D D Maharaj
email: maharadd@mcmaster.ca

G Sala

M B Stone

Edwin Kermarrec

C Ritter

F Fauth

C A Marjerrison

J E Greedan

A Paramekanti

B D Gaulin

Octupolar vs Néel Order in Cubic 5d 2 double perovskites

Keywords: numbers: 75.25.j, 75.40.Gb, 75.70.Tj

come

Introduction -Ordered double perovskite (DP) magnets, with the chemical formula A 2 BB O 6 , provide a fascinating avenue for the study of geometric frustration and its interplay with strong spin-orbit coupling (SOC) [1]. Here, B and B sublattices individually form an FCC lattice of edge-sharing tetrahedra, an archetype for geometric frustration in three dimensions. Furthermore, the flexibility of the DP lattice to host heavy ions at the B site allows the study of spin-orbit driven physics, as the strength of SOC scales ∼ Z 2 , where Z is the atomic number of the magnetic ion. This interplay of SOC and frustration has been predicted to yield exotic ground states in a host of these DP systems [1][2][3][4][5].

Strong SOC splits the t 2g levels associated with the magnetic B in an octahedral crystal field, resulting in a four-fold degenerate, j = 3/2, ground state and a doubly degenerate, j = 1/2, excited state. Famously, for a d 5 electronic configuration, as occurs for Ir 4+ or Ru 3+ , this results in a single hole in a j = 1/2 state, leading to extreme quantum magnetism, and bond-dependent Kitaev interactions in appropriate geometries [6][7][8][9][10][11][12].

In the case of d 2 ions on the B site, the subject of this Letter, the combination of SOC and Hund's coupling results in an effective J = 2 angular momentum. Studies of such interacting d 2 ions on the FCC lattice have highlighted the importance of orbital repulsion, in addition to the conventional magnetic exchange. This leads to complex multipolar interactions, and large regimes of quadrupole order in the predicted phase diagram [2][3][4][5].

Here we report new magnetic neutron powder diffraction (NPD), inelastic neutron scattering (INS), and high resolution X-ray diffraction (XRD) results on three cubic DPs with a 5d 2 electronic configuration at the B site: Ba 2 M OsO 6 , with M = Zn, Mg, Ca (respectively referred to henceforth as BZO, BMO, and BCO). All three display clear thermodynamic signatures of a phase transition [13][14][15] at relatively high temperatures ∼ 30-50 K, which is associated with time-reversal symmetry breaking based on oscillations observed in zero field muon spin relaxation (µSR) experiments [15]. Our INS results show strong, gapped, magnetic spectral weight at wavevectors typical of type I antiferromagnetic (AF) order, but we find no indication of an ordered moment in the diffraction data, allowing us to place upper limits of between 0.13-0.06 µ B per B site. Furthermore, our NPD and XRD results show no deviation from cubic symmetry expected from quadrupolar ordering. We present arguments showing that these results may be understood as arising from octupolar order in these cubic 5d 2 DP materials. Multipolar ordered phases have been extensively studied in f -electron compounds [16]. Examples include NpO 2 [17][18][19][20], where experiments have been interpreted in terms of a primary rank-5 magnetic multipolar order (triakontadipoles) driving secondary quadrupolar order, the "hidden order" state of URu 2 Si 2 which has been proposed to host hexadecapolar order [21], and recent discoveries of quadrupolar and octupolar orders in PrX 2 Al 20 (X = Ti, V) [22][23][24]. In stark contrast, multipolar orders in d-electron systems are rare [25][26][27][28][29][30]; our work appears to be the first report of octupolar order.

BZO, BMO and BCO have been previously studied in powder form. In all three materials, neutron and x-ray diffraction confirm that they remain in the cubic F m 3m space group down to the lowest temperature. They all display Curie-Weiss-like magnetic susceptibilities (χ) at high temperatures, with large AF Curie-Weiss constants (Θ CW ∼ 130 K), and anomalies at T * in the form of a splitting between field-cooled and zero-field cooled results. They all exhibit peaks in their heat capacity, or in the related measure d(χT )/dT , at T * ∼ 50 K (BMO, BCO) or T * ∼30 K (BZO), indicating a phase transition [13,15]. These findings are summarized in Table I.

The entropy released up to ∼ 2T * in all three materials appears to be ∼ R ln(2) per mole, much smaller than R ln(5) expected for a J = 2 moment [13][14][15]. This points to part of the entropy being quenched at T T * . This is in contrast to the ∼ R ln(5) entropy released up to ∼ 2T N for the tetragonal counterpart Sr 2 MgOsO 6 , which has a high Néel ordering temperature T N ∼ 100 K [31].

These three cubic samples have also been previously studied using µSR techniques [13,15], and it is primarily on the basis of these zero longitudinal field (ZLF) µSR oscillations for T < T * , indicative of a time-reversal broken state, that the transition at T * was associated with AF order. However, no magnetic neutron diffraction peaks could be identified in this earlier study at low temper- [13,15]. θCW is the Curie-Weiss temperature extracted from high temperature susceptibility data [13,15]. µ ord is the upper limit on the ordered dipolar moment associated with type-I AF order, as determined from neutron diffraction in this work.

atures, with a sensitivity to ordered moment of ∼ 0.7 µ B . In the present work, we significantly improve on this bound, still finding no evidence of magnetic Bragg peaks.

The corresponding 5d 3 osmium-based DP magnets, both cubic Ba 2 YOsO 6 and monoclinic Sr 2 ScOsO 6 and La 2 LiOsO 6 , show clear Néel transitions to AF ordered states, with large ordered moments of ∼ 1.7µ B [14,[32][33][34][35]. These observed ordered moments are reduced from the 3µ B value expected for an orbitally-quenched moment, pointing to strong SOC effects, or covalency, or both. Nonetheless magnetic Bragg scattering at the (100) and (110) positions is easily observed, along with strong, gapped inelastic magnetic scattering centred at these two ordering wavevectors. Previously studied 5d 2 DPs such as monoclinic Sr 2 MgOsO 6 and cubic Ba 2 LuReO 6 (with Re 5+ ) also show transitions to Type I AF order, as seen via neutron diffraction, albeit with much smaller ordered moments, 0.6(2) and 0.34(4) µ B , respectively [31,36].

Below we present our experimental findings on powder samples of the cubic systems, BZO, BMO and BCO. Details of experimental and analysis methods can be found in the Supplemental Material (SM) [37]. Our new NPD measurements on D20 [38] at the Institut Laue Langevin have ∼ 10 to 20 times more sensitivity to magnetic Bragg scattering as compared with previous NPD measurements taken at the C2 instrument of the Chalk River Laboratories. No magnetic Bragg scattering is observed at 10 K, factors of 3-5 below T * for any of these materials. We do however observe gapped, inelastic magnetic spectral weight centred on wavevectors characteristic of type I AF order, leading us to conclude that the dominant broken symmetry below T * in these three cubic DP d 2 magnets must involve multipolar ordered phases. Results -Time-of-flight INS measurements which were performed at SEQUOIA [39] The data sets for all three samples in Fig. 2 are similar, with gapped magnetic spectral weight at low |Q|'s, typical of the 100 (0.78 Å-1 ) and 110 (1.1 Å-1 ) Bragg positions. The full bandwidth of the magnetic excitation spectrum appears to be ∼ 6 meV. From Fig. 2 b),c), e), and f), this magnetic spectral weight overlaps in energy with strong phonon scattering near ∼ 18 meV and 14 meV for BMO and BCO respectively. Even though our low |Q|-integration favours magnetic scattering at the expense of scattering from phonons, whose intensity tends to go like |Q| 2 , we still pick up a sizeable contribution from this high phonon density of states (DOS), especially at high temperatures where the thermal population of the phonons is large. A redshift in the peak of the phonon DOS from ∼ 17 meV in BMO to ∼ 14 meV for BCO is observed. This is expected, as Ca 2+ is isoelectronic to Mg 2+ but heavier, hence, all other factors being equal, BCO will display lower frequency phonons. While the Zn 2+ in BZO is heavier still than Ca 2+ , it is not isoelectronic, instead possessing a filled 3d shell. This might lead to its higher energy phonon.

As the high phonon DOS is well separated from the magnetic spectral weight in BZO, shown in Figs. 2(a) and 2(d), this is where the nature of the gapped magnetic scattering can be most easily appreciated. The energy cuts in Fig. 2(d) clearly show a well developed gap of ∼ 10 meV and a bandwidth of ∼ 6 meV. This structure collapses by 25 K, where T * = 30 K for BZO, at which point the gap has largely filled in and only a vestige of an overdamped spin excitation at ∼ 10 meV remains. This is very similar to what occurs in the d 3 DPs on the approach to their T N s, except that there is no obvious temperature dependent Bragg scattering at the 100 or 110 positions, as would be expected for type I AF order.

The absence of evidence for magnetic Bragg scattering is seen in Fig. 3. Fig. 3(a) shows neutron diffraction data taken at T = 10 K, well below T * = 30 K in BZO, using the D20 diffractometer at the Institut Laue-Langevin [38]. This data and the corresponding NPD data on BMO and BCO refine in the cubic F m 3m space group at all temperatures measured. Figure 3 b),c) and d) then show a subtraction of high temperature (50 K for BZO; 70 K for BMO and BCO) data sets from low temperature data sets for each of BZO, BCO, and BMO, respectively. A calculated neutron diffraction profile appropriate for a type I AF structure below T * is shown as the red line in Fig. 4 b),c) and d), where the assumed ordered moment in the calculation is 0.06µ B for BZO (b), 0.11µ B for BMO (c), and 0.13µ B for BCO (d). Taking the case where the evidence against long range magnetic order below T * is most stringent, BZO, we can eliminate conventional type I AF order of magnetic dipoles with an ordered moment greater than ∼ 0.06µ B . This upper limit for magnetic dipole order is a factor of ∼ 12 more stringent than previous limits on magnetic Bragg scattering for this family of cubic DP materials. This upper bound for µ ord in BCO is ∼ 35% lower than the value, 0.2µ B , which was previously extracted from a comparison of the µSR internal fields of BCO and Ba 2 YRuO 6 [13,40].

