Magnetic properties and magnetic structure of the frustrated quasi-one-dimensional antiferromagnet SrCuTe 2 O 6
P. Saeaun, Y. Zhao, P. Piyawongwatthana, T. J. Sato, F. C. Chou, M. Avdeev, G. Gitgeatpong, K. Matan
MMagnetic properties and magnetic structure of the frustrated quasi-one-dimensionalantiferromagnet SrCuTe O P. Saeaun, Y. Zhao,
2, 3
P. Piyawongwatthana, T. J. Sato, F. C. Chou,
5, 6, 7, 8
M. Avdeev, G. Gitgeatpong,
10, 11 and K. Matan
1, 11, ∗ Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Department of Materials Science and Engineering,University of Maryland, College Park, Maryland 20742, USA NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA IMRAM, Tohoku University, Sendai, Miyagi 980-8577, Japan Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan Taiwan Consortium of Emergent Crystalline Materials,Ministry of Science and Technology, Taipei 10622, Taiwan Center of Atomic Initiative for New Materials, National Taiwan University, Taipei 10617, Taiwan Australian Nuclear Science and Technology Organisation,ANSTO, Locked Bag 2001, Kirrawee DC, NSW, Australia Department of Physics, Faculty of Science and Technology,Phranakhon Rajabhat University, Bangkok 10220, Thailand Thailand Center of Excellence in Physics, Ministry of Higher Education, Science,Research and Innovation, 328 Si Ayutthaya Road, Bangkok 10400, Thailand (Dated: September 28, 2020)Magnetization measurements on single-crystal cubic SrCuTe O with an applied magnetic fieldof along three inequivalent high symmetry directions [100], [110], and [111] reveal weak magneticanisotropy. The fits of the magnetic susceptibility to the result from a quantum Monte Carlo simula-tion on the Heisenberg spin-chain model, where the chain is formed via the dominant third-nearest-neighbor exchange interaction J , yield the intra-chain interaction ( J /k B ) between 50.12(7) K forthe applied field along [110] and 52.5(2) K along [100] with about the same g -factor of 2.2. Single-crystal neutron diffraction unveils the transition to the magnetic ordered state as evidenced by theonset of the magnetic Bragg intensity at T N1 = 5 . T N2 reported previously. Based on irreducible representation theory and magnetic spacegroup analysis of powder and single-crystal neutron diffraction data, the magnetic structure in theShubnikov space group P
32, where the Cu S = 1 / . µ B ,which represents 48% reduction from the expected value of 1 µ B , suggests the remaining influenceof frustration resulting from the J and J bonds. I. INTRODUCTION
During the past few decades, quantum magnetismin low-dimensional and frustrated systems have cap-tured the interest of condensed-matter physicists be-cause of their potential to exhibit exotic magnetic groundstates such as spin ice , a quantum valence bond(dimer) solid , and the most sought-after quantumspin-liquid . Among these unconventional states, aquantum spin liquid has gained the most attention be-cause its discovery and fundamental understanding couldpotentially yield a better understanding of other phe-nomena in physics such as high- T c superconductivity ,topological states, and anyonic physics , and lead to ap-plications in quantum computing .A search for this elusive quantum spin liquid hasso far focused on low-dimensional and frustrated lat-tices, which include triangular-based lattices. Topo-logically, triangular (two-dimensional (2D) edge-sharing-triangle), kagome (2D corner-sharing-triangle), hyper-kagome (3D corner-sharing triangle), and pyrochlore(3D edge-sharing-tetrahedron) lattices are considered as possible hosts of the quantum-spin-liquid groundstate due to their high degree of frustration, giv-ing rise to a macroscopically degenerate ground state,which could prompt the formation of a highly en-tangled quantum state. In recent years, significantprogress has been made in the search, and manypossible realizations of the quantum spin liquid werediscovered and exhaustively studied . A group oftriangular-based materials that show quantum-spin-liquid traits includes ZnCu (OH) Cl , Na Ir O ,YbMgGaO , Ca Cr O , Ce Zr O , andPbCuTe O (Cu , S = 1 / .Studies of PbCuTe O revealed the absence of a long-range λ -like transition down to 2 K and possible emer-gence of quantum spin liquid at low temperatures .On the other hand, SrCuTe O with an almost identicalcrystal structure exhibits two successive magnetic phasetransitions, one of which is to a magnetically orderedstate at the N´eel temperature T N1 of 5.5 K . In or-der to decipher the underlying mechanism that gives riseto different ground states in these two seemingly similarsystems, a detailed study on a single-crystal sample is a r X i v : . [ c ond - m a t . s t r- e l ] S e p required. Hence, in this article we report the studies ofmagnetic properties and a magnetic structure on powderand single-crystal SrCuTe O .Noncentrosymmetric SrCuTe O crystallizes in thechiral cubic space group P
