Structure, Spin Correlations, and Magnetism of the S = 1/2 Square-Lattice Antiferromagnet Sr2CuTe1–xWxO6 (0 ≤ x ≤ 1)

Quantum spin liquids are highly entangled magnetic states with exotic properties. The S = 1/2 square-lattice Heisenberg model is one of the foundational models in frustrated magnetism with a predicted, but never observed, quantum spin liquid state. Isostructural double perovskites Sr2CuTeO6 and Sr2CuWO6 are physical realizations of this model but have distinctly different types of magnetic order and interactions due to a d10/d0 effect. Long-range magnetic order is suppressed in the solid solution Sr2CuTe1–xWxO6 in a wide region of x = 0.05–0.6, where the ground state has been proposed to be a disorder-induced spin liquid. Here, we present a comprehensive neutron scattering study of this system. We show using polarized neutron scattering that the spin liquid-like x = 0.2 and x = 0.5 samples have distinctly different local spin correlations, which suggests that they have different ground states. Low-temperature neutron diffraction measurements of the magnetically ordered W-rich samples reveal magnetic phase separation, which suggests that the previously ignored interlayer coupling between the square planes plays a role in the suppression of magnetic order at x ≈ 0.6. These results highlight the complex magnetism of Sr2CuTe1–xWxO6 and hint at a new quantum critical point between 0.2 < x < 0.4.


INTRODUCTION
Spin-1/2 square-lattice antiferromagnets have been of significant scientific interest since the discovery of high-Tc superconductivity in cuprates. 1These antiferromagnetic and insulating parent phases become superconducting upon hole or electron doping. 2 The S = 1/2 square-lattice Heisenberg model is also one of the foundational models of frustrated magnetism. 3This model has two magnetic interactions: the nearestneighbor J1 interaction along the side of the square and the next-nearest-neighbor J2 along the diagonal.A dominant antiferromagnetic J1 leads to Néel antiferromagnetic order as observed in the high-Tc parent phases, while dominant J2 leads to columnar antiferromagnetic order.2][13] A number of S = 1/2 square-lattice antiferromagnets are known with either Néel or columnar antiferromagnetic order, [14][15][16][17][18][19][20] but the predicted quantum spin liquid state has never been observed.4][25] 48,50 .The black squares represent measured Néel temperatures and the blue circles represent the lowest temperatures measured, where the magnetism remains dynamic.A spin-liquid-like state without magnetic order is observed between x = 0.05 and x = 0.6.
The suppression of magnetic order in such as wide region in Sr2CuTe1-xWxO6 is likely related to the significant Te/W disorder on the B''-site.This results in a special type of bond disorder, where the J1 and J2 interactions are effectively switched on and off depending on the local B''-cation in the middle of each Cu 2+ square. 52,53The ground state in the spin-liquid-like region has been proposed to be a random-singlet state 23- 25,54,55 .The random-singlet state is a disorder-induced spin liquid, where spin singlets of different strengths are formed based on the underlying quenched disorder. 23Partial spin freezing into either a "spin jam" state 56 or patches of Néel and columnar-type correlated spins 53 have also been proposed for this region.
Here we present an average and local scattering investigation of Sr2CuTe1-xWxO6 revealing new insights into this unique magnetic system.Using neutron diffraction, we show that the average crystal structure of x = 0.5 with the spin-liquid-like state is tetragonal at low temperatures retaining an undistorted square of S = 1/2 Cu 2+ cations.Our combined neutron and synchrotron X-ray total scattering experiments show the local structure of x = 0.5 is well described by the average structure.However, we are unable to resolve effects of any potential clustering or ordering of Te and W on the B''-site.Our neutron diffraction study of the magnetic order in x = 0.9, 0.8 and 0.7 reveals columnar magnetic order with k = (0, ½, ½) as expected.However, an additional reflection belonging to the propagation k = (0, ½, 0) is observed for x = 0.8 and 0.7.This suggests that Te-doping in the columnar region disturbs the interlayer magnetic interactions and is responsible for the suppression of magnetic order at x ≈ 0.6.Finally, our polarized neutron study of the compositions x = 0.2 and 0.5 reveal distinctly different local spin correlations related to the short-range correlated states above TN in the parent phases.This suggests x = 0.2 and 0.5 have different ground states, despite both being in the spinliquid-like region, and a quantum critical point is expected around x ≈ 0.3.

EXPERIMENTAL METHODS
Polycrystalline powder samples of Sr2CuTe1-xWxO6 with 0 ≤ x ≤ 1 were synthesized using a conventional solidstate synthesis method as described previously. 48Stoichiometric quantities of SrCO3, CuO, TeO2 and WO3 (≥99.995%,Alfa Aesar) were ground in an agate mortar.The precursor mixture was calcined in air at 900 °C for 12 hours.Synthesis was carried out in air at 1050 °C for 72 h with intermittent grindings.