Competing multipolar orders. -Our study shows all or most of the static 5d 2 moment associated with Os 6+ in BZO, BMO and BCO is not visible to neutron diffraction below the transition at T * . Nonetheless, strong inelastic magnetic scattering is easily observed at all temperatures, and it is most clearly gapped at low temperature. One possible scenario to explain these results is that the ground state has dominant quadrupolar ordering, accompanied by weak dipolar ordering, as proposed in certain models [2,3,5]. In this scenario, the observed entropy is most naturally accounted for by assuming a twostep transition, with quadrupolar ordering at T T * quadrupolar ordered state would lower the crystal symmetry due to orbital ordering, which is at odds with our high resolution NPD data shown for BCO in Fig. 4(a). We have carried out even higher-resolution synchrotron X-ray diffraction (XRD) measurements on BCO, the family member which best exhibits undamped ZF µSR oscillations. The sensitivity of these measurements to possible weak splittings of the cubic Bragg peaks is ∼ 10 times greater than the NPD measurements, as the inset to Fig. 4(a) demonstrates. These XRD results, in Figs. 4 (b)-(d), show no splitting of the cubic Bragg peaks, confirming that BCO remains cubic even for T T * , inconsistent with quadrupolar ordering.

Here, we propose a distinct scenario -an octupolar ordered ground state -that provides the most promising vehicle to explain all the salient observations. For an effective J = 2 moment, a residual octahedral crystal field Hamiltonian is H CEF = -V eff (O 40 + 5O 44 ), where the

O 44 = 1 2 (J 4 + + J 4 -). (2)
For V eff > 0, this results in a non-Kramers ground state doublet, and an excited triplet with a gap ∆ = 120V eff . This naturally accounts for partial entropy quenching for T ∆, without a phase transition, with the residual R ln The ground state manifold thus has vanishing matrix elements for the dipole operators J, precluding magnetic dipole ordering. However, J can induce transitions into the excited triplet, thus accounting for the spin gap seen in INS. Defining pseudospin-1/2 operators τ within the ground state doublet, the quadrupolar operators (J 2

x -J 2 y ) ≡ 2 √ 3τ x , (3J 2 z -J 2 ) ≡ -6τ z , while the octupolar operator J x J y J z ≡ -√ 3τ y (where the overline denotes symmetrization). Thus, the ground state doublet can lead to broken time-reversal symmetry below T * , while preserving cubic symmetry, if τ y = 0. Further details are given in the SM [37] and in Ref. [41].

To conclude, the low temperature phases of the cubic 5d 2 DPs BZO, BMO, and BCO are best described as arising from a ground state non-Kramers doublet with octupolar symmetry breaking. This exotic ground state appears to require the perfect FCC structure as non-cubic d 2 DPs display more conventional AF ground state selection. Tools such as magnetostriction, which has recently been proposed as a useful way to detect octupolar ordering [42], or other experiments, could be needed to provide a smoking gun signature of octupolar symmetry breaking in these 5d 2 materials. Details relevant to both experimental and theoretical aspects of the main paper are discussed in this supplemental material. Experimental methods for the time-offlight inelastic neutron scattering (INS), neutron powder diffraction (NPD) and high angular resolution x-ray powder diffraction (XRD) measurements are fully explained in section I., A, B and C. An outline of the calculations performed in order to estimate the upper bound on the ordered magnetic moment in Ba 2 CaOsO 6 , Ba 2 MgOsO 6 , and Ba 2 ZnOsO 6 is provided in section I B. part 2. Section II provides further detail for the theoertical argumemnt which determines the ground and excited state wavefunctions for the d 2 ions in Ba 2 CaOsO 6 , Ba 2 MgOsO 6 , and Ba 2 ZnOsO 6 .

I. EXPERIMENT DETAILS A. Time-of-flight Inelastic Neutron Scattering

Methods -The INS measurements were conducted using the direct geometry chopper spectrometer SEQUOIA, which is located at the Spallation Neutron Source 1 of Oak Ridge National Laboratory. Powder samples weighing 8 grams of Ba 2 MgOsO 6 , Ba 2 ZnOsO 6 , and Ba 2 CaOsO 6 were packed tightly in aluminum foil and loaded in identical annular cans, 3 cm in diameter. An identical empty can was also prepared for background measurements. The sample cans and empty can were sealed in a glove box containing helium gas to improve thermalization of samples at low temperatures. The cans were loaded on a three-sample carousel mounted in a closed-cycle refrigerator with a base temperature of 7 K. Inelastic neutron scattering (INS) measurements were carried out on Ba 2 MgOsO 6 and Ba 2 ZnOsO 6 at a variety of temperatures above and below T * , using an incident energy of E i = 60 meV while in the case of Ba 2 CaOsO 6 , the incident energy chosen was E i = 90 meV. The chopper settings utilized for both incident energies were the same. The chopper settings selected for the fast neutron chopper (T 0 ) and the high flux fermi chopper (F C 1 ) were T 0 = 240 Hz, and F C 1 = 120 Hz, respectively. Empty can background data sets were collected at base temperature, and T = 200 K, and correspondingly subtracted from the data sets. The data sets were reduced using Mantid 2 and analyzed using neutron scattering software DAVE 3 .

B. High Intensity Neutron Powder Diffraction

Methods

NPD data were collected on the two-axis high intensity diffractometer D20 (Institut Laue Langevin, Grenoble, France) 4 using a wavelength of λ = 2.41 Å selected by a Graphite monochromator using a take-off angle of 42 • . The powder samples of Ba 2 CaOsO 6 and Ba 2 MgOsO 6 were those utilized for the INS measurements while a new sample of Ba 2 ZnOsO 6 was synthesized for the NPD measurements. Measurements were taken at a base temperature of 10 K for 10 hrs on Ba 2 ZnOsO 6 for 11 hrs on Ba 2 MgOsO 6 and at 1.5 K for 11 hrs on Ba 2 CaOsO 6 . A higher base temperature was selected for the former compounds to avoid probing any features which could arise from the ferromagnetic ordering of the impurity phase Ba 11 Os 4 O 24 present at low levels in these compounds. High temperature data sets with the same statistics were taken at T = 50 K for Ba 2 ZnOsO 6 and at T = 70 K for both Ba 2 MgOsO 6 and Ba 2 CaOsO 6 . Data analysis was performed using the Rietveld refinement program FULLPROF 5 .

Magnetic Neutron Profile Calculations

The NPD measurements performed at D20 show no evidence for magnetic order of the Os 6+ ions in Ba 2 CaOsO 6 , Ba 2 MgOsO 6 and Ba 2 ZnOsO 6 . A limit for the upper bound on the magnetic moment potentially arXiv:1909.03113v3 [cond-mat.str-el] 19 Sep 2019 present at low temperatures in these materials can still be performed, and used for comparison with other double perovskites. This involves assuming a magnetic structure, and we employed a simple type 1 AF structure, as is observed in a variety of d 3 double perovskites. FULL-PROF was utilized to calculate the diffraction pattern due to the assumed magnetic structure alone, for each compound. Each calculation was performed utilizing the lattice parameters determined from fits to the low temperature data sets for each compound. The magnetic form factor for Os 6+ was also included in this calculation, using parameters extracted from previous work done by Kobayashi et al 6 . The result from this calculation was then compared with the plots showing the difference between the high temperature data set from the low temperature data set. This calculation produced a strong magnetic Bragg peak near ∼ 0.76 Å-1 , corresponding to the type I AF structure, whose intensity depended on the size of the ordered moment in the structure. The ordered moment was then adjusted until this magnetic peak was just observable above background, thus setting the upper bound to the size of a possible ordered magnetic moment in these materials. Upper bounds for the ordered Os 6+ moments of 0.13µ B , 0.11µ B , and 0.06µ B were obtained for Ba 2 CaOsO 6 , Ba 2 MgOsO 6 and Ba 2 ZnOsO 6 , respectively.

C. High Angular Resolution X-ray Diffraction

Methods -Synchrotron experiments were performed on the powder diffraction station of the BL04-MSPD beamline 7 of the ALBA Synchrotron Light facility (Barcelona, Spain). Data were collected in transmission mode using the 13 channel multi-analyzer (MAD) setup (220 Bragg reflection of silicon crystal) which offers the highest possible instrumental angular resolution (∆2θ = 0.005 • ). In order to minimize absorption, the patterns were recorded at a short wavelength, λ = 0.31713 Å, with the sample filled in a 0.5 mm diameter borosilicate capillary. Data were recorded in the temperature range between 20 K and 250 K using the liquid Helium flow cryostat Dynaflow 8 .