32 (No. 213) with latticeparameter a = 12 . . The crystal structure con-sists of CuO square plates, units of TeO , and oxygenoctahedra surrounding strontium (Sr) atoms as shownin Fig. 1(a). The Cu ions in the CuO plaquettescarry spin 1/2 and give rise to the magnetic propertiesof this system. As with PbCuTe O , the spin networkof SrCuTe O consists of three types of intertwined net-works connected by the nearest-, second-nearest-, andthird-nearest-neighbor exchange interactions J , J , and J , respectively , as shown in Fig. 1(b). The Cu ions connected by J form an isolated triangle with Cu-O-Sr-O-Cu superexchange pathways. The second-nearest-neighbor network formed by J via Cu-O-Te-O-Cu path-ways connect the Cu ions to form a hyper-kagomenetwork. Finally, J links the Cu ions to form spinchains along the cubic crystallographic axes. First prin-ciple calculations performed to estimate the strength ofthese exchange interactions showed that for SrCuTe O , J is the most dominant with J about 10% of J and J about 1% of J , and established that SrCuTe O isa spin-chain system with relatively weak and intricateinter-chain couplings . In contrast, for PbCuTe O , J is the most dominant interaction, and hence this systemcan be characterized as the 3D hyper-kagome antiferro-magnet, in which geometric frustration can potentiallysuppress the N´eel state and prompt the emergence of thequantum spin state .Previous magnetic susceptibility and heat capacitymeasurements on a SrCuTe O powder sample re-vealed magnetic transitions at T N1 = 5 . T N2 , which issmaller than T N1 and field-dependent . The magneticsusceptibility data also showed a broad maximum, whichis typical in a spin-chain system, at around 32 K suggest-ing short-range correlation. A fit of the high-temperaturedata to the Curie-Weiss law yields the Curie-Weiss tem-perature Θ CW = −
35 K. Assuming that the spin-chainnetwork is connected by J , the dominant exchange in-teraction ( J /k B ) is estimated to be 49 K as comparedto the first-principle-calculation value of 45 K . Fur-thermore, multiple magnetic transitions with a nontrivial H − T phase digram were observed for magnetizations upto 9 T .In this work, magnetic properties of SrCuTe O arestudied using magnetization measurements on a single-crystal sample, and the magnetic transition at T N1 andmagnetic structure of the ordered state are investigatedusing neutron diffraction. This article is organized as fol-lows. In Section II, the single-crystal synthesis and ex-perimental methods are described. The result of the sin-gle crystal X-ray diffraction is discussed in III A, macro-scopic magnetic properties on the single crystals are in- vestigated and analyzed using quantum Monte Carlo sim-ulations in III B, and powder and single-crystal neutrondiffraction data are discussed in III C. The article endswith the conclusion in Section IV II. EXPERIMENTAL DETAILS
A powder sample of SrCuTe O was first synthe-sized by the standard solid state reaction of high puritySrCO , TeO and CuO. The preparation method is de-scribed elsewhere . The obtained pure phase powder ofSrCuTe O was then used as a starting material for sin-gle crystal growth using the vertical gradient freeze tech-nique. The powder was loaded into a pointed-bottomalumina tube (recrystallized alumina 99.8%). The cru-cible was then sealed in an evacuated quartz tube, whichis crucial to minimize the formation of SrCuTe O as animpurity phase. The sample was melted at 800 ◦ C andheld at this temperature for 24 hours to ensure a homo-geneous and complete melt, before subjecting the sampleto a 20 ◦ C/cm temperature gradient in a vertical furnaceat a rate of 1 cm/day. After the sample reached the po-sition with T = 650 ◦ C, the furnace was cooled to roomtemperature at a rate of 200 ◦ C/h. The sample was thenmechanically extracted from the crucible. Single crystalswith the largest mass of about 1 g were obtained.Small pieces of the crystals were collected and groundthoroughly for a powder X-ray diffraction measurementto confirm sample purity. The powder X-ray diffractiondata was fitted using the Rietveld method implementedin fullprof . Single-crystal X-ray diffraction was alsoperformed using a Bruker X8 APEX II CCD Diffractome-ter with Mo Kα radiation at room temperature. The re-finement on the single crystal diffraction data for frac-tional coordinates was done using ShelXle .In order to investigate the macroscopic magnetic prop-erties of SrCuTe O in the single crystal sample, themagnetic susceptibility was measured with the appliedfield aligned along the three inequivalent directions [100],[110], and [111]. A single crystal was aligned using a four-circle X-ray diffractometer and cut into a cube with di-mensions of 1 × × ( ∼
20 mg). The aligned crystalwas then attached to a Teflon rod using GE-7031 var-nish and placed inside a measuring stick. The magneticsusceptibility was measured as a function of temperaturefrom 2 K up to 300 K with the applied magnetic fieldof 1.0 T using a superconducting quantum interferencedevice (MPMS-XL, Quantum Design). To analyze thesusceptibility data, quantum Monte Carlo (QMC) simu-lations were performed to capture the broad maximumof the magnetic susceptibility data in order to extractthe value of the leading exchange interaction. The QMCsimulations were performed using the loop algorithm in the alps simulation package on a cluster of 100 spinsfor the spin-chain model and 96 000 spins (20 × × J model of SrCuTe O in the temper-ature range of 0 . (cid:54) t (cid:54)
10 ( t = k B T /J ) with 100 000
SrO TeO3
CuTeSrO (a) (b)
FIG. 1. (a) The crystal structure of SrCuTe O consists of CuO square plaquettes (blue), TeO triangular plaquettes (yellow),and SrO octahedra (purple). (b) The intertwined spin network of Cu S = 1 / J (yellow), J (green), and J (red) to form isolated triangles, a hyper-kagome lattice, and spin chains, respectively. Monte Carlo steps for thermalization and 500 000 MonteCarlo steps after thermalization. The numerical resultwas fitted with the experimental data by a Pad´e approx-imant and the leading exchange interaction and Land´e g -factor were finally obtained.To study the magnetic structure, powder neutrondiffraction was performed at the high-resolution neutronpowder diffractometer BT1 at the NIST Center for Neu-tron Research (NCNR), USA. The Ge(311) monochro-mator was used to select neutrons with λ = 2 .