Time-of-flight neutron total scattering experiments were carried out at the POLARIS diffractometer 57 at the ISIS Neutron and Muon Source.Approximately 11 g of Sr2CuTe0.5W0.5O6(x = 0.5) powder was sealed in an 8mm vanadium can.Experiments were carried out in a cryostat at temperatures of 5 K, 100 K and 300 K.The empty sample can and cryostat were also measured for background correction.The data were reduced using standard procedures in Mantid 58 to obtain the Bragg scattering patterns for individual detector banks.Rietveld refinement was carried out using FULLPROF 59 and the structural figures were made with VESTA. 60ynchrotron X-ray total scattering experiments were carried out at the I15-1 diffractometer at Diamond Light Source using a wavelength of 0.161669 Å (76 keV).A 2D Perkin Elmer XRD4343CT detector was positioned 20 cm away from capillary to maximize the Q-range for optimal PDF data quality.A dark (without X-rays) detector image was collected to determine the dark current contribution and subsequently subtracted from the data; the detector was kept in a constant read-out state and air-cooled with fans to maintain a constant temperature, which led to negligible changes to the dark current contribution during the experiment.
The pair-distribution functions (PDFs) were obtained using the GUDRUN software package.Neutron data collected at 300 K on POLARIS were corrected for absorption, multiple scattering, and background from the sample environment.X-ray scattering data collected at room temperature on I15-1 were corrected for scattering from the empty capillary, Compton scattering, and incident beam polarisation.The neutron PDF was obtained with a maximum wavevector magnitude Qmax = 40 Å -1 and the X-ray PDF was obtained with Qmax = 25 Å -1 .The PDFs were expressed as D(r), in the notation of ref. 61 .
Constant-wavelength neutron diffraction was measured at the high-intensity D20 diffractometer at the Institut Laue-Langevin.2g of Sr2CuTe1-xWxO6 powders with x = 0.7, 0.8 and 0.9 were enclosed in 6 mm vanadium cans.These compositions are known to magnetically order at TN = 7 K, 11 K and 15 K, respectively. 48he data were collected at a wavelength of 2.41 Å at temperatures of 2 K and 30 K. The exact wavelength was determined by refining room-temperature neutron diffraction data against laboratory X-ray data for the x = 0.9 composition.The magnetic Bragg scattering was extracted by subtracting the nuclear scattering observed at 30 K from the 2 K data.The magnetic structure of the x = 0.9 sample was refined using FULLPROF.59 Potential k-vectors in the I4/m space group were considered based on the Brillouin zone database 62 of the Bilbao Crystallographic Server [63][64][65] and the k-search program included in the FULLPROF Suite.59 Magnetic phase fractions for x = 0.8 and 0.7 were evaluated by refining the two main magnetic peaks while fixing the ordered moment to be the same in both magnetic phases.The scale factor was fixed by first refining the crystal structure of the corresponding sample.The full width at half maximum (FWHM) was evaluated by fitting a single peak Voigt function.
Diffuse magnetic scattering was investigated at the D7 diffuse scattering spectrometer 66,67 at the Institut Laue-Langevin.11-19g of sample powder was sealed in aluminum cans with inserts to form an annulus shape.The samples were measured using cold neutrons with a wavelength of 4.8 Å (Ei = 3.55 meV).An orange cryostat was used for temperature control.The samples Sr2CuTeO6 (x = 0) and Sr2CuWO6 (x = 1), which magnetically order at 29 K and 24 K, respectively, were measured at 40 K in the short-range correlated state.Sr2CuTeO6 was also measured at 1.5 K, 60 K and 100 K.The spin-liquid-like x = 0.2 and 0.5 samples 47,48 were measured at 1.5 K.The collected data were reduced using LAMP. 68The data were corrected for polarizer efficiency with a quartz standard and for detector efficiency with a vanadium standard.The vanadium standard was also used to normalize the data to an absolute intensity scale.The magnetic signal was isolated using xyz polarization analysis, which removes the non-magnetic signal (including background). 69We have previously presented the raw data with limited analysis for x = 0.2 and 0.5 in ref. 53 .The magnetic diffuse scattering was fitted to an equation described later using the non-linear curve fitting tool in OriginPro.The diffuse scattering was also modelled with a Reverse Monte-Carlo (RMC) approach as implemented in SPINVERT using 8 × 8 × 6 supercells. 70Each SPINVERT analysis was repeated 10 times in order to reduce statistical noise.2][73] .

RESULTS AND DISCUSSION
Low-temperature crystal structure of x = 0.5 Sr2CuTeO6 and Sr2CuWO6 crystallize in the B-site ordered double perovskite structure with the tetragonal space group I4/m. 30,32,34,35The tetragonal symmetry is retained at low temperatures based on neutron diffraction measurements. 34,35Therefore, the square-lattice of the S =1/2 Cu 2+ cations remains undistorted at low temperatures, where the quantum magnetism occurs.The d 10 /d 0 doping does not have a significant effect on the room-temperature crystal structure, as the Sr2CuTe1-xWxO6 solid solution retains tetragonal symmetry for the full range 0 < x < 1. 47,48 However, the low-temperature crystal structure has not been reported for the doped samples.220) and (004) reflections showing that there is no peak splitting or anisotropic line broadening, which confirms the structure is tetragonal.