II. THEORY A. Single-site model, multipole moments

We begin with an effective J = 2 moment and incorporate the impact of a residual octahedral crystal field via

H CEF = -V eff (O 40 + 5O 44 ), (1)
where the Steven's operators

O 40 = 35J 4 z -(30J(J + 1) -25)J 2 z + 3J 2 (J + 1) 2 -6J(J + 1) (2)
O 44 = 1 2 (J 4 + + J 4 -), (3)
For V eff > 0, we find that the J = 2 multiplet splits into a ground doublet and an excited triplet with a gap ∆ = 120V eff . Choosing V eff = 0.2 meV, for example, leads to a doublet-triplet gap ∆ ∼ 24 meV. The ground state wavefunctions in the |m basis are

|ψ g,↑ = |0 (4) |ψ g,↓ = 1 √ 2 (|2 + | -2 ) (5)
while the excited state wavefunctions are given by |ψ e,± = | ± 1 ( 6)

|ψ e,0 = 1 √ 2 (|2 -| -2 ) (7)
This sequence of level splittings for the 5d 2 configuration is schematically described in Fig. 1. The rightmost level scheme results from the familiar splitting of the 5d levels in an octahedral crystal electric field (CEF). Strong spin orbit coupling (SOC) then splits the t 2g levels into an higher energy j = 1/2 doublet and a lower energy quartet, corresponding to j = 3/2. The two d electrons then occupy the j = 3/2 levels, and combine to give a J eff = 2 degree of freedom. A residual crystal field, arising from electron-interaction induced change in the CEF splitting (denoted as CEF') gives the final non-Kramers doublet ground state with octupolar and quadrupolar moments (discussed below), and an excited state triplet, some 15 to 30 meV above the octupolar doublet. It is clear that the ground state manifold has vanishing matrix elements for the dipolar vector operators J. However, J can induce transitions between the ground doublet and the excited triplet. We thus attribute the observed spin gap to this doublet-triplet gap ∆.

Defining the pseudospin-1/2 operators τ within the ground state doublet, we find that the quadrupolar operators (J 2

x -J 2 y ) ≡ 2 √ 3τ x , (3J 2 z -J 2 ) ≡ -6τ z , while the octupolar operator J x J y J z ≡ -√ 3τ y where the overline denotes symmetrization.

In the single-site limit, we can easily compute the following results for the entropy and the magnetic susceptibility. Here, we pick V 0 = 0.2 meV as above, and include a local octupolar symmetry breaking field, in the spirit of mean field theory, via H = H CEFφ(T ) J x J y J z .

(8)

As an illustrative example, we choose the symmetry breaking field φ(T ) to be given by the simple form φ(T < T * ) = T * 1 -T /T * and φ(T > T * ) = 0.

The entropy, shown in Fig. 2, starts at R ln(5) at high temperature, before decreasing to R ln(5) for T 100 K. This residual low temperature entropy of near R ln(2) is quenched by ordering of the doublet states induced by intersite interactions for T < T * . The inverse spin susceptibility, in units where a free spin-S leads to the Curie susceptibility S(S + 1)/3T , is shown in Fig. 3. It is clear that the spin gap induces an effective 'antiferromagnetic Curie-Weiss' type behavior, with an intercept at T ≈ -30 K. The larger Curie-Weiss temperature Θ CW ≈ -150 K inferred from experiments must thus stem from residual intersite interactions.

FIG. 1 .

 1 FIG. 1. (a) Structure of cubic double perovskites with space group F m 3m. (b) The frustrated FCC network of edgesharing tetrahedra of the B site magnetic moments that are generated in the cubic Ba2BB O6 structure.

  are shown in Fig. 2. Panels (a)-(c) show the INS spectra well below (top panel) and above T * for BZO, BMO, and BCO respectively. Panels (d)-(f) show cuts through this data as a function of energy, integrating all |Q| < 1.15 Å-1 and as a function of temperature, again for BZO, BMO and BCO respectively.

FIG. 2 .

 2 FIG. 2. (a) -(c): Neutron scattering intensity contour plots for BZO, BMO, and BCO shown as a function of energy transfer, E and momentum transfer |Q| at base temperature (top) and at T > T * (bottom), respectively. Below T * , clear gapped magnetic inelastic spectral weight develops around (100) and (110) wavevectors (∼ 0.78 Å-1 ) and 110 (1.1 Å-1 ) in each case. (d) and (e): Low |Q| integrated cuts of the neutron scattering intensity as a function of energy transfer E as a function of temperature for BZO, BMO, and BCO, respectively. A gap in the magnetic excitation spectum is clearly revealed for each compound for T < T * .

FIG. 4

 4 FIG. 4. a) The NPD profile for BCO is shown at T = 1.5 K in the main panel, while the inset shows a comparison of neutron vs synchrotron x-ray diffraction data taken on BCO at 20 K. Panels b) and c) show synchrotron x-ray scattering data on BCO at T = 20 K (b), and T = 250 K (c), along with corresponding cubic structural refinements, in red.

2 z

 2 Steven's operators (dropping constant terms) are O 40 = 35J 4 z -(30J(J + 1) -25)J

( 2 ) 1 √ 2 ( 1 √ 2 (

 21212 entropy being quenched by ordering within the doublet sector at T * . In the |J z = m basis, the ground state wavefunctions are |ψ g,↑ = |0 and |ψ g,↓ = |2 + | -2 ), with excited triplet wavefunctions |ψ e,± = | ± 1 and |ψ e,0 = |2 -| -2 ).

FIG. 1 .

 1 FIG.1. The splitting of the energy levels of the 5d 2 Os 6+ ion is schematically illustrated. The 5d levels are acted on by the octahedral crystal-electric field (CEF) of the lattice, strong spin-orbit coupling (SOC), and finally a residual cubic CEF due to electron-interaction induced change in the CEF splitting (denoted here as CEF') that results in a non-Kramers doublet ground state with quadrupolar and octupolar moments, and a weakly split off triplet of excited states some 15-30 meV higher in energy. The ground state and excited state wavefunctions of the d 2 ion are explicity stated in the figure.

FIG. 2 .

 2 FIG.2. Entropy of the single-site model (solid) line in units of R as a function of temperature, showing the quenching of the entropy from R ln(5) at high temperature to R ln(5) (dashed lines denote R ln(5) and R ln(2)) for T ∼ 100 K, which is further quenched by octupolar ordering at T * = 50 K.

FIG. 3 .

 3 FIG.3. Plot of the inverse susceptibility 1/χ of the singlesite model as a function of temperature, showing its apparent 'Curie-Weiss' like behavior at high temperature (indicated by the dashed fit) and its eventual saturation at low temperature, arising from the spin-gap, with the kink denoting the octupolar transition.

ACKNOWLEDGMENTS

This work was supported by the Natural Sciences and Engineering Research Council of Canada. It was also supported in part by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics. We also acknowledge the hospitality of the Telluride Science Research Center. We gratefully acknowledge useful conversations with G. M. Luke and G. Chen. We are very grateful for the instrument and sample environment support provided during our inelastic neutron scattering measurements at SE-QUOIA. The experiments which were performed at the Spallation Neutron Source at Oak Ridge National Laboratory was sponsored by the US Department of Energy, Office of the Basic Energy Sciences, Scientific User Facilities Division. The authors would also like to acknowledge ILL for beam time allocation experiment code 5-31-2577, doi:10.5291/ILL-DATA.5-31-2577. We acknowledge the BL04-MSPD staff of ALBA for the x-ray synchrotron powder diffraction data collection


J Gaudet

A M Hallas

D D Maharaj

C R C Buhariwalla

E Kermarrec

N P Butch

T J S Munsie

H A Dabkowska

G M Luke

B D Gaulin

Magnetic dilution and domain selection in the XY pyrochlore antiferromagnet Er 2 Ti 2 O 7

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

Below TN = 1.1 K, the XY pyrochlore Er2Ti2O7 orders into a k = 0 non-collinear, antiferromagnetic structure referred to as the ψ2 state. The magnetic order in Er2Ti2O7 is known to obey conventional three dimensional (3D) percolation in the presence of magnetic dilution, and in that sense is robust to disorder. Recently, however, two theoretical studies have predicted that the ψ2 structure should be unstable to the formation of a related ψ3 magnetic structure in the presence of magnetic vacancies. To investigate these theories, we have carried out systematic elastic and inelastic neutron scattering studies of three single crystals of Er2-xYxTi2O7 with x = 0 (pure), 0.2 (10%-Y) and 0.4 (20%-Y), where magnetic Er 3+ is substituted by non-magnetic Y 3+ . We find that the ψ2 ground state of pure Er2Ti2O7 is significantly affected by magnetic dilution. The characteristic domain selection associated with the ψ2 state, and the corresponding energy gap separating ψ2 from ψ3, vanish for Y 3+ substitutions between 10%-Y and 20%-Y, far removed from the 3D percolation threshold of ∼60%-Y. The resulting ground state for Er2Ti2O7 with magnetic dilutions from 20%-Y up to the percolation threshold is naturally interpreted as a frozen mosaic of ψ2 and ψ3 domains.

The network of corner-sharing tetrahedra that make up the pyrochlore lattice is one of the canonical architectures for geometrical frustration in three dimensions [1]. In the pyrochlore magnets with chemical composition A 2 B 2 O 7 , each of the A 3+ and B 4+ sublattices independently form such a network, that can be decorated by many ions. The rare earth titanate family, R 2 Ti 2 O 7 , which features a single magnetic R 3+ site and non-magnetic Ti 4+ on the B-site, has been of specific interest. This family displays a great variety of exotic magnetic ground states, including classical [2][3][4] and quantum spin ice [5][6][7][8], as well as spin liquid states [9][10][11]. Such exotic states arise from different combinations of magnetic anisotropies with differing exchange and dipolar interactions, depending on the nature of the R 3+ magnetic ion.

