 5 :

 5 FIG. 5: Energy vs. | Q| slice of the mean-field calculation of S(| Q|, E) for Yb2Ti2O7.-S(| Q|, E) has been computed using an anisotropic spin 1/2 exchange Hamiltonian with the exchange parameters determined by Ross et al.12 . This calculation successfully accounts for the spin wave spectrum in the high magnetic field, polarized state at all wavevectors, but does not resemble the inelastic neutron scattering seen in Fig.4. In particular the measured inelastic magnetic scattering is gapless at all wavevectors, while the calculated S(| Q|, E), shown here, possesses a minimum gap of ⇠ 0.25 meV.

FIG. 6 :

 6 FIG. 6: Energy cuts of S(| Q|, E) for the stoichiometric powder sample of Yb2Ti2O7 and the CSC Yb2+xTi2 xO7+y with x=0.046.-Top panels : These 4 di↵erent panels show cuts through S(| Q|, E) of Yb2Ti2O7, taken from Fig. 4 a), for di↵erent ranges of | Q| and at di↵erent temperatures. Four di↵erent integrations in | Q| going from smaller | Q| in a) to larger | Q| in d) have been applied, as indicated in the individual panels. No di↵erences in the scattering is observed from T=2.5 K down to T=0.1 K. Bottom panels : Same data and -Q-integrated cuts as in a) -d) but for Yb2+xTi2 xO7+y with x=0.046 taken from Fig. 4 b).

  Energy cuts through the S(| Q|, E) data sets for stoichiometric powder Yb 2 Ti 2 O 7 and lightly stu↵ed CSC with approximate composition Yb 2+x Ti 2 x O 7+y and x=0.046 are shown in Figs.6. While di↵erences between the four data sets of Fig.4can be seen at Q  0.4 Å 1 , this low-Q region corresponds to the edge of the neutron detector bank closest to the incident beam, and is thus the part of the data set that is most sensitive to the precise details of the background subtraction. This low-Q region of scattering is thus avoided in the following quantitative analysis shown in the energy scans of Fig.6. These energy scans employ di↵erent Q-integrations, going from smaller Q to larger Q in panels Fig.6 a) -d) for Yb 2 Ti 2 O 7 and Fig.6 e) -h) for Yb 2+x Ti 2 x O 7+y and x=0.046. Consistent with the detailed S(| Q|, E) maps, we see that pronounced inelastic shoulders are observed at intermediate Q's, as seen in Figs.6 b) and c), and these are the vestiges of the ferromagnetic spin waves discussed earlier. Well defined ferromagnetic spin waves would show inelastic peaks which would disperse as E ⇠ Q 2 .

  and which are known to be stoichiometric Yb 2 Ti 2 O 7 and the CSC with composition Yb 2+x Ti 2 x O 7 y with x=0.046. In addition, we report a comparison between the spin dynamics measured on a stoichiometric powder of Yb 2 Ti 2 O 7 , the CSC powder, a single crystal of Yb 2 Ti 2 O 7 grown by the floating zone technique, and a powder sample of Yb 2 Ti 2 O 7 's sister pyrochlore magnet, Yb 2 Sn 2 O 7 .

Acknowledgments

We wish to thank Jan Kycia for collaborations and discussion related to heat capacity measurements. We thank C. Wiebe and H. Zhou for making their data available to this work. The neutron scattering data were reduced using Mantid 38 and analyzed using DAVE software package 39 . Research using ORNL's Spallation Neutron Source was sponsored by the Scientific User Facilities Division, O ce of Basic Energy Sciences, U.S. Department of Energy. The NIST Center for Neutron Research is supported in part by the National Science Foundation under Agreement No. DMR-094472. Works at McMaster University was supported by the National Sciences and Engineering Research Council of Canada (NSERC).