077 ˚A andcollimations of 60 (cid:48) − (cid:48) − (cid:48) were employed. In addi-tion, elastic neutron scattering on a small 130 mg singlecrystal was conducted using the Double Focusing Triple-Axis Spectrometer BT7 at NCNR. Neutron scatteringmeasurement was performed using fixed incident energyof 14.7 meV and pyrolytic graphite PG(002) was used asa monochromator. The sample was cooled to the basetemperature using a closed cycle He cryostat. A posi-tion sensitive detector (PSD) was employed in the two-axis mode with collimations of open − (cid:48) − (cid:48) − PSD.The sample was aligned so that the hhl plane was in thescattering plane.
III. RESULTS AND DISCUSSIONA. X-ray diffraction
The X-ray diffraction data, which were measured onthe powder sample obtained by grinding small pieces ofthe single crystals, along with the result of the Rietveldrefinement are shown in Fig. 2(a). The results show thatthe obtained SrCuTe O single crystals are single-phasewithout any trace of impurity. However, we will latershow that from the neutron diffraction data measured onan as-grown powder sample, SrCuTe O is present as animpurity. The absence of this impurity phase in the single crystal X-ray diffraction is due to the fact that only smallpieces of the SrCuTe O single crystals were selected forgrinding, and hence SrCuTe O , which also showed upas having a different color, was selected out. The latticeparameter obtained from the refinement is a = 12 . P
32 in agreement withprevious reports . The residual parameters for theRietveld refinement are R p = 6 . R wp = 8 . θ -2 θ scan shown in theinset of Fig. 2(a) on the naturally cleaved facet confirmssingle-crystallinity and shows that the cleaved facet isthe 111 plane. To further investigate the quality of thesingle crystals, Laue photography was also performed inthe transmission mode. The result shown in the inset ofFig. 2(b) reveals clear Bragg peaks confirming the qualityof the crystal.Single crystal X-ray diffraction at room temperaturewas also performed and the data were refined to extractthe atomic fractional coordinates. The refinement wasdone on 1192 unique reflections with F obs > σ ( F obs )yielding the fitted lattice parameter a = 12 . R = 2 . wR = 6 . R for the refinementperformed on all (1263) reflections is 2.79%. The frac-tional coordinates of SrCuTe O obtained from the re-finements of the powder (the crushed crystalline sample)and single-crystal X-ray diffraction data are summarizedin Appendix A. B. Magnetic susceptibility
Figure 3 shows the magnetic susceptibility measuredwith the applied field along three inequivalent directions[100], [110], and [111]. The susceptibility data are plot-
FIG. 2. (a) Powder X-ray diffraction pattern measured on the crushed single crystals of SrCuTe O at room temperature. Theblack circle denotes the observed data, the red line the calculated pattern, the vertical green line the Bragg positions, and theblue line the difference between the observed and calculated patterns. The inset shows the θ -2 θ scan on the cleaved (1 , , O . (b) The refinement result of the single crystal X-ray diffraction data shows the agreementbetween the measured and calculated scattering intensities. Error bars represent one standard deviation. The inset shows aphotograph of a crystal and Laue X-ray diffraction measured on the cleaved surface.TABLE I. The fitted parameters of magnetic susceptibilitiesdata along [100], [110], and [111] with the mean-field Curie-Weiss law (Eq. 1) H (cid:107) χ (cm / mol Cu) (fixed) Θ CW (K) µ eff ( µ B )[100] 1 . × − − . . × − − . . × − − . ted against the temperature in a log scale to emphasizethe broad maximum at ∼
30 K, and the kink at the N´eeltemperature T N1 = 5.25(5) K ( T N1 was obtained fromthe neutron scattering data shown in Fig. 4). Below T N1 ,the magnetic susceptibility decreases, which is indicativeof the antiferromagnetic arrangement of the spins. Thebroad maximum is a sign of short-range spin correlation,typical for low-dimensional antiferromagnetic systems ,and can be used to estimate the leading exchange interac-tions of the system. This broad peak was consistent withthat previously observed in the powder sample . Wenote that for the data measured in the applied field of 1.0T on the powder sample in Ref. 27 and 28, the anomaly at T N2 is absent, possibly because it is very weak at low field;for the magnetization measurements, this anomaly, whichis field-dependent, appears below T N1 for H ≥ . .Consistent with the previous work on the powder sample,we were able to observe the second magnetic transitionat higher magnetic fields (not shown here).To estimate the dominant exchange coupling betweenthe Cu S = 1 / T (cid:38) T N1 ) datato the mean-field and QMC calculations. The magnetic TABLE II. The fitted parameters of the magnetic susceptibil-ities data along [100], [110], and [111] with the QMC simula-tion H (cid:107) χ (cm / mol Cu) J/k B (K) g -factor[100] 1 . × − . . × − . . × − . susceptibility data above 70 K along the three axes werefirst fitted with the mean-field Curie-Weiss law, χ ( T ) = χ + CT − Θ CW , (1)where C and Θ CW are the Curie constant and Curie-Weiss temperature, respectively. χ , which is fixedto the values obtained from the fits of the QMCresults (later discussed), represents the temperature-independent background, which is relatively high (abouthalf of the measured susceptibility at high temperatures)due to the Teflon used to fix the crystal. We note that χ is positive and its magnitude is about an order ofmagnitude larger than the core diamagnetic and VanVleck paramagnetic susceptibilities . Teflon is diamag-netic and hence gives rise to negative magnetic suscep-tibility. However, in the measurements, the signal fromthe Teflon rods was treated as background by MPMS.As a result, the empty gap between the Teflon rods givesrise to positive magnetic-susceptibility background. Thedifference in the value of χ could be due to the differencein the gap between the two Teflon rods that were usedto fix the aligned crystal in different orientations. The FIG. 3. Magnetic susceptibility measured on a single crystal of SrCuTe O with applied magnetic fields of 1.0 T along threecrystallographic directions, (a) [100], (b) [110] and (c) [111]. The blue and red lines represent the fits to the 1D-chain modeland our QMC simulations, respectively. The green line denotes the Curie-Weiss-law fit. The lower panels show the residualsbetween the data and the fits. Curie-Weiss fitted curves are shown by the green linesin Figs. 3(a)-(c). The fitted parameters along each fielddirection are shown in Table I. The effective magneticmoment, µ eff = (cid:112) k B C/N A , obtained for the three in-equivalent directions does not significantly deviate fromone another, which is indicative of relatively weak spinanisotropy with a similar g − factor. All are found to beslightly higher but still close to the spin-only value of µ eff = gµ B (cid:112) S ( S + 1) = 1 . µ B for g = 2 and S = 1 / µ eff from the spin-only value could indicate the presence of the spin-orbitcoupling and the orbital contribution to the magneticmoment. For S = 1 /
2, the measured µ eff of 2.0 im-plies g = 2 .
3. The obtained Curie-Weiss temperature,which is close to the previous reported values measuredon the powder sample , as shown in Table I, indicatesno significant deviation along the three field directions.The negative Curie-Weiss temperature indicates that thedominant exchange interaction is antiferromagnetic. Theorder of frustration f = | Θ CW /T N1 | for T N1 = 5 .
25 K andthe average Curie-Weiss temperature of −
44 K, yields f ∼ J and J that give rise tothe frustrated spin networks, isolated triangles and thehyper-kagome lattice, respectively.The decrease of the magnetic susceptibility for T 32 symmetry of the under-lying crystal structure can lead to the uniform DM in-teractions, which, in combination with the complex spinnetwork, can result in the nontrivial H − T phase diagramobserved in previous work . It was discovered that a chi-ral material with space group P 32 can host magneticskyrmions , the existence of which can be explained bythe DM interaction. In addition, the similar drop of themagnetic susceptibility for T (cid:46) T N1 along all three fielddirections [Figs. 3(a)-(c)] further suggests that there isno global easy axis and the antiferromagnetic alignmentmust be along a local easy axis determined by the localenvironment around the Cu ions.To go beyond the mean-field approximation, we per-formed a QMC simulation to calculate the magnetic sus-ceptibility that was subsequently used to fit the experi-mental data. The previous work showed that the broadmaximum observed in SrCuTe O can be well capturedby the 1D spin-chain model . For this work, we per-formed QMC simulations with the loop algorithm basedon the one-dimensional spin-chain model and on the spinnetwork connected by J [shown in Fig. 1(b)]. We notethat the two models without any inter-chain coupling arein fact equivalent. However, the J spin network was usedfor possible inclusion of J and J , which serve as inter-chain interactions. The simulated result was then fittedto the experimental data using χ ( T ) = χ + χ QMC ( T ) , (2)where χ QMC ( T ) = N A µ g k B J χ ∗ ( k B T /J ) . (3) N A , µ B , and k B are the Avogadro constant, Bohr magne-ton, and Boltzmann constant, respectively. J and g arefitted to the experimental data. The susceptibility χ ∗ ( t )as a function of the reduced temperature t = k B T /J was obtained by fitting the QMC results using a Pad´eapproximant . The fitting was performed on the ex-perimental data from T = 300 K down to 10 K, whichis slightly above T N1 and contains the broad