Much current interest is focused on the XY antiferromagnetic pyrochlore Er 2 Ti 2 O 7 , which displays an antiferromagnetic Curie-Weiss susceptibility with θ CW = -22 K [12]. Crystal field effects on Er 3+ in this environment give rise to a g-tensor with XY anisotropy [13][14][15]. In contrast to the exotic ground states displayed by other members of the R 2 Ti 2 O 7 family, Er 2 Ti 2 O 7 has a rather conventional k = 0, Γ 5 antiferromagnetic ground state below T N = 1.1 K [12,[16][17][18][19]. Γ 5 is made up of two basis vectors, referred to as ψ 2 and ψ 3 , and sophisticated magnetic crystallography was used to identify the ordered structure in Er 2 Ti 2 O 7 as ψ 2 [16]. The microscopic spin Hamiltonian for Er 2 Ti 2 O 7 was determined through measurements of the spin wave dispersions within its field polarized state [20,21]. This work, amongst others [12,22,23], has provided an explanation for the selection of ψ 2 within the Γ 5 manifold in terms of a quantum order-by-disorder mechanism. The order-by-disorder scenario has been further strengthened by works showing that thermal fluctuations also select ψ 2 over ψ 3 [24][25][26][27]. A spin wave gap of 0.053±0.006 meV has been successfully measured in Er 2 Ti 2 O 7 , consistent with the order-by-disorder mechanism [28]. Recently, there has been a proposal that the selection of the ψ 2 ordered state could also occur via higher multipolar interactions, originating from virtual crystal field transitions [21,29].

The phase transition to the ψ 2 antiferromagnetic ordered state in Er 2 Ti 2 O 7 is unusual amongst the R 2 Ti 2 O 7 family by virtue of how conventional it is. In contrast to Yb 2 Ti 2 O 7 [30][31][32] and Tb 2 Ti 2 O 7 [33,34], there appears to be no sample dependence to its low temperature phase diagram. Also, systematic studies of magnetic dilution at the rare earth site in Er 2 Ti 2 O 7 are consistent with three dimensional (3D) percolation theory [START_REF] Niven | Proc. R. Soc. A[END_REF]. It was therefore surprising that two recent, independent theoretical works predicted that, upon magnetic dilution, the ψ 2 ground state selection in Er 2 Ti 2 O 7 should be unstable to the selection of its ψ 3 partner within the Γ 5 manifold [START_REF] Maryasin | [END_REF]37].

Such a possible ψ 2 to ψ 3 phase transition upon magnetic dilution in Er 2 Ti 2 O 7 has yet to be explored experimentally and is the topic of this manuscript. The phase transition between ψ 2 and ψ 3 in Er 2 Ti 2 O 7 is predicted to occur near a critical dilution of ∼10% [START_REF] Maryasin | [END_REF]37]. As Y 3+ is non-magnetic and comparable in size to Er 3+ , it is an ideal ion to employ for magnetic dilution studies. Hence, we have grown single crystal samples of Er 2-x Y x Ti 2 O 7 with x = 0 (pure), 0.2 (10%-Y) and 0.4 (20%-Y) to investigate a possible change in the magnetic ground state. We performed time-of-flight inelastic neutron scattering measurements on these three single crystals at various temperatures and magnetic fields. Our results show that the signature for ψ 2 long range magnetic order disappears upon dilution. Instead, we observe a softening of the zero field, low energy excitations at the (220) ordering wavevector, consistent with a closing of the spin gap. At a relatively high level of magnetic dilution, on the order of 20%-Y, the system cannot globally select the ψ 2 state over ψ 3 , and we suggest that it forms a frozen mosaic of both ψ 2 and ψ 3 domains. The three Er 2-x Y x Ti 2 O 7 samples were characterized by dc magnetic susceptibility, and these results are summarized in Table I. In each case, the transition to the ordered phase is marked by a cusp and a bifurcation of the field cooled and zero field cooled susceptibilities, as shown in the Supplemental Material [38]. Both the Néel temperature and the paramagnetic moment per formula unit systematically decrease as a function of the magnetic dilution. The Curie-Weiss temperature (θ CW ) does not vary appreciably in these three materials, ranging from -17.2(3) K to -18.0(2) K.

We performed time-of-flight elastic and inelastic neutron scattering on these three single crystals using the Disc Chopper Spectrometer at the NIST Center for Neu-tron Research [39]. Measurements were performed in a dilution refrigerator in both zero magnetic field and an applied magnetic field along the [1,-1,0] direction, perpendicular to the (HHL) scattering plane. For each of the three samples, magnetic Bragg peaks form below their respective Néel transitions (see Table I). The relative intensities of the magnetic Bragg reflections in the 10%-Y and 20%-Y crystals are unchanged from pure Er 2 Ti 2 O 7 . That is to say, all three order into the Γ 5 irreducible representation, which is dominated by an intense magnetic Bragg reflection at (220). In large magnetic fields, all three enter a field polarized state, characterized by an intense (111) magnetic Bragg reflection and the disappearance of the (220) magnetic reflection [40,41].

The Γ 5 irreducible representation is comprised of two basis vectors, ψ 2 and ψ 3 , which are indistinguishable in an unpolarized elastic neutron scattering experiment. However, the distinction between ψ 2 and ψ 3 ground state selection can be resolved by considering the magnetic field dependence of the (220) magnetic Bragg peak. Both ψ 2 and ψ 3 display six distinct domains that are degenerate in zero field. This degeneracy is lifted by the application of a small magnetic field along the [1,-1,0] direction [20,42,43]. For ψ 2 , the two domains selected in a magnetic field are the ones that maximize the (220) magnetic Bragg peak, resulting in a twofold increase of its intensity. Domain selection in a [1-10] field for the ψ 3 state is less clear. Two scenarios are proposed; one selecting two domains which reduce the intensity of (220) [20,42] while the other selects four domains that resulting in a factor of 1.5 increase for (220) [43,44]. We note these domain effects occur in small [1,-1,0] fields, significantly below the onset of the field polarized state. The effect of the canting angle on the (220) Bragg peak intensity have been well characterized for Er 2 Ti 2 O 7 [40,41,45] and little effect is seen on it's intensity between 0.1T and 1T. Hence, we used a field of 0.5T to investigate domain effects, which, is sufficiently large to overcome pinning of domains by disorder in a small field. We first consider the elastic scattering from the three crystals at the (220) Bragg position in zero field, 0.5 T and 3 T, as shown in Fig. 1. The intensities have been normalized using the Bragg intensity at H = 3T, which corresponds to the polarized state where the scattered intensity at (220) is purely structural. As the field is increased to 0.5 T, we observe a marked increase in the peak intensity for the pure sample as well as for the 10%-Y doped sample, but not for the 20%-Y diluted crystal. More striking is the evolution of the shape of the (220) Bragg peak with magnetic dilution. In zero field, the undoped sample exhibits a resolution-limited Bragg peak with a small amount of diffuse scattering extending out along [HH0] (Fig. 1(a)). Upon dilution, the relative contribution of the resolution-limited Bragg scattering weakens with a corresponding increase in the diffuse scattering, such that the lineshape of the 20%-Y sample is dominated by a Lorentzian form in both zero field and at 0.5 T (Fig. 1(c)).

To determine the origin of the diffuse scattering around (220), we look to the full inelastic neutron spectra in Fig. 2. These data are shown along three reciprocal space directions for each of the three samples in four different magnetic fields applied along the [1,-1,0] direction. In zero field (top row of Fig. 2), the spin waves of the three samples are qualitatively similar, consisting of one flat branch near 0.4 meV and quasi-Goldstone modes which soften at (111) and (220). A previous high resolution inelastic neutron scattering study of the pure sample determined that the quasi-Goldstone modes are gapped by 0.053 ± 0.006 meV [28]. As the energy resolution associated with the present inelastic measurements is 0.09 meV, elastic cuts of the form shown in Fig. 1 necessarily integrate over some of the spectral weight of these quasi-Goldstone modes. Thus, for the pure sample, this low energy inelastic scattering is the origin of the diffuse scattering observed in our cuts over (220) in Fig. 1. However for the 10%-Y and especially the 20%-Y sample, our data suggests that the spin gap is reduced from 0.053 meV, allowing for additional quasi-elastic and elastic magnetic scattering, characteristic of a frozen mosaic of ψ 2 and ψ 3 domains.

In order to understand the contributions to the scattering in the cuts of Fig. 1, we fit each elastic data set to the sum of a Gaussian and a Lorentzian lineshape, quantifying the magnetic long-range order (LRO) and the dynamic, quasi-elastic or frozen spin contributions, respectively. An example of such a fit for each sample is shown in the Supplemental Material [38]. The resulting fits show that the relative contribution of the diffuse, Lorentzian lineshape grows as a function of doping and accounts for ∼75% of the Q-integrated scattering near (220) in the 20%-Y sample at T = 0.1 K and zero field. We therefore suggest that with increasing magnetic dilution, x, the spin excitations near (220) in Er 2-x Y x Ti 2 O 7 soften to lower energies and freeze. This is likely the result of a collapsing spin gap, a direct measure of the selection of ψ 2 over ψ 3 .

We can now isolate the LRO component of the elastic scattering and study its field dependence at T = 0.1 K. Once again, we use the fits to the scattering around (220), wherein a resolution-limited Gaussian lineshape represents the LRO and a broadened Lorentzian represents the dynamic, quasi-elastic and frozen spin response captured by our finite energy resolution. The LRO integrated intensity at (220) is shown as a function of a [1,-1,0] magnetic field for each sample in the insets of Fig. 1. Comparing the LRO intensity at zero field and 0.5 T, we observe a twofold increase for both the pure and 10%-Y samples, while the LRO is unchanged for the 20%-Y sample. This indicates, via the domain selection scenario described above, that the pure and 10%-Y samples order into ψ 2 , but the 20%-Y sample does not. Furthermore, these results are inconsistent with the detailed theoretical predictions that low levels(∼7%) of magnetic dilution should induce a transition from ψ 2 to ψ 3 [START_REF] Maryasin | [END_REF]37].