We investigated the structure of the main spin liquid composition x = 0.5 using neutron diffraction at 5 K, 100 K and 300 K.The space group remains tetragonal I4/m at all temperatures and no structural transition is observed.The refined structure at 300 K (Supporting Information) is essentially identical to the previously published structure based on laboratory X-ray diffraction. 47Figure 2 shows the refined time-of-flight neutron data for x = 0.5 at 5 K.If the symmetry is lowered from tetragonal, one would expect either the (220) reflection to split or anisotropic broadening of this reflection if the splitting is too small to be resolved.The (220) and (004) reflections are highlighted in the inset.The (220) peak does not split nor is there any anisotropic line broadening.This confirms the average structure remains tetragonal down to 5 K.This means that the Cu 2+ cations are arranged in a perfect square even at low temperatures.The refined parameters are listed in Table 1.The bond lengths and angles are very similar to the parent phases Sr2CuTeO6 and Sr2CuWO6 at low temperatures 34,35 due to the similar ionic radius of W 6+ and Te 6+ .We do not observe any superlattice reflections arising from Te/W ordering and therefore there is no long-range order of Te and W occupancies on the B''site.
Pair-distribution function analysis of x = 0.5 Our Bragg diffraction data for the x = 0.5 composition are well-described by the average structure model with no long-range ordering on the W/Te site.However, this result does not rule out short-range ordering of W and Te.Such short-range ordering would give rise to broad (diffuse) scattering features, which are not modeled in Rietveld refinement.However, such features can be apparent at small distances r in the pair-distribution function (PDF), which is the Fourier transform of the diffracted intensity, appropriately normalized and corrected for background scattering.
To investigate this possibility, we analysed our PDF data for x = 0.5 using the Topas Academic software 74 .Co-refinements were performed against X-ray and neutron PDF collected at T ≈ 300 K. We first performed refinements of the average structure over a large r-range (rmax = 40 Å) to determine the instrumental parameters Qdamp and Qbroad, 75 which were fixed in subsequent refinements.We tested three models of local Te/W occupancies against the neutron+X-ray PDF data: (i) the average-structure model, corresponding to locally random Te/W occupancies; (ii) clustering of Te and W into domains, so that the measured PDF is the average of the PDFs of Sr2CuTeO6 and Sr2CuWO6 with identical lattice constants and structural parameters; (iii) a model of local anti-clustering of Te and W, such that W-Te neighbours are favoured over W-W or Te-Te as far as possible.This last model corresponds to the X1 + irrep for Te/W ordering, and was generated using the ISODISTORT 76 program.For each model, the refined parameters were the scale factors for the two data sets, a and c lattice parameters, O fractional coordinates, isotropic displacement parameters for all atoms, and the low-r peak-sharpening function d1 defined in ref. 75 (13 refined parameters for each model).Refinements were performed for each model over the low-r region (1.0 ≤ r ≤ 9.8 Å), and the wide-r region (1.0 ≤ r ≤ 40 Å).
Our neutron PDF data and fits for the three models described above are shown in Fig. 3c over the 1.0 ≤ r ≤ 9.8 Å range.Good agreement with the data is observed for all models.Unfortunately, however, the goodness-of-fit is not distinguishable between different models, over either the low-r or wide-r regions.That is, the PDF data are equally consistent with correlated vs. random Te/W occupancies.This result is perhaps surprising, since Te and W have reasonable scattering contrast for both neutrons ((bW/bTe) 2 = 0.70) and X-rays ( (fW/fTe ) 2 = 2.03), with different contrast ratios for each experiment.However, the situation of physically different models giving rise to essentially identical PDFs is not unknown in the literature. 77We hypothesise that this situation is more likely in materials where the disordered atoms occupy a high-symmetry site, such as the distorted face-centred cubic lattice of We/Te atoms, where the limitations of powder data are likely to be more important.
Overall, the average-structure model yields good agreement with the measured PDF at low r, which suggests that the O1 and Cu atoms do not undergo large "size effect" displacements depending on their local Te/W coordination.In support of this conclusion, we do not observe any anomalously large displacement parameters in our Rietveld refinement, see Table 1.The displacement parameter for O1 is somewhat elongated along c, and the Cu displacement parameter is highly cigar-shaped, yet not anomalously large, with u33(Cu) = 0.0051(5) Å 2 at 100 K.For comparison, a well-studied quantum-spin-liquid candidate material with occupational disorder, YbMgGaO4, has a much larger u33(Yb) = 0.0240(4) Å 2 at 100 K, 78 which implies that local Cu displacements in Sr2CuTe0.5W0.5O6 are small compared to local Yb displacements in YbMgGaO4 .The observation of minimal local distortions in Sr2CuTe0.5W0.5O6 is consistent with the fact that Sr2CuTeO6 and Sr2CuWO6 have nearly identical crystal structures.
Magnetic order in the W-rich materials x = 0.9, 0.8, 0.7 One of the main reasons the Sr2CuTe1-xWxO6 system is so interesting is the fact that the parent phases Sr2CuTeO6 (x = 0) and Sr2CuWO6 (x = 1) have different magnetic structures: x = 0 has the Néel structure with k = (½, ½, 0) and x = 1 has the columnar structure with k = (0, ½, ½). 34,35The compositions x = 0.9, 0.8 and 0.7 are also known to magnetically order at TN = 15 K, 11 K and 7 K based on muon measurements. 48The magnetic structure for these compositions was proposed to be columnar, 48 which was supported by later neutron diffraction experiments revealing the presence of the (0½½) reflection around |Q| = 0.68 Å -1 . 51However, diffraction data was only presented for this one reflection.We have reinvestigated the type of magnetic order in x = 0.9, 0.8 and 0.7 using high-intensity neutron diffraction.