peak, alongthe three field directions as shown in Fig. 3. The QMCresult appears to fit the data for the applied field alongthe [110] direction very well, where the QMC calcula-tions and the experimental data are in good agreementaround the broad peak and the consistency extends downto the temperature just above T N1 . In contrast, for the[100] and [111] data, the QMC results appear to de-viate from the data around the broad peak. We can-not explain why the result of the QMC calculations fitsthe [110] data much better than it does the other twofield directions. The fitted parameters are summarizedin the Table II. The fitted values of the Land´e g -factorare consistent with those obtained from the Curie-Weissfit discussed in the previous section. It can be seen thatthe obtained exchange interactions along three axes donot deviate much and the values are very close to thoseobtained from the mean-field approximation. This sug-gests that the 1D spin-chain model along third-nearestneighbors adequately describes the macroscopic magneticproperties. The obtained Land´e g -factor along the threeinequivalent field directions from the QMC fit are alsoconsistent with the spin network where the chains arerunning along the three crystallographic axes giving riseto the relatively isotropic magnetic susceptibility in thiscubic system. We attempted to include J and J in ourQMC calculations but encountered the sign problem dueto the frustration of the J and J bonds. As a result, wewere unable to obtain reliable J and J values especiallyaround the broad peak and the low-temperature region. C. Single-crystal and powder neutron diffraction To investigate the microscopically magnetic propertiesof SrCuTe O , elastic neutron scattering was performedon a small crystal ( ≈ 130 mg) at BT7. The inset of Fig. 4shows a θ -scan (rotating only the sample) around the 220structural Bragg peak with a Gaussian fit yielding a full-width-at-half-maximum (FWHM) equal to 0.226(5) ◦ , in-dicative of good crystallinity. The scattering intensity ofthe 003 and 113 reflections was measured as a function oftemperature. The results show the onset of the intensityincrease at T N1 . The extra scattering intensity below T N1 is indicative of magnetic scattering resulting from the or-dering of the magnetic moments. Figure 4 shows the 003scattering intensity data, which also serve as a measure ofthe order parameter below T N1 , along with the power lawfit. The fit to I ( T ) ∝ (1 − T /T N1 ) β for the order param-eter yields the critical exponent β = 0 . T N1 = 5 . I n t e n s it y ( × c oun t s / s ec ) T = 3.0 K220 0.226(5) º I n t e n s it y ( c oun t s / m i n ) T (K) I ( T ) ∝ − T/T N1 ) β T N1 = 5.25(5) K β = 0.23(3) FIG. 4. Neutron diffraction intensity of the 003 reflection asa function of temperature. The red line denotes a power-lawfit for the magnetic scattering intensity representing the orderparameter. Error bars represent one standard deviation. Theinset shows the θ -scan of the 220 structural reflection. Wenote that error bars are smaller than the plot symbol. Thesolid line represents the Gaussian fit to capture the line-shapeof the peak. sent one standard deviation. We note that the magneticscattering intensity is proportional to M , where M isthe sublattice magnetic moment, and hence the factor oftwo in the exponent. For the 113 data (not shown), thecritical exponent and N´eel temperature are 0.27(4) and5.12(5) K, respectively, which are consistent with thoseobtained from the 003 data. The fitted value of β is typ-ical for low-dimensional magnetic systems , the 1Dspin chain in this case. The obtained value of T N1 is con-sistent with magnetic susceptibility data and with previ-ous reports on the powder sample . However, giventhe resolution of the data, the order parameter in Fig. 4,which was measured at zero magnetic field, does not showan anomaly of the second magnetic transition around T N2 . It is possible that the second transition can onlybe detectable at high field where the anomaly becomesstronger as suggested by the magnetization data. Hencein order to investigate the magnetic structure change at T N2 , future in-field elastic neutron scattering is required.To determine the magnetic structure of SrCuTe O ,powder neutron diffraction was performed at BT1 andthe data were collected at 10 K and 1.5 K, above andbelow T N1 , respectively. The refinement of the nuclearstructure was first performed on the 10-K data with thefitted lattice parameter a = 12 . R p = 3.72%,shows results that are consistent with the crystal struc-ture of SrCuTe O obtained from the X-ray diffractiondata (Appendix A). The sample, however, containedsome impurities, the majority of which was identified tobe SrCuTe O that constitutes roughly 2.1 wt.