A striking feature of Fig. 1, is the almost complete absence of diffuse scattering around (220) at H = 0.5 T for both the pure and 10%-Y samples, but not the 20%-Y sample. Examination of the detailed spin wave spectra in Fig. 2, shows that a 0.5 T field gaps the quasi-Goldstone mode at (220) in the pure and 10%-Y samples (second row of Fig. 2). Meanwhile, quasi-elastic scattering persists around (220) in the 20%-Y sample to fields greater than 1 T. To quantify this effect, we integrated a small portion of reciprocal space around (220) and plotted the energy dependence of this scattering in zero field and 0.5 T for all three samples (Fig. 3). It is clear that the quasi-elastic scattering from the quasi-Goldstone modes in the pure and 10%-Y samples are gapped by ∼0.3 meV in a 0.5 T field. However, for the 20%-Y sample, the zero field excitations are soften and the application of a 0.5 T magnetic field makes little difference to the low energy spectral weight near (220). Both of these observations are consistent with the absence of a spin wave gap and the associated domain selection in a magnetic field in the 20%-Y sample .

We propose the schematic phase diagram shown in Fig. 4 to describe the magnetic state in magnetically diluted Er 2-x Y x Ti 2 O 7 . The phase transition and ψ 2 ground state selection are already well-established in the pure material [12,16], along with the concomitant opening of a 0.053 ± 0.006 meV spin wave gap at T N = 1.1 K [28]. Heat capacity measurements as a function of magnetic dilution are consistent with conventional 3D percolation theory [START_REF] Gaulin | Chapter 8 : experimental studies of frustrated pyrochlore antiferromagnets in Frustrated spin systems[END_REF], and a percolation threshold near 60%-Y has also been established [START_REF] Niven | Proc. R. Soc. A[END_REF]. The present measurements on magnetically diluted single crystal samples shows that ψ 2 domain selection is observed at low temperatures for the 10%-Y sample but not for the 20%-Y sample. This strongly suggests the presence of a phase boundary that mirrors the collapse of the spin gap as a function of dilution, indicated by the dashed line in Fig. 4(a). Weak dilution to the left of the dashed line produces a stable set of ψ 2 domains (Fig. 4(b)), at least above our minimum T = 0.1 K. However, to the right of the dashed line, the spin gap collapses and there is no mechanism for selection of ψ 2 over ψ 3 . A ground state characterized by a linear combination of ψ 2 and ψ 3 forming the full U(1) manifold is consistent with our data. However, in light of the theoretical study of ref [START_REF] Maryasin | [END_REF]37], it is then natural to think of the diluted system forming a frozen mosaic of ψ 2 and ψ 3 domains, with the ψ 3 domains (Fig. 4(c)) pinned by locally high concentrations of the quenched vacancies (Fig. 4(d)). It is interesting to note that a similar mixed ψ 2 and ψ 3 state is also observed in NaCaCo 2 F 7 but believed to originate from bond disorder [47].

We find that the ψ 2 ordered state in Er 2 Ti 2 O 7 displays remarkable fragility induced by the presence of magnetic vacancies. Measurements as a function of temperature in our most magnetically dilute sample, 20%-Y, provides no evidence for an additional phase transition between T N = 0.74(8) K and our base temperature of 0.1 K, as shown in Fig. 3 of the Supplemental Material [38]. Nonetheless, we do identify a change in phase behavior between ψ 2 and a ground state characterized by no selection of ψ 2 over ψ 3 , likely a frozen mosaic of the two, all at dilution concentrations far below the 3D percolation threshold. Clearly, the preceding theoretical work correctly identified the sensitivity of the ψ 2 ground state selection to the presence of this form of quenched disorder, if not the detailed manifestation of the disorder on the phase behavior. As such, we hope our characterization of the spin statics and dynamics in single crystal Er 2-x Y x Ti 2 O 7 , and low temperature phase behavior as a function of dilution, will motivate a complete understanding of these exotic, and fragile, ordered states. Magnetic susceptibility measurements were performed on a small portion of the crushed single crystal of undoped Er 2 Ti 2 O 7 and on crushed single crystal of the 10%-Y and 20 %-Y samples. The dc susceptibility was measured on warming using both a field cooled and a zero field cooled protocol with an external field of 0.01 T. In each case, the transition to long range magnetic order is marked by a cusp in the susceptibility as well as a bifurcation of the field cooled and zero field cooled susceptibilities. As the cusp and the bifurcation in the susceptibility do not occur concomitantly, we define T N as the cusp maxima, and the difference between the two as the error bar. The undoped Er 2 Ti 2 O 7 orders at T N = 1.1(1) K, consistent with previous studies [? ]. The effect of diluting magnetic Er 3+ with non-magnetic Y 3+ is to suppress the magnetic ordering transition to lower temperature. The 10%-Y and 20%-Y samples have Néel temperatures of T N = 1.04(5) K and T N = 0.74(8) K, respectively. This suppression of T N is consistent with previous studies of yttrium-dilution in Er 2 Ti 2 O 7 [? ].

At high temperatures, Curie-Weiss behavior is observed in both undoped and yttrium-doped Er 2 Ti 2 O 7 . Curie-Weiss fits were performed for each sample between 50 K and 300 K, and the results of these fits are summarized in Table I of the main manuscript. The Curie-Weiss temperature (θ CW ) does not change appreciably in these three materials, ranging from -17.2(3) K to -18.0(2) K. The paramagnetic moment extracted from the high-temperature Curie-Weiss fits, 9.555(4)µ B for Er 3+ as well as prior bulk magnetization studies [? ]. However, the paramagnetic moment per Er 3+ systematically decreases with yttrium doping, likely a consequence of differences in the high energy crystal electric field schemes in these three materials. We note, however, that our inelastic scattering study revealed no low-lying crystal electric field levels in either the 10%-Y or 20%-Y samples up to 3 meV.

LONG RANGE ORDERED AND DIFFUSE CONTRIBUTIONS TO THE (220) BRAGG PEAK IN Er2-xYxTi2O7

To extract the long-range order as well as the dynamic, quasi-elastic and frozen contributions to the (220) magnetic Bragg peak, we performed an integration of the neutron scattering data set of -0.2 to 0.2 in the [00L] direction and -0.1 to 0.1 meV in energy transfer. The results of such integration for the three single crystals are shown in Fig. 2 for T = 0.1 K and zero applied field. Each Bragg peak has been fit to a Gaussian function to account for the long-range order (blue shaded region in Fig. 2) and a Lorentzian function for the dynamic, quasi-elastic and frozen contributions to the scattering (yellow shaded region in Fig. 2). The width of the Gaussian function was fixed by the width of a strictly nuclear reflection in the 3T data set. An additional Gaussian contribution to the scattering was accounted for by the 3T data set itself, representing the nuclear contribution (red shaded region in Fig. 2) and kept fix for the low field data sets. Finally, the width of the Lorentzian function was allowed to refine freely. The results for all samples show that the relative contribution of the Lorentzian scattering over the Gaussian (long-range order) contributions is strongly enhanced as a function of doping. As discussed in the main manuscript, this indicates that upon dilution, the spins do not order in the discrete states belonging to ψ 2 but instead freeze into a mosaic of ψ 2 and ψ 3 domains, whose characteristic size is approximated by the inverse of the Q-width of the Lorentzian component to the scattering, which gives 35 ± 7 Å.

TEMPERATURE DEPENDENCE OF THE (220)

BRAGG PEAK FOR 20%-Y Er2Ti2O7.

The temperature dependence of the (220) magnetic Bragg peak in Er 2-x Y x Ti 2 O 7 with x = 0.4 (20%-Y) has also been explored to investigate a possible phase transition between ψ 2 and ψ 3 states at temperatures between the base temperature of our neutron experiment (T = 0.1 K) and the Néel temperature (T = 0.74(8) K) of this sample. Fig. 3 shows elastic cuts around the (220) Bragg position obtained by an integration of -0.2 to 0.2 in the [00L] direction and -0.1 to 0.1 meV in energy transfer.

The same procedure followed in the previous section was again employed to fit the (220) magnetic Bragg peak at every temperature collected in our neutron experiment in zero applied field. In this figure, the structural contribu- tion of the long-range order scattering at (220) has been removed using intensity obtained from the paramagnetic, T = 2 K data set. The temperature dependence of the Lorentzian and the long-range order scattering are shown in the inset of Fig. 3. The long-range order contribution shows typical order parameter behavior, correlated with T N and no anomaly is observed below that temperature. In contrast, the Lorentzian scattering has little temperature dependence but is enhanced near T N , consistent with critical scattering near a continuous phase transition.

FIG. 1 .

 1 FIG. 1. Magnetic Bragg peak intensity for Q = (220) as a function of field for the pure, 10%-Y and 20%-Y diluted samples of Er2-xYxTi2O7. The elastic scattering was isolated by integrating from -0.2 to 0.2 in the [00L] direction and from -0.1 to 0.1 meV in energy transfer. The insets show the integrated intensity of the long range ordered component of the scattering for each of the three samples. Examples of the fits used to obtain these intensities are shown in Fig.1 of the Supplemental Material. The dashed line in the inset of (b) indicates the expected behavior for the 10%-Y sample at 1.75 T.

FIG. 2 .

 2 FIG. 2. Spin wave spectra along three different directions in the HHL plane: [HH0], [22L], and [HHH], all at T =0.1 K. The spectra are shown for each of the three samples in zero magnetic field, and with magnetic fields of 0.5 T, 1 T, and 3 T applied along the [1-10] direction. The contour plots along the [HH0], [22L] and [HHH] directions are obtained by integrating the 3D data set, S( Q, E), over -0.2 ≤ [00L] ≤ 0.2, 1.8 ≤ [HH0] ≤ 2.2 and 0.2 ≤ [HH -2H] ≤ 0.2, respectively.