The refined magnetic Bragg scattering for x = 0.9 is shown in Figure 4a.Clear magnetic Bragg peaks are observed at positions corresponding to the propagation vector k = (0, ½, ½).The magnetic scattering is almost identical to previous reports on x = 1. 35The propagation vector has only one irreducible representation Γ2, which has three basis vectors along a, b and c.Similar to x = 1, setting the moment along a resulted in the best fit.The refined Cu 2+ ordered moment was found to be 0.45(1) μB, which is slightly lower than the 0.57(1) μB reported for x = 1. 35This magnetic structure is shown in the Figure4b inset.A comparison of the magnetic Bragg scattering for x = 0.9, 0.8 and x = 0.7 is presented in panel (b).The main peak at |Q| = 0.68 Å corresponding to (0½½) is still observed in x = 0.8 and 0.7, but the magnetic scattering is significantly weaker and the other reflections of corresponding to k = (0, ½, ½) can no longer be resolved.Surprisingly, we observe significant magnetic scattering at |Q| = 0.58 Å -1 corresponding to (0½0).This reflection is not allowed for the propagation vector k = (0, ½, ½), and therefore this reflection must belong to another propagation vector.Very weak scattering at this position also occurs in the x = 0.9 sample.Figure 4. (a) Rietveld refinement of the magnetic neutron diffraction data for Sr2CuTe0.1W0.9O6(x = 0.9) at 2 K obtained by subtracting the nuclear scattering measured at 30 K. The propagation vector is k = (0, ½, ½) with a refined moment of 0.45(1) μB along a direction (Rmag = 13.3%).(b) Comparison of the magnetic neutron diffraction data for Sr2CuTe1-xWxO6 samples x = 0.9, 0.8 and 0.7 at 2 K.The magnetic scattering has been normalised using the nuclear scale factor to account for differences in sample masses and counting times.A number of magnetic Bragg peaks are observed for x = 0.9 with the main peak being (0½½).For x = 0.8 and 0.7, intensity of the magnetic reflections is significantly reduced.The (0½½) peak is retained for these compositions, but a new peak at (0½0) is observed.Panel (b) inset: Magnetic structure of x = 1 and 0.9 with propagation vector k = (0, ½, ½) and the proposed magnetic structure for the second magnetic phase in x = 0.8 and 0.7 with propagation vector k = (0, ½, 0).The ordering along c changes from antiferromagnetic to ferromagnetic.Bottom panels: two-phase magnetic refinement of Sr2CuTe1-xWxO6 samples (c) x = 0.9, (d) x = 0.8 and (e) x = 0.7.The x = 0.9 sample is almost entirely in the k = (0, ½, ½) magnetic structure that is also observed in x = 1.For x = 0.8 we obtain phase fractions of 65(5)% k = (0, ½, 0) and 35(5)% k = (0, ½, ½) and for x = 0.7 we find 55(8)% k = (0, ½, 0) and 45(8)% k = (0, ½, ½).
The magnetic propagation vectors of materials often correspond to high-symmetry points of the Brillouin zone, and therefore these are an excellent starting point for searching for reasonable k-vectors.The columnar magnetic structure of x = 1 and 0.9 with k = (0, ½, ½) corresponds to the high-symmetry point X.While one might expect k = (0, ½, 0) observed in x = 0.8 and 0.7 to also be a high-symmetry point of the Brillouin zone, it is not one in the space group I4/m.This is due to the I-centering of the lattice and the relationship between the conventional unit cell and the primitive cell.As a consequence, this magnetic structure belongs to the line SM with k = (0, a, 0) and a = 0.500 (2).
Symmetry-allowed magnetic structures for k = (0, ½, 0) were evaluated using BASIREPS. 59Two irreducible representations were found: Γmag = Γ1 + Γ2.Γ1 corresponds to magnetic moments along c, while Γ2 corresponds to moments within the ab plane.Given that we were only able to resolve the main magnetic peak, we are unable to determine the moment directions.Our proposed magnetic structure for x = 0.8 and 0.7 with k = (0, ½, 0) is presented in Figure 4b inset with magnetic moments along a.This structure corresponds to columnar antiferromagnetic order in the square-layers in the ab plane similar to x = 1.However, the interlayer coupling along c is now ferromagnetic instead of antiferromagnetic.
The presence of the forbidden (0½0) reflection in the magnetic scattering of Sr2CuTe1-xWxO6 samples x = 0.8 and 0.7 can be interpreted in two ways.It can be due to magnetic phase separation such that parts of the sample have the k = (0, ½, ½) and parts have the k = (0, ½, 0) magnetic structure.The other possibility is that there is a complex multi-k structure that accounts for all the observed magnetic scattering.The weak magnetic scattering makes this distinction complicated, since we can only observe two magnetic Bragg peaks.In the case of magnetic phase separation, we would expect the intensity of the two observed magnetic peaks to vary freely between samples.If the materials have a multi-k structure, the relative intensities should remain the same and be an integer ratio.
The widths of the magnetic Bragg peaks are wider than the nuclear peaks in these materials.For x = 0.9 we obtain FWHMs of 0.64(2)° for the main magnetic peak (0½½) and 0.46(1)° for the first nuclear peak (011).The FWHMs for x = 0.8 are 0.69(3)° and 0.68(5)° for the (0½0) and (0½½) magnetic peaks and 0.47(1)° for (011) and for x = 0.7 we obtain 0.51(8)°, 0.55(6)° and 0.47(1)° respectively.The nuclear peak widths are dominated by instrumental broadening as opposed to sample broadening.This supports the presence of size broadening for the magnetic peaks: the size of the magnetic domains is smaller than the crystallite size.Clustering into Te-rich and W-rich regions could lead to the presence of both magnetic structures within a single crystallite.Unfortunately, we were unable to determine the presence or absence of clustering in the Xray PDF analysis.