%. The FIG. 5. (a) The difference data between 1.5 K and 10 K show the magnetic Bragg scattering at 012, 003, 013, and 113. Thelines represent the magnetic-structure refinements based on magnetic space groups P (cid:48) (cid:48) , P P (cid:48) (cid:48) , and P (cid:48) (cid:48) 2. Thevertical red lines denote the magnetic Bragg reflections. The powder neutron diffraction data were measured at 10 K (the inset).The black symbols denote the observed data, the grey line the calculated nuclear-scattering pattern, the vertical green lines theBragg positions, and the blue line the difference between the observed and calculated data. (b) The calculated single-crystalmagnetic intensities for the magnetic space group P 32 are shown as a function of ordered moment ( M ) for the reflections003 (black circles) and 113 (blue squares). The red lines denote the fits to the quadratic function. The black (blue) horizontalline represents the experimental value of the ratio between the magnetic scattering intensity measured at 003 and 113, andthe structural scattering intensity measured at 220. The gray (blue) shaded region indicates a range of the measured orderedmoment for 003 and 113. proximity of impurity reflections to some of the magneticBragg reflections and weak magnetic intensity hinder therefinement of the magnetic structure from the powderneutron diffraction data. As a result, the fitted orderedmagnetic moment has large error as will be discussed be-low.The magnetic structure of SrCuTe O was first ana-lyzed using the irreducible representation theory. Thedetail was described in Appendix B. We note that dueto the large number of free parameters, we were unableto perform full refinement for Γ , Γ , and Γ with 6,12 and 15 free parameters, respectively. Therefore, wehave to rely on the magnetic space group analysis in or-der to further sub-classify possible magnetic structures ofSrCuTe O and reduce the number of free parameters.Based on the Landau-type transition with a single or-der parameter, the magnetic Shubnikov space groupscan be derived from the paramagnetic parent space-group P (cid:48) giving rise to 14 Shubnikov space groupsas shown by the graph of subgroups, which was gen-erated using k - subgroupsmag , in Fig. 6. Out ofthese 14 subgroups, there are a total of five maximalmagnetic subgroups, P 32 (No. 213.63), P (cid:48) (cid:48) (No.213.65), P (cid:48) (cid:48) (No. 92.114), P (cid:48) (cid:48) C (cid:48) (No. 20.34). P 32 and P (cid:48) (cid:48) correspondto Γ and Γ , respectively, whereas P (cid:48) (cid:48) , P (cid:48) (cid:48) 2, and C (cid:48) (cid:48) correspond to Γ and Γ . Since some of the basisvectors of Γ and Γ are absent for the magnetic spacegroups P (cid:48) (cid:48) , P (cid:48) (cid:48) 2, and C (cid:48) (cid:48) , the number of fit- ting parameters is reduced.Assuming that symmetry reduction at the magnetictransition to the ordered state is minimal, we performedthe refinement of the magnetic structure on the 1.5-Kdata for the four (out of five) maximal subgroups, namely P 32 (Γ ) with one free parameter, P (cid:48) (cid:48) (Γ ) with2 free parameters, P (cid:48) (cid:48) with 5 free parameters, and P (cid:48) (cid:48) C (cid:48) (cid:48) with 9 free parameters was unsuccessful.All fitting parameters including the lattice parameter,atomic positions, and peak profile parameters, were keptconstant and the same as those obtained from the fittingof the 10-K data. However, the background was adjusteddue to the difference in the incident neutron flux. Sincethe magnetic scattering was observed at low momentumtransfer, and hence low 2 θ angles, the magnetic-structurerefinement was performed for 20 . ◦ < θ < . ◦ and28 . ◦ < θ < . ◦ , where four magnetic Bragg reflec-tions were observed. The inset of Fig. 5(a) shows thedifference pattern between 1.5 K and 10 K for thesefour magnetic reflections, 012, 003, 013 and 113, two ofwhich [003 and 013] appear next to the impurity peaks(not shown). The figure also shows that the calculatedintensity based on the Shubnikov space group P P (cid:48) (cid:48) with the R-factor of7.02%. The fitted magnetic moment for P 32 is 0.8(7) µ B , where the error was estimated from fitting the 10-K (2) (2)(6)(6) (12) FIG. 6. A diagram of subgroups shows a hierarchy of possible subgroups of the paramagnetic parent space group P (cid:48) .The maximal subgroups are indicated by eclipses, and those that were used in the refinement are highlighted by shading. Asubgroup index between the parent space group and a maximal subgroup is shown in a parenthesis. The diagram is generatedusing k - subgroupsmag . data; the large error is due to the significant contribu-tion from the nearby impurity peaks and weakness of themagnetic signal. The refinement based on P (cid:48) (cid:48) and P (cid:48) (cid:48) P P (cid:48) (cid:48) , P (cid:48) (cid:48) , and P (cid:48) (cid:48) P 32, the fitted results for P (cid:48) (cid:48) , P (cid:48) (cid:48) ,and P (cid:48) (cid:48) O magnetically ordersin the magnetic space group P 32. Furthermore, wewere unable to rule out C (cid:48) (cid:48) , nor, under the assump-tion of minimal symmetry reduction at the transition,examine non-maximal subgroups. As previously noted,since the magnetic scattering in SrCuTe O is weak andsome magnetic Bragg reflections are in close proximityto impurity peaks, the refinement of the powder neutrondiffraction data yields an inconclusive result with largeerror. Hence, in this work, P 32 is proposed as themost likely candidate based on the magnetic space groupanalysis.The resulting magnetic structure for P 32 is shownin Fig. 7. The magnetic moments of the Cu S = 1 / O antiferromagnetically align in the di-rection perpendicular to the chain formed by J , which isthe most dominant exchange interaction, consistent withproposed spin network deduced from the DFT calcula-tions . This antiferromagnetic spin structure is con-sistent with the magnetization data (discussed above)where no weak ferromagnetism, which could result from spin canting, was observed. Interestingly, we observedthat the spins on the corners of an isolated triangleconnected by J , i.e. the weakest exchange interactionamong the three considered, form a co-planar 120 ◦ config-uration [Fig. 7(b)], which relieves, to some degree, the ge-ometrical frustration inherent in the triangle-based spinnetwork. However, the J interactions, which form thehyper-kagome spin network, remain highly frustrated.The magnetic order in this quasi-1D system is most likelystabilized by this intricate network of further-nearest-neighbor interactions [Figs. 1(b) and 7(b)].In order to better extract the value of the ordered mo-ment, we performed the analysis on the single-crystalneutron diffraction data measured at BT7. In Fig. 4,we were able to clearly observe the magnetic Bragg in-tensities at 003 and 113 (not shown), which we will de-note as I M and I M , respectively. The base tempera-ture of 3 K for the single-crystal experiment might notbe low enough relative to T N to give a good estimateof the ordered moment as suggested by the increasingtrend of the scattering intensity in Fig. 4. Hence, I M and I M were obtained by extrapolating the order pa-rameter fitted curve of the scattering intensity measuredat 003 and 113, respectively, to 1.5 K, at which thepower neutron diffraction was measured. In compari-son with the nuclear Bragg intensity at 220, I N , wecalculated the ratio between the magnetic and nuclearscattering intensity as I M /I N = 6 . × − and I M /I N = 5 . × − . These obtained values of themagnetic to nuclear intensity ratio are very weak i.e. ,roughly of the same order of magnitude as the statistical (a) J J J (b)(c) Cu3Cu2 Cu1Cu6 Cu4Cu11Cu9 Cu8Cu7Cu5 Cu12Cu10 FIG. 7. (a) The magnetic structure of SrCuTe O belongsto the Shubnikov space group P 32 (Γ ). The dominant J antiferromagnetic exchange interactions form spin chainsalong the crystallographic axes. (b) The spin network, whichis formed by J (yellow), J (dashed green) and J (blue), isshown along with the spin structure. (c) The Cu atoms in aunit cell are labeled according to Table IV. error in the powder data, and hence small contributionsfrom the nearby impurity scattering can cause a largeerror in the refinement. As a result, the magnetic scat-tering is barely noticeable in the difference plot betweenthe 1.5-K and 10-K data (Fig. 5(a)), and the magneticstructure refinement on the powder neutron diffractiondata fails to yield a reliable result.To obtain the value of the ordered magnetic momentfrom the single-crystal data, we compare the magneticintensities I M /I N and I M /I N from the single crys-tal data and the magnetic scattering intensity calculatedfrom fullprof for P 32. We convert the integrated in-tensities calculated from fullprof for powder to thosefor single-crystal by multiplying sin θ/m hkl , where m hkl is the multiplicity of the hkl reflection . The result isshown in Fig. 5(b); the red curves in Fig. 5 denote a fitto I mag /I ∝ M . We note that M is the only freeparameter for P 32. Given the values of I M /I N and I M /I N from above, we estimate the ordered moment tobe 0 . µ B and 0 . µ B , respectively. The horizontalblack and blue solid lines denote the values of the inten-sity ratios for 003 and 113, respectively, with the dashedlines representing the range of the error. We have donea similar analysis for P (cid:48) (cid:48) , which has two free param-eters, C and C . We found that if C is equal to zero,the calculated magnetic intensity of 113 will be greaterthan that of 003, which is inconsistent with the experi-mental data. With increasing C , the 003 intensity canbecome larger than the 113 intensity but they are stillinconsistent with the experimental data. In addition, re- finement of the single-crystal data was also performedusing Jana2006 for magnetic space groups P 32. Wewere unable to check P (cid:48) (cid:48) , P (cid:48) (cid:48) , and P (cid:48) (cid:48) wR = 4 . 