FIG. 3 .

 3 FIG. 3. Inelastic intensity as a function of energy at the (220) Bragg position for the pure, 10%-Y and 20%-Y diluted samples of Er2-xYxTi2O7. These plots are obtained by integrating the 3D data sets S( Q, E) at T = 0.1 K over 1.6 ≤ [HH0] ≤ 2.4 and -0.5 ≤ [00L] ≤ 0.5.

7 FIG. 1 .

 71 FIG. 1.Magnetic susceptibility data obtained for Er2-xYxTi2O7 with x =0 (pure), 0.2 (10%-Y) and 0.4 (20%-Y). Powder samples from crushed single crystals used in the neutron scattering measurements were employed. For each sample, both the zero field cooled (ZFC -closed symbols) and the field cooled (FC -open symbols) susceptibility are shown. The inset shows the Curie-Weiss fits performed for each sample.

FIG. 2 .

 2 FIG. 2. Typical fits of the (220) magnetic Bragg peak performed in this work for Er2-xYxTi2O7 with x = 0 (pure), 0.2 (10%-Y) and 0.4 (20%-Y)) at T = 0.1 K in zero field. The yellow areas represent the elastic, resolution-limited (Gaussian) magnetic contribution to the scattering and the blue area represents the dynamic, quasi-elastic or frozen (Lorentzian) contributions to the scattering. The red area shows the structural scattering obtained via a fit of the 3T data set. The black lines shows the resulting fit which are obtained by the addition of the three curves (blue,yellow and red).

FIG. 3 .

 3 FIG. 3. Elastic cut showing the (220) magnetic Bragg peak for varying temperature in Er2-xYxTi2O7 with x=0.4 (20%-Y) in zero applied field. The black line represents the resulting fit for the T = 0.1 K data set. The inset panel shows the temperature dependence of the Gaussian, long range ordered, and Lorentzian contributions to the (220) scattering, where the dashed line marks the Néel transition temperature, TN = 0.74(8) K.

We wish to acknowledge useful conversations with Leon Balents, Kate Ross, Mike Zhitomirsky, Michel Gingras, and Jeff Rau. We would also like to thank Juscelino Leao for his assistance with the sample environment. This work was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Foundation for Innovation. Work at the NIST Center for Neutron Research is supported in part by the National Science Foundation under Agreement No. DMR-0944772.


Edwin Kermarrec

D D Maharaj

J Gaudet

K Fritsch

D Pomaranski

J B Kycia

Y Qiu

J R D Copley

M Couchman

A Morningstar

H A Dabkowska

B D Gaulin

Gapped and gapless short range ordered magnetic states with ( 1 2 , 1 2 , 1 2 ) wavevectors in the pyrochlore magnet Tb 2+x Ti 2-x O 7+δ

Keywords: numbers: 75, 25, -j 75, 10, Kt 75, 40, Gb 75, 40, -s

HAL is

INTRODUCTION

Geometric frustration is a natural route towards the stabilization of exotic quantum states in magnetic materials. [START_REF]Introduction to Frustrated Magnetism[END_REF] Considerable effort has been directed to the study of the cubic pyrochlore lattice, one of the canonical architectures supporting geometrical frustration in three dimensions. [START_REF] Gardner | [END_REF] Such materials have the chemical composition A 2 B 2 O 7 with both the A 3+ sublattice and the B 4+ sublattice, independently, occupying the vertices of corner-sharing tetrahedra. The rare earth titanate family R 2 Ti 2 O 7 (R =rare-earth) remains a source of surprise and excitement even after more than a decade of extensive study. The Ti 4+ ions are non-magnetic within this structure and the magnetism originates from trivalent rare-earth ions on the A pyrochlore sublattice. These magnetic moments can display varied interactions and anisotropies resulting in very rich and exotic phenomena such as emergent magnetic monopole excitations and a range of interesting phase behavior at low temperatures displaying spin ice [3,4] and spin liquid ground states [5][6][7], as well as order selected by order-by-disorder mechanisms. [8] The pyrochlore antiferromagnet Tb 2 Ti 2 O 7 has long been a promising quantum spin liquid candidate as it fails to find a conventional magnetically-ordered state down to at least 20 mK. [9] However, the exact nature of its disordered ground-state remains of some debate [10][11][12]. It displays an antiferromagnetic Curie-Weiss susceptibility with a Θ CW ∼ -17 K, and the crystalline electric field J = 6 eigenstates of the Tb 3+ ion give rise to a ground state and a first excited state doublets that are separated by only ∼ 1.5 meV. While the corresponding eigenfunctions correspond primarily to m J =±4 and ± 5, [13][14][15] and therefore to Ising-like anisotropy, the proximity in energy of the ground and first excited state doublets makes it difficult to consider a S eff = 1 2 description of the moments, and therefore precludes the relatively simple spin Hamiltonian which can result from a wellisolated ground state doublet.

At very low temperatures, Tb 2 Ti 2 O 7 is known to display a peak in the real ac susceptibility between T g ∼ 0.15 K and 0.25 K depending on both sample and frequency, and this is consistent with a glassy ground state below T g . [16][17][18] Anomalies in the heat capacity, C P , indicative of a thermodynamic phase transition, are also observed at higher critical temperatures, T C between 0.4 K and 0.5 K in a subset of samples. Our own earlier neutron scattering work on a single crystal of Tb 2 Ti 2 O 7 with a large C P anomaly at T C = 0.45 K (labelled from now on as sample B), was observed to display quasi-Bragg peaks at ordering wavevectors of the form 1 2 , 1 2 , 1 2 below T g ∼ 0.275 K, with a concomitant gap in its spin excitation spectrum of ∼ 0.08 meV. [19] This state is stabilized at zero magnetic field only under a magnetic field-cooled protocol, and can be understood in terms of a short-ranged antiferromagnetic spin ice (AFSI) structure, wherein an ordered correlated region is comprised of ∼ 8 conventional unit cells which individually adopt ordered spin ice domains, with two spins pointing in and two spins pointing out of each tetrahedron. These domains are then antiferromagnetically correlated between neighbouring unit cells. Another symmetry allowed spin structure made of consecutive layers of one-in/three-out, two-in/two-out and three-in/one-out tetrahedra was recently found to share the same powder-averaged structure factor. However, the elastic diffuse scattering measured on single crystals suggests that the AFSI structure is stabilized at low temperature. [20] Recent work on polycrystalline samples of Tb 2+x Ti 2-x O 7+δ , with the off-stoichiometry x tightly controlled, has demonstrated a strong sensitivity of T C to the precise Tb concentration. [21] In particular this work showed that the size of the C P anomaly is maximal at x ∼ 0.005 and T C ∼ 0.5 K and that no such C P anomaly and corresponding phase transition is observed at any temperature for x ≤ 0. We were then motivated to investigate the x-dependence of the AFSI ground state and its gapped spin excitation spectrum using single crystals, and that is what we report here. To this end, we grew two new single crystal samples, hereafter referred to as samples A and C, which correspond to Tb 2+x Ti 2-x O 7+δ with x = -0.0010 and x = +0.0147. In this paper we examine the systematics of the elastic and inelastic magnetic neutron scattering in these two new crystals as compared with our previous results for sample B, with x = +0.0042.

EXPERIMENTAL DETAILS

Two single-crystal samples of Tb 2 Ti 2 O 7 were grown using the optical floating zone technique at McMaster University [START_REF] Dabkowska | Floating Zone Growth of Oxides and Metallic Alloys[END_REF][START_REF] Gardner | [END_REF]. These crystals, which are referred to as samples A and C, were intentionally grown off-stoichiometry; with target compositions of Tb 2+x Ti 2-x O 7+δ with x = -0.005 and +0.005, under an absolute pressure of 3 atm of O 2 . We compare our results on these single crystal samples with sample B, which was previously grown by the same optical floating zone technique at McMaster University, but with a tar- get stoichiometric x = 0 composition. We carried out characterization studies with the intent of accurately estimating x for each of the three samples.

High resolution x-ray scattering measurements were carried out at McMaster University, using a PANalytical powder diffractometer equipped with a X'Celerator detector [24]. Cu-K α1 x-rays (λ = 1.540598 Å) were generated from a Cu anode. For each of the three single crystals, small discs were cut from the samples used for neutron scattering measurements and then ground in an agate pestle and mortar into a ∼ 100 mg uniform powder. A small amount of pure silicon powder (∼ 10 mg) was added to serve as a common reference for the determination of the lattice parameters. Diffraction patterns were measured at room temperature with the 2θ range extending from 10 to 120 degrees, with a step size of 0.00835 degrees, and then refined using the FULLPROF package. [25] New specific heat measurements were performed on samples A and C with a 3 He/ 4 He dilution refrigerator at the University of Waterloo. The single crystals used for neutron scattering measurements were cut and sectioned into ∼ 30 mg masses with dimensions 2 × 2 × 2 mm 3 TABLE I. Summary of the characterization of the three single crystal samples A, B and C, showing the Curie constant of the magnetic susceptibility (CCurie), the lattice constant a, the value of the off-stoichiometry x in the chemical formula Tb2+xTi2-xO 7+δ , the ratio of the elastic neutron scattering intensity of the ( 12 , 1 2 , 5 2 ) magnetic quasi-Bragg peak to that of the (113) nuclear Bragg peak, the moment related to the AFSI order and the correlation length.