The magnetic interactions in Sr2CuTe1-xWxO6 materials are highly two-dimensional.The interlayer interaction Jc is an extended Cu -O -W/Te -O -Cu superexchange similar to J2, but much weaker due to the fully occupied Cu dz 2 orbitals.In Sr2CuWO6 (x = 1), the interlayer exchange is antiferromagnetic with Jc = -0.01meV, while the dominant in-plane exchange is J2 = -9.5 meV.As a result of the weak Jc, inelastic scattering in the forbidden (0½0) position is observed even in undoped Sr2CuWO6. 27The Jc interaction is ferromagnetic in Sr2CuTeO6 (x = 0) following the trend observed in the sign of J2.As a result, it appears that Te-doping (decreasing x) disrupts the interlayer interactions leading to the appearance of the competing k = (0, ½, 0) structure.Similar magnetic phase separation into structures with different interlayer orderings is also observed in the solid solution Sr2Cr1.85Mn1.15As2O2,which has a square-layer of Cr 2+ (S = 2) cations. 79The weak interlayer interaction is ultimately responsible for magnetic ordering in the Sr2CuTe1-xWxO6 materials, since magnetic order in two dimensions at non-zero temperature is forbidden by the Mermin-Wagner theorem when the interactions are isotropic. 80As such, the disorder in Jc could be the cause of the quantum phase transition from columnar magnetic order in the W-rich side to the spin-liquid-like state at x ≈ 0.6.This is supported by our diffuse magnetic scattering analysis on x = 0.5 and x = 0.2 samples (Supporting Information), which show that the average interlayer spin correlations are weak and antiferromagnetic in x = 0.5, but change to weakly ferromagnetic for x = 0.2.
Diffuse magnetic scattering in the spin-liquid-like materials x = 0.2 and 0.5 The Sr2CuTe1-xWxO6 parent phases x = 0 (Sr2CuTeO6) and x = 1 (Sr2CuWO6) are known to have shortrange correlated magnetic states above TN based on inelastic neutron scattering studies. 26,27Spin correlations persist up at least 2TN ≈ 60 K in both compounds.We previously proposed that the spin-liquid-like state in x = 0.5 could be related to the columnar-type short-range correlated state in x = 1 based on inelastic neutron scattering data. 52Similarly, the spin-liquid-like state between x = 0.05 and 0.2 has been proposed to have Néeltype short-range correlations. 51We can test this hypothesis by measuring the diffuse magnetic scattering in the spin-liquid-like phases and in the short-range correlated states of the parent phases.This allows us to model the local spin correlations in these materials for the first time.
The diffuse magnetic scattering of Sr2CuTe1-xWxO6 samples extracted from our D7 experiment is shown in Figure 5.The parent phases x = 0 (a) and x = 1 (b) were measured above TN at 40 K, where they are in a short-range correlated magnetic state.The other samples measured were x = 0.2 (c) and x = 0.5 (d) at 1.5 K, which are in the spin-liquid-like region and lack long-range magnetic order.The diffuse scattering for x = 0 at 40 K has a peak around |Q| ≈ 0.85 Å -1 .This is related to the Néel magnetic order at low temperatures with the Bragg reflection (½½0) at |Q| = 0.82 Å -1 . 34The diffuse scattering for x = 1 at 40 K is very different, with a main peak around |Q| ≈ 0.6 Å -1 .This is related to the columnar magnetic order below TN, which has a magnetic Bragg reflection (0½½) at |Q| = 0.68 Å -1 and inelastic scattering at the forbidden (0½0) position at |Q| = 0.58 Å -1 . 27The scattering from the x = 0.2 sample at 1.5 K is similar to x = 0, although the peak at |Q| ≈ 0.85 Å -1 is significantly broader in x = 0.2 with FWHMs of 0.27(8) Å -1 and 0.61(15) Å -1 respectively.This supports the hypothesis that spin correlations in x = 0.2 are Néel-like.The diffuse scattering for x = 0.5 is similar to x = 1 with a peak around |Q| ≈ 0.6 Å -1 .This supports the hypothesis that spin correlations in x = 0.5 are columnarlike.
The low incident energy of Ei = 3.55 meV is a limitation in our D7 experiment.Ideally, the incident energy would be high enough to capture all features of the inelastic neutron scattering spectrum. 81In the case of Sr2CuTe1-xWxO6, significant inelastic scattering is observed up to at least 15 meV. 26,27,53Fitting the hightemperature (100 K) diffuse scattering of x = 0 to the Cu 2+ magnetic form factor results in  eff 2 = 0.91(2) μ B 2 , which is only one third of the expected  eff 2 = 3 μ B 2 for a S = 1/2 cation.We can estimate the effect of the missing magnetic scattering by integrating inelastic neutron scattering data up to higher energies.These cuts for x = 0.2 and 0.5 (Supporting Information) show that our D7 experiment does capture the essential features of the diffuse scattering.