Sample

CCurie(cm and these smaller single crystals were used in the heat capacity study. The relaxation method was employed with a thermal weak link of manganin wire, with conductance 5.0 × 10 -7 J/K/s at 0.80 K. The resulting time constant was greater than 600 s at the highest temperature measured. The average step size for the relaxation measurement was 3.5% of the nominal temperature, with a minimum equilibration time window of five times the thermal relaxation constant. These results are combined here with our earlier measurements on sample B, which employed very similar protocols. Magnetic susceptibility measurements were performed on ∼ 4 mg and 100 mg powder samples taken from each of single crystals A, B and C, and using an applied magnetic field of µ 0 H = 1 T. The measurements employed a Quantum Design SQUID magnetometer operating between 1.8 K and 300 K [24].

Time-of-flight neutron scattering measurements were performed using the Disk Chopper Spectrometer (DCS) at the NIST Center for Neutron Research. [26] Two different incident energies were employed; for lower energy resolution measurements we employed E i = 3.27 meV, giving an energy resolution of 0.1 meV, while the higher energy resolution measurements employed E i = 1.28 meV with a resulting resolution of 0.02 meV. The sample was carefully aligned with the [110] direction vertical to within 0.5 degrees, and mounted in a dilution refrigerator magnet cryostat such that the (H,H,L) plane was coincident with the horizontal scattering plane. Measurements were performed in a temperature range between 80 mK and 650 mK, and employed magnetic fields of up to 0.2 T applied along the vertical [110] direction.

RESULTS AND DISCUSSION

Bulk Characterization and Structure

Heat capacity measurements carried out on single crystal samples A, B, and C of Tb 2+x Ti 2-x O 7+δ are shown in Fig. 1b) for temperatures between 0.2 K and 0.6 K for samples A and C, and between 0.12 K and 0.7 K for sample B. As mentioned above, sample B is the single crystal sample that was grown with starting constituents aimed at producing a stoichiometric, x = 0, sample. Both neu-tron scattering and heat capacity measurements have already been reported for it [27], and its C P data is reproduced in Fig. 1b) for ease of comparison. Clearly it shows a large C P anomaly at T C = 0.45 K, indicative of a thermodynamic phase transition. Samples A and C are new single crystals that were grown from starting constituents aimed at producing off-stoichiometry crystals, x = -0.005 and +0.005, respectively. As can be seen in Fig. 1b), neither of the samples A and C displays a C P anomaly within this field of view.

In Fig. 1a) we compare the integrated heat capacity, or release of entropy, in our three single crystals, over the temperature range from T = 0.38 K to 0.55 K. This temperature range covers the range of the C P anomalies previously reported in polycrystalline samples. The integrated C P from our single crystal samples is also compared with C P integrated over the same temperature range from a set of powder samples from Taniguchi et al., with stoichiometries estimated to range from -0.013 to +0.005 (grey disks in Fig. 1a)). [21,28] This comparison allows us to estimate our actual single crystal stoichiometries, as opposed to those expected on the basis of the masses of the starting materials in the single crystal growth, relative to those determined by Taniguchi et al. From this comparison, we estimate the stoichiometry for single crystal sample B as x = +0.0042 [START_REF] Gardner | [END_REF]. The stoichiometries of samples A and C are then inferred from the values of their lattice parameters determined by xray diffraction measurements (see below), and are found to be x = -0.0010(2) and +0.0147(2), respectively. We also note that only one of our single crystal samples, sample B with x = 0.0042, displays a C P anomaly. In Fig. 1 a) we therefore label the approximate range in stoichiometry from x = 0 to +0.01 as corresponding to an ordered ground state with some sort of thermodynamic phase transition, while those stoichiometries outside of this narrow bound display the corresponding disordered ground states, at least for temperatures above T = 0.2 K. Figure 1c) shows the temperature dependence of the Curie constant extracted from a Curie-Weiss fit of the magnetic susceptibility χ = χ 0 + C Curie /(T -θ CW ), measured in a µ 0 H = 1 T magnetic field for crushed powders of each of our single crystal samples. The temperature independent term χ 0 comprises the core diamagnetic and the Van Vleck paramagnetic contributions. Its common value of 1.5 [START_REF]Introduction to Frustrated Magnetism[END_REF].10 -3 cm 3 /mol Tb is shared between the three data sets. The Curie-Weiss temperatures θ CW vary between -12 K and -14 K, while the Curie constants themselves, extracted from the fits, are 11.58(2), 11.90(2) and 11.95(2) in units of cm 3 .K/mol Tb (Table I). It is clear that the Curie constant increases from sample A to B to C, reflecting an increase in the off-stoichiometry x, and hence the Tb-content.

Figure 1d) shows high resolution x-ray powder diffraction data taken on small pieces of single crystal samples A, B, and C, which were subsequently crushed to form a powder and then mixed with a small amount of pure Si powder. The room temperature powder xray diffraction measurements were taken with Cu-K α1 radiation, and inclusion of the Si powder, with a very accurately known lattice parameter, allows an accurate room temperature measurement of the cubic lattice parameters for our single crystal Tb 2+x Ti 2-x O 7+δ samples, on an absolute scale. Figure 1d) highlights the powder diffraction pattern around the (662) Bragg peak of Tb 2 Ti 2 O 7 . No shift is observed in the reference Bragg signal from the Si powder, and the Rietveld refinements of the entire diffraction patterns give the room temperature (293 K) lattice parameters of our A, B and C single crystals of Tb 2+x Ti 2-x O 7+δ as 10.15784(2) Å, 10.15849(2) Å and 10.15980(2) Å, respectively. While these values are slightly larger than those determined for the powder samples prepared by Taniguchi et al., the relative change in lattice parameter with stoichiometry x is approximately the same. Assuming that the slope of the linear relationship between the cubic lattice parameter and the off-stoichiometry x reported in Ref. 28 holds, the compositions of our samples A, B and C are respectively x = -0.0010(2), +0.0042(2) and +0.0147(2) (Table I).

Neutron Scattering

New time-of-flight neutron scattering data on single crystal Tb 2+x Ti 2-x O 7+δ samples A and C are shown in Figs. 2 and3, where Fig. 2 shows elastic scattering data with an energy resolution of 0.1 meV, and Fig. 3 shows both elastic and inelastic neutron scattering with an energy resolution of 0.02 meV. Figure 2 shows reciprocal space maps of the elastic scattering within the (HHL) plane for both samples A and C, under zero field cooled (ZFC) and field cooled (FC). These data sets show intense magnetic quasi-Bragg peaks to appear at ( 1 2 , 1 2 , 1 2 ) and symmetry related positions for both samples A and C, under FC conditions only. For these FC data sets, the samples were cooled from T = 4 K in a µ 0 H = 0.2 T field aligned along the [110] direction, while the ZFC data sets were obtained while cooling in nominal zero field, where only the residual magnetic field of the superconducting magnet cryostat (typically 0.01 T) is present. The quasi-Bragg peaks are seen to form below T AFSI ∼ 275 mK, consistent with earlier measurements on sample B. The low temperature elastic reciprocal space maps obtained under ZFC conditions resemble the FC maps, although we observe only vestiges of the quasi-Bragg peaks at ( 1 2 , 1 2 , 1 2 ) positions. These features are much more diffuse than those observed under FC conditions, and a "checkerboard" pattern of ZFC diffuse scattering is observed such that ( 12 , 1 2 , 1 2 ) reciprocal space positions occupy the corners of the diffuse scattering ZFC checkerboard. This is again consistent with what was observed earlier on sample B of Tb 2+x Ti 2-x O 7+δ under ZFC conditions, and from other studies which did not employ FC protocols [12,29].

Measurements using different FC cooling protocols (in [110]] field from high temperatures; ZFC to low temperatures, then apply a field and remove it), demonstrate that it is only important to have applied a magnetic field along [110] µ 0 H 0.2 T while the sample is at low temperatures, below T AFSI ∼ 275 mK, in order to observe intense quasi-Bragg peaks at ( 1 2 , 1 2 , 1 2 ) positions. With the elastic neutron scattering data in hand, we can take cuts of the diffuse scattering for the purpose of extracting correlation lengths of the FC AFSI state in all three of samples A, B, and C. Typical reciprocal space cuts of this data for sample B are shown in Fig. 2e), where we show diffuse elastic scattering along the [H,H,0] direction, with an integration along L from L=2.3 to L=2.8, which is sufficient to cover the reciprocal space extent of the quasi-Bragg peaks at (-1 2 , -1 2 , 5 2 ) and ( 12 , 1 2 , 5 2 ). For comparison we have overplotted low energy inelastic scattering, integrating from 0.1 meV to 0.3 meV under both FC and ZFC conditions (averaged together). Consistent with the reciprocal space maps in Fig. 2, we see much sharper quasi-Bragg peaks present at low temperature under FC conditions only.

Figure 2f) shows the difference between FC and ZFC elastic scattering along [HH0], as defined in Fig. 2e), and from this we can extract the low temperature correlation length induced by FC protocols relative to ZFC protocols. This difference scattering profile is shown for all three Tb 2+x Ti 2-x O 7+δ crystals, and we obtain somewhat longer correlation lengths, ξ = 28(2) Å for samples A and B, than for sample C where we obtain ξ = 21(4) Å. For ease of comparison, we have listed these correlation lengths for the three samples in Table I. These correlation lengths are larger than those previously estimated for sample B, which was ξ ∼ 8 Å. However that correlation length corresponded to the inverse width in reciprocal space of the full diffuse elastic scattering around ( 1 2 , 1 2 , 1 2 ) type positions, as opposed to the inverse width of the difference between the FC and ZFC elastic diffuse scattering that we are presenting in Fig. 2f). Nonetheless we conclude that the local AFSI structure at low temperatures under FC conditions is a robust characteristic of Tb 2 Ti 2 O 7 , and the local correlation volume corresponds to a small number (∼ 2) of unit cells of the local AFSI structure, independent of the precise stoichiometry.