In order to quantify the spin correlations in Sr2CuTe1-xWxO6, the observed magnetic scattering crosssections were fitted to: 82 where F(Q) is the magnetic form factor of Cu 2+ ,  eff 2 =  2 ( + 1), Zi is the number of neighboring spins at distance ri and 〈 0 ⋅   〉 is the average spin correlation between a central spin and its neighbors at distance ri.The spin correlations have been normalised such that 〈 0 ⋅   〉 = 1 (-1) corresponds to complete (anti-)ferromagnetic alignment of spins.We considered only the nearest-neighbor spins at r1 ≈ 5.4 Å and the in-plane next-nearest-neighbor spins at r2 ≈ 7.6 Å.These correspond to the spins along the side (J1 interaction) and diagonal (J2 interaction) of the square, respectively.Thus, we have three fitting parameters:  eff 2 and the spin correlations 〈 0 ⋅  1 〉 and 〈 0 ⋅  2 〉.For complete Néel antiferromagnetic order 〈 0 ⋅  1 〉 = -1 and 〈 0 ⋅  2 〉 = 1.For columnar order 〈 0 ⋅  1 〉 = 0 and 〈 0 ⋅  2 〉 = -1.1).The magnetic diffuse scattering has a peak at |Q| ≈ 0.85 Å -1 for x = 0 and at |Q| ≈ 0.6 Å -1 for x = 1.These are related to the Néel and columnar magnetic ordering below TN, respectively.The x = 0.2 data has a peak at |Q| ≈ 0.85 Å -1 similar to x = 0 above TN.For x = 0.5, the peak is observed at |Q| ≈ 0.6 Å -1 similar to x = 1 above TN.The diffuse magnetic scattering in the two spinliquid-like samples x = 0.2 and x = 0.5 is clearly different, but also clearly related to the two parent phases above TN.
The fits to equation ( 1) are shown in Figure 5.The fitting captures the main features of the scattering for all samples, but the model is missing significant intensity at the main peak positions at |Q| ≈ 0.85 Å -1 (x = 0) and |Q| ≈ 0.6 Å -1 (x = 0.5 and 1).The magnetic diffuse scattering becomes narrower and more Bragg-like when the temperature approaches TN, which is not captured by this model.This could explain why the main peak intensity does not fit well for x = 0 and x = 1.It should be noted that equation ( 1) is a simple model including only the two nearest-neighbor in-plane correlations, which are assumed to be fully independent of each other, and Heisenberg spins. 70Moreover, our experiment does not capture the full spectral weight, which makes the fitting more difficult.
For comparison, we also fitted the diffuse magnetic scattering using a Reverse Monte Carlo (RMC) method as implemented in SPINVERT 70 , see Supporting Information.The analysis was complicated by the low quality of the data, which is why we mainly present results from fitting equation (1).The main spin correlations obtained using SPINVERT broadly support our fitting results with the exception of x = 0.5, where the weaker 〈 0 ⋅  1 〉 is moderately antiferromagnetic in the equation (1) fit, but zero within experimental error (as expected) in the SPINVERT fit.The same main conclusion of Néel-like correlations in x = 0.2 and columnar-like in x = 0.5 is observed using both fitting methods.
One advantage of the RMC approach is that spin correlations up to further neighbors can easily be evaluated.The obtained spin correlations for x = 0.2 are Néel-type: strong antiferromagnetic nearest-neighbor 〈 0 ⋅  1 〉 correlations (r1 ≈ 5.4 Å) and ferromagnetic in-plane next-nearest neighbour correlations 〈 0 ⋅  2 〉 (r2 ≈ 7.6 Å).The spin correlations at distance 2r1 are weakly ferromagnetic, whereas at 2r2 they are nearly zero.The first three in-plane spin correlations are as expected for Néel-type correlations.The interlayer spin correlations are weak and ferromagnetic.For the x = 0.5 sample, we obtain 〈 0 ⋅  1 〉 ≈ 0 and strong antiferromagnetic 〈 0 ⋅  2 〉 correlations.The spin correlations at 2r1 and 2r2 are ferromagnetic.All four main in-plane spin correlations in x = 0.5 are precisely as expected for columnar-type correlations.The interlayer correlations are weak and antiferromagnetic.A comparison of the obtained and expected spin correlations is provided in the Supporting Information.