It is also possible to estimate the magnetic moment participating in the ( 12 , 1 2 , 1 2 ) quasi-Bragg peaks and to examine how that varies with off-stoichiometry, x. To do this we integrate over the magnetic ( 1 2 , 1 2 , 5 2 ) quasi-Bragg peak obtained under FC protocols, in a transverse scan which covers the full extent of the quasi-Bragg peak, and we do the same thing for the nuclear-allowed (113) Bragg peak. We can then calculate the static magnetic moment involved, assuming the AFSI local structure and the cubic pyrochlore structure. Previous mea- I.

Previous measurements on sample B, Tb 2+x Ti 2-x O 7+δ with x = +0.0042, showed that the appearance of the ( 12 , 1 2 , 1 2 ) quasi-Bragg peaks under FC protocols is coincident with the opening of a ∼ 0.08 meV gap in the spin excitation spectrum. As our new measurements on samples A and C shown in Fig. 3 demonstrate, a very similar phenomenon is relevant to other stoichiometries as well, both with and without a low temperature C P anomaly. Figure 3 shows high energy-resolution inelastic neutron scattering, integrating around the ( 12 , 1 2 , 3 2 ) quasi-Bragg peak in sample C, Tb 2+x Ti 2-x O 7+δ with x = +0.0147 [START_REF] Gardner | [END_REF]. Figure 3 a) shows inelastic data taken under ZFC protocols at T = 0.1 K as well as T = 0.3 K. One sees that both the elastic and inelastic spectra are identical, and the inelastic magnetic scattering shows gapless, quasi-elastic scattering, down to our resolution limit, ∼ 0.02 meV. In contrast, data sets on sample C at T = 0.1 K taken under FC and ZFC protocols show dramatic differences. Larger elastic magnetic scattering is clearly seen near zero energy, and the spin excitation spectrum is gapped below ∼ 0.1 meV. Both of these results are consistent with earlier results on sample B [19,27]. Thus the appearance of ( 12 , 1 2 , 1 2 ) quasi-Bragg peaks and a concomitant gapping of the spin excitation spectrum under FC protocols for temperatures less than ∼ 0.275 K, are robust features of Tb 2 Ti 2 O 7 , and are not correlated with the observation (or lack thereof) of C P anomalies at ∼ 0.4 K to 0.5 K.

These results reaffirm our original assertion that the thermodynamic phase transition characterized in Tb 2+x Ti 2-x O 7+δ with 0 ≤ x ≤ 0.01 by a C P anomaly near T C = 0.4 K to 0.5 K, is different from and independent of the transition to a short ranged AFSI state below T AFSI ∼ 0.275 K. Our original assertion was made on the basis of the difference between T C and T AFSI . The current work explicitly shows the ( 12 , 1 2 , 1 2 ) quasi-Bragg peaks and concomitant gap in the spin excitation spectrum occur in the absence of T C . As neutron scattering sees magnetic dipole degrees of freedom only, this implies that T C corresponds to degrees of freedom which are of higher order than dipole. This then begs the following question: why is T C and the ordered state connected with such quadrupoles or octupoles, etc., so sensitive to the weak disorder connected with stoichiometry on the 0.5% scale or less ? The observation that the spin dipole degrees of freedom are robust to such off-stoichiometry is typical of most magnetic materials. Higher-than-dipole order is likely associated with aspherical f -electron charge distributions, and that could be sensitive to the nature of the 4f electrons on the "stuffed" B-site in pyrochlores of the form A 2 B 2 O 7 . Such stuffed, B-site, Tb ions would experience different crystal field effects from the A-site Tb ions, and these may act as impurities with a different aspherical f -electron charge distribution that are effective in frustrating higher-than-dipole order. Such a scenario of very different crystal field eigenfunctions for rare earth ions on A and B sublattices has recently been discussed for Yb 2+x Ti 2-x O 7+δ [30]. Additionally, there is the question of why the ordered states of Tb 2+x Ti 2-x O 7+δ are not centered at x = 0; indeed T C appears to be maximal at x ∼ +0.005. This would suggest that excess Ti is very detrimental to such a higher-than dipole ordered state.

CONCLUSIONS

We have performed new heat capacity, susceptibility and neutron scattering measurements on two single crystal samples (samples A and C), which we characterize as Tb 2+x Ti 2-x O 7+δ with x = -0.0010(2) and +0.0147 [START_REF] Gardner | [END_REF]. Neither of these new single crystals displays a peak in C P above ∼ 0.2 K. Neutron scattering measurements on these crystals show the same ( 12 , 1 2 , 1 2 ) quasi-Bragg peaks and concomitant 0.1 meV gap in the spin excitation spectrum to appear below T AFSI ∼ 0.275 K when cooled using a FC protocol. This phenomenology is very similar to that previously observed in sample B, which is characterized by Tb 2+x Ti 2-x O 7+δ with x = +0.0042(2), and which does display a large C P anomaly at T C =0.45 K. These results show that the ( 1 2 , 1 2 , 1 2 ) quasi-Bragg peaks and associated AFSI local structure are robust features of Tb 2 Ti 2 O 7 , and are not sensitive to precise stoichiometry. In contrast, the ordered phase corresponding to T C is distinct from that associated with T AFSI , and is located in a small range of stoichiometry, near, but not coincident with perfect stoichiometry: it is found for Tb 2+x Ti 2-x O 7+δ with 0 ≤ x ≤ 0.01. Very recent work carried out on single crystal samples reached similar conclusions regarding the existence of a finite region for the ordered phase in the x -T phase diagram. [31] These results are consistent with and substantially amplify our original suggestion that the thermodynamic phase transition below T C is associated with higher-than dipole order (quadrupole, octupole etc.), while the AFSI state observed under FC protocols is associated with the magnetic dipoles.

FIG. 1 .

 1 FIG. 1. (Color online) (a) The phase diagram for Tb2+xTi2-xO 7+δ showing the release of entropy between 0.38 and 0.55 K as a function of the off-stoichiometry x. The grey disks are from measurements on polycrystalline samples, from Ref.21. Dashed line is a guide to the eye. (b) Temperature dependence of the specific heat (CP ) for samples A, B and C. The dashed lines delimit the integration range used for the entropy calculation. (c) Temperature dependence of the Curie constant (CCurie) for samples A, B and C. (d) High resolution x-ray powder diffraction data showing the region around the (622) Bragg peak for samples A, B and C.

FIG. 2 .

 2 FIG. 2. (Color online) Panels a) and b) show reciprocal space maps of elastic neutron scattering (± 0.1 meV) within the HHL plane for sample A, Tb2+xTi2-xO 7+δ with x = -0.0010(2) taken at T = 0.1 K under ZFC (a) and FC (b) protocols. Panels c) and d) show similar data for sample C, x = +0.0147(2). (e) Q-scans through the elastic neutron data shown in panels a) and b) of the form HHL, where L is integrated from 2.3 to 2.8, for sample B at T = 0.1 K. Elastic scattering (± 0.1 meV) scans are shown under both FC and ZFC protocols. A similar Q-scan of the inelastic scattering (0.1 meV ≤ E ≤ 0.3 meV) is shown for comparison. Panel f) shows the difference between the FC and ZFC Q scans at T = 0.1 K for all three single crystals of Tb2+xTi2-xO 7+δ , A, B, and C. Lorentzian fits to the resulting lineshape, from which correlation lengths can be estimated, are shown, along with a horizontal bar representing the resolution limit of such a measurement. Error bars in all figures represent one standard deviation.

FIG. 3 .

 3 FIG. 3. (Color online) High resolution energy scans of an integration around the ( 1 2 , 1 2 , 3 2 ) quasi-Bragg peak in sample C, which displays no low temperature CP anomaly. Panel a) shows data taken at T = 0.1 K with a ZFC protocol, and at T = 0.3 K. Both data sets show identical elastic scattering, and display a gapless quasi-elastic inelastic spectrum. Panel b) shows data taken at T = 0.1 K under both a FC and a ZFC protocol. One clearly sees the increased elastic scattering giving rise to the ( 1 2 , 1 2 , 3 2 ) quasi-Bragg peak under FC conditions, along with the concomitant gapped spin excitation spectrum with a gap of ∼ 0.1 meV. Error bars in both panels represent one standard deviation.

  surements on powder samples by Taniguchi et al. reported the existence of a true (resolution-limited) long range order for their x = +0.005 sample that showed a C P anomaly at ∼ 0.5 K, involving a very small static moment of 0.08 µ B /Tb[21]. With our new measurements it is qualitatively clear that a much larger static moment is present in a short range AFSI structure for all three single crystals of Tb 2+x Ti 2-x O 7+δ , as the (12 , 1 2 , 1 2 ) quasi-Bragg peaks are strong and easy to observe in all three single crystals. Structure factor calculations comparing the integrated intensity of the (12 , 1 2 , 5 2 ) quasi-Bragg peak and the (113) nuclear Bragg peak produce static Tb moments of 1.8(2) µ B , 2.5(3) µ B , and 1.5(2) µ B , for single crystals A, B, and C of Tb 2+x Ti 2-x O 7+δ with x = -0.0010(2), +0.0042(2) and +0.0147(2), respectively. These estimates for the static moment in the quasi-Bragg peak are listed in Table

ACKNOWLEDGMENTS

This work utilized facilities supported in part by the National Science Foundation under Agreement No. DMR-0944772, and was supported by NSERC of Canada. The DAVE software package [32] was used for data reduction and analysis of DCS data.