To summarise, the spin-liquid-like states in Sr2CuTe1-xWxO6 with x = 0.2 and x = 0.5 have short-range spin correlations very similar to the parent phases x = 0 and x = 1, respectively, above TN.The spin correlations in x = 0 and 0.2 are Néel-like while the correlations in x = 0.5 and 1 are mainly columnar-like.The cross-over from Néel to columnar correlations is likely to occur around x ≈ 0.3, because the inelastic neutron scattering data for x = 0.4 and 0.5 are very similar. 53It is unclear whether x = 0.2 and x = 0.5 have the same ground state given the significant differences in spin correlations.Significant Néel-type correlations are expected in the random-singlet state 55 as we observe for x = 0.2.The presence of Néel-type correlations in x = 0.5 is less certain.It is clear that the main spin correlations are columnar-type in x = 0.5, but some weak Néel-like correlations were found in the fits to equation (1).However, we did not observe these correlations in the SPINVERT analysis of the same data.As such, our experiment suggests x = 0.2 and x = 0.5 have a different ground state and that there is quantum critical point between them.This could be a quantum critical point between two different types of random-singlet states or a random-singlet state and a state with weakly frozen moments. 56We expect this quantum critical point, should it exist, to occur around x ≈ 0.3, where the spin correlations change from Néel to columnar.This could be further investigated by muon spectroscopy as the scaling behaviour of the muon spin relaxation rate should change in the presence of a critical point. 51

Conclusions
We have investigated different compositions of the S = 1/2 square-lattice antiferromagnet Sr2CuTe1-xWxO6 using neutron and X-ray scattering.The average crystal structure of the spin-liquid-like x = 0.5 sample remains tetragonal down to 5 K confirming a square magnetic lattice.Our PDF analysis showed the local structure is overall well described by the average structure, although we were unable to distinguish between different models of Te and W correlations (random, clustering or ordering).Our neutron diffraction experiments of the W-rich magnetically ordered materials reveal the presence of columnar magnetic order for x = 0.7-1.Surprisingly, magnetic phase separation was observed for x = 0.7 and 0.8 with part of the sample ordering ferromagnetically and part antiferromagnetically along c, while the columnar order in the ab-plane was preserved.This shows replacing W with Te leads to disorder in the interlayer interactions, which could be the origin of the transition to the spin-liquid-like state at x ≈ 0.6.
The spin correlations of the spin-liquid-like phases x = 0.2 and 0.5 were investigated using polarized neutrons.The spin-correlations in x = 0.2 are Néel-like, and very similar to the short-range correlated state observed above TN in Sr2CuTeO6.The spin correlations in x = 0.5 are mainly columnar-type with potentially some weak Néel-like correlations remaining, and similar to the short-range correlated state above TN in Sr2CuWO6.Despite both compositions being in the spin-liquid-like region, the spin correlations are very different.This suggests the ground states are also different, because Néel-type correlations are expected in the random-singlet state.If so, a quantum critical point would be expected around x ≈ 0.3, where the spin correlations change from Néel to columnar-like.
Our results highlight the complexity of Sr2CuTe1-xWxO6 as a S = 1/2 square-lattice antiferromagnet with frustration and disorder in the local in-plane and interlayer interactions.The strong suppression of Néel order at x ≈ 0.05 and the related quantum critical point has previously garnered significant attention. 50,51,55Our results show that further investigation is warranted at higher doping levels both at the other known quantum critical point at x ≈ 0.6 and our proposed new quantum critical point near x ≈ 0.3.Given that the d 10 /d 0 substitution approach for tuning magnetism is applicable to many 3d transition metal oxides, 37 our insights have relevance to other systems such as SrLaSb1-xNbxCuO6 or the spin ladder Ba2CuTe1-xWxO6. 40,46

Table of contents
Crystal structure of Sr2CuTe0.5W0.5O6(x = 0.5) 2 Magnetic scattering in the ordered W-rich phases 5 Magnetic scattering of Sr2CuTeO6 at various temperatures 5 Integrated inelastic neutron scattering data 6 SPINVERT fits of diffuse magnetic scattering 7 Crystal structure of Sr 2 CuTe 0.5 W 0.5 O 6 (x = 0.5) Table S3.The refined low-temperature crystal structure of Sr2CuTe0.5W0.5O6(x = 0.5) at 5 K based on POLARIS time-of-flight neutron data.Space group I4/m with lattice parameters a = 5.41025(10) Å and c = 8.41718(18) Å. Rp = 2.60% and Rwp = 2.05% for the high-resolution bank 5 (2θ = 146.72°).This table is reproduced here for convenience from Table 1 in the article.6) μB previously reported in ref. 1 , but consistent with the 0.57(1) μB reported for Sr2CuWO6. 2nset: The Néel antiferromagnetic structure of Sr2CuTeO6.The nearest-neighbor spins order antiferromagnetically along the side of the square, while the in-plane next-nearest-neighbor (square diagonal) and interplane ordering along c are ferromagnetic.SPINVERT fits of diffuse magnetic scattering  Table S5.In-plane spin correlations obtained from the SPINVERT fits of Sr2CuTe1-xWxO6 diffuse magnetic scattering and the expected spin correlations for complete Néel or columnar magnetic order.J1 corresponds to the side of the square, J2 to the diagonal, J3 to a Chess Knight move (two lengths along one side and one along a perpendicular side) and Jc to the interlayer distance.

Figure 1 .
Figure 1.(a) The B-site ordered double perovskite structure of Sr2CuTe1-xWxO6, where the green, blue, gray and red spheres represent Sr, Cu, Te/W and O, respectively.The magnetic interactions are highly twodimensional in ab plane, where the S = 1/2 Cu 2+ cations form a square lattice.(b) The Néel antiferromagnetic structure of Sr2CuTeO6 (x = 0) stabilised by a dominant antiferromagnetic J1 interaction.(c) The columnar magnetic structure of Sr2CuWO6 (x = 1) stabilised by a dominant antiferromagnetic J2 interaction.(d) Magnetic phase diagram of the solid solution Sr2CuTe1-xWxO6 based on muon spectroscopy from refs.48,50 .The black squares represent measured Néel temperatures and the blue circles represent the lowest temperatures measured, where the magnetism remains dynamic.A spin-liquid-like state without magnetic order is observed between x = 0.05 and x = 0.6.

Figure 2 .
Figure 2. Rietveld refinement of the time-of-flight neutron diffraction data for Sr2CuTe0.5W0.5O6(x = 0.5) at 5 K from bank 4 on POLARIS (2θ = 92.59°)with Rp = 2.38% and Rwp = 2.29%.The low-temperature structure remains tetragonal in the space group I4/m.Inset: Close-up of the (220) and (004) reflections showing that there is no peak splitting or anisotropic line broadening, which confirms the structure is tetragonal.

Figure 3 .
Figure 3. Small-box modelling of the combined neutron and synchrotron X-ray pair-distribution function data for Sr2CuTe0.5W0.5O6(x = 0.5) at T ≈ 300 K.The wide-r (1 ≤ r ≤ 40 Å) (a) neutron and (b) X-ray PDF data are well described by the average structure.In panels (c) and (d), we fitted the neutron and X-ray PDF in the low-r range (1 ≤ r ≤ 9.8 Å) using three different models: a random distribution of Te and W (average structure), clusters of Te and W and finally Te-W ordering (X1+) on the B''-sites.Our experiment is not sufficient to differentiate between these models and the fitted lines overlap.

Figure 5 .
Figure 5. Magnetic diffuse scattering of the Sr2CuTe1-xWxO6 parent phases (a) x = 0 (Sr2CuTeO6) and (b) x = 1 (Sr2CuWO6) at 40 K in the short-range correlated state above TN and the spin-liquid-like phases (c) x = 0.2 and (d) x = 0.5 at 1.5 K.The red lines are fits to equation (1).The magnetic diffuse scattering has a peak at |Q| ≈ 0.85 Å -1 for x = 0 and at |Q| ≈ 0.6 Å -1 for x = 1.These are related to the Néel and columnar magnetic ordering below TN, respectively.The x = 0.2 data has a peak at |Q| ≈ 0.85 Å -1 similar to x = 0 above TN.For x = 0.5, the peak is observed at |Q| ≈ 0.6 Å -1 similar to x = 1 above TN.The diffuse magnetic scattering in the two spinliquid-like samples x = 0.2 and x = 0.5 is clearly different, but also clearly related to the two parent phases above TN.
Figure S5.Refined magnetic structure of Sr2CuTeO6 at 1.5 K. Propagation vector k = (½, ½, 0), a moment of 0.58(4) μB per copper and Rmag = 23.5 %.The refined moment is slightly smaller than the 0.69(6) μB previously reported in ref.1  , but consistent with the 0.57(1) μB reported for Sr2CuWO6.2Inset: The Néel antiferromagnetic structure of Sr2CuTeO6.The nearest-neighbor spins order antiferromagnetically along the side of the square, while the in-plane next-nearest-neighbor (square diagonal) and interplane ordering along c are ferromagnetic.

Figure S6 .
Figure S6.Integrated inelastic neutron scattering between 2.5 meV and 7.5 meV energy transfer for Sr2CuTe1-xWxO6 samples a x = 0.2 and b x = 0.5.Data from refs. 3,4collected on MERLIN at T = 5 K with Ei = 18 meV.The positions of the maximums and the overall shapes of the curves are very similar to the D7 measurements with Ei = 3.55 meV.This shows that despite not capturing the full spectral weight, the D7 data is representative of the overall scattering at least up to 7.5 meV.The limited low-|Q| coverage of the MERLIN data at higher energies prohibits comparisons up to 15-20 meV, where features are still observed in the spectra.

Figure
FigureS7.a SPINVERT fits of the diffuse magnetic scattering of Sr2CuTeO6 (x = 0) and Sr2CuWO6 (x = 1) at 40 K in the short-range correlated state above TN.The quality of the fits is limited by the very weak magnetic scattering and limited energy coverage.The magnetic scattering of x = 0 has a peak at |Q| ≈ 0.85 Å -1 , which is related to the (½½0) reflection of the Néel magnetic structure below TN.For x = 1, a peak is observed at |Q| ≈ 0.6 Å -1 , which is related to the main (0½½) magnetic Bragg peak of the columnar structure and inelastic scattering from the forbidden (0½0) reflection due to the weak interlayer coupling.b Spin correlation functions of Sr2CuTeO6 and Sr2CuWO6 obtained from SPINVERT fits.The spin correlations for x = 0 are Néel-type with antiferromagnetic J1 correlations and ferromagnetic J2 correlations.The correlations in x = 1 are columnar-type with very weak J1 and antiferromagnetic J2 correlations.

Figure
Figure S8. a SPINVERT fits to diffuse magnetic scattering of x = 0.2 and 0.5 samples at 1.5 K.The quality of the fits is limited by the very weak magnetic scattering and limited energy coverage.The maximum in scattering is observed at different |Q| positions in the two samples: at |Q| ≈ 0.85 in x = 0.2 and at |Q| ≈ 0.6 Å -1 in x = 0.5.This indicates a significant difference in the spin correlations of these two materials.There is a feature in the x = 0.2 data at |Q| ≈ 1.13 Å -1 that is not captured in the model.b Spin correlation functions of the x = 0.2 and 0.5 samples obtained from SPINVERT fits.Spin correlations in the x = 0.2 are Néel-type with strong antiferromagnetic J1 and ferromagnetic J2.The correlations are columnar-type in x = 0.5 with negligible J1 and strong antiferromagnetic J2 as expected.The interlayer spin correlation Jc is ferromagnetic in x = 0.2 and antiferromagnetic in x = 0.5.