63% and goodness of fit (GoF) of2.46. The fitted ordered magnetic moment of 0 . µ B is in good agreement with the above values obtained fromthe graph in Fig. 5(b). Hence, we reach the same con-clusion as from the analysis of the powder data that themagnetic space group for the magnetically ordered stateof SrCuTe O below T N1 is P . µ B is abouthalf of the expected value of 1 µ B for S = 1 / J and J bondspotentially induces spin fluctuations and significantly re-duce the ordered moment. The reduction of ordered mo-ments has been observed in ordered frustrated systems,KFe (OH) (SO ) ( S = 3 / 2) with 24% reduction , andCs Cu SnF ( S = 1 / 2) with 32% reduction . In com-parison, for PbCuTe O , where the dominant J formsthe frustrated hyper-kagome lattice, spin fluctuations areso large that the N´eel state is totally suppressed and aquantum spin liquid possibly emerges at low tempera-tures . From the DFT calculations, the exchange in-teractions in PbCuTe O are close to one another, whichcould enhance the frustration whereas the intra-chain in-teraction J in SrCuTe O is, respectively, one order andtwo orders of magnitude larger than J and J , whichcould place SrCuTe O away from the quantum spin liq-uid state even though structurally it is almost identicalto PbCuTe O . Nevertheless, even though SrCuTe O magnetically orders at low temperatures, the residue ef-fect of the frustrated bonds remains and evidences in thereduced ordered moment. It would be interesting to in-vestigate this subtle effect of frustration in spin dynamicsof this system. IV. CONCLUSION Magnetization measurements on single-crystalSrCuTe O reveal highly isotropic magnetic suscep-tibility along the three inequivalent directions [100],[110], and [111] in this cubic system. The value of theleading exchange interaction ( J /k B ) estimated using aquantum Monte Carlo simulation on the 1D spin-chainmodel is between 50.1 and 52.5 K. The order parametermeasured by neutron scattering confirms that the systemmagnetically orders below T N1 = 5 . T N , which was previously observedin magnetization and heat capacity measurements.Further in-field neutron scattering measurements arerequired to investigate this second transition. Basedon the neutron diffraction data on the powder andsingle-crystal samples, the magnetic structure in theShubnikov space group P 32, where the Cu S = 1 / . µ B . This worksuggests the dominance of the intra-chain interaction J over the frustrated J and J bonds, and sheds lighton the difference in magnetic ground states betweenSrCuTe O and PbCuTe O . The 48% reduction of theordered moment in SrCuTe O points to the residual ef-fect of frustration, which could have nontrivial influenceon spin dynamics in this magnetically ordered system. Note: After submitting this manuscript, we becameaware of similar work , which was published in a publicarchive. Magnetic susceptibility measured on a single-crystal sample reported in Ref. 48 is consistent with ourresults. Ref. 48 also confirms our reported magneticstructure with the reduced ordered magnetic moment.However, their measured value of the ordered moment is slightly lower than that reported in this work. ACKNOWLEDGMENTS Work at Mahidol University was supported in partby the Thailand Research Fund (TRF) Grant NumberRSA6180081 and the Thailand Center of Excellence inPhysics. PS was supported by the RGJ-PhD scholar-ship (Grant No. PHD/0114/2557) from TRF. FCC ac-knowledges funding support from the Ministry of Sci-ence and Technology (108-2622-8-002-016 and 108-2112-M-001-049-MY2) and the Ministry of Education (AI-MAT 108L900903) in Taiwan. 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Refined values of fractional coordinates ofSrCuTe O from powder and single crystal X-ray diffractionmeasured at room temperature, and powder neutron diffrac-tion measured at 10 K.Atom Site x/a y/a z/a powder X-ray diffractionTe 24e 0.3380(1) 0.9187(1) 0.0588(1)Sr(1) 8c 0.0545(2) 0.0545(2) 0.0545(2)Sr(2) 4b 0.375 0.625 0.125Cu 12d 0.4762(1) 0.875 0.2738(3)O(1) 24e 0.6635(9) 1.1271(9) 0.1761(9)O(2) 24e 0.4404(9) 1.0205(9) 1.2210(8)O(3) 24e 0.2222(9) 0.9781(10) 0.1302(11) R p = 0.0666, R wp = 0.0849, GoF = 2.1single-crystal X-ray diffractionTe 24e 0.33827(3) 0.91872(3) 0.05938(3)Sr(1) 8c 0.05469(5) 0.05469(5) 0.05469(5)Sr(2) 4b 0.375 0.625 0.125Cu 12d 0.47567(7) 0.875 0.27433(7)O(1) 24e 0.6710(4) 1.1273(4) 0.1785(4)O(2) 24e 0.4382(4) 1.0171(4) 1.2284(4)O(3) 24e 0.2224(5) 0.9767(6) 0.1305(5) R = 0.0247, wR = 0.0601, GoF = 1.042powder neutron diffractionTe 24e 0.3379(1) 0.9192(1) 0.0589(1)Sr(1) 8c 0.0536(1) 0.0536(1) 0.0536(1)Sr(2) 4b 0.375 0.625 0.125Cu 12d 0.4760(1) 0.875 0.2741(1)O(1) 24e 0.6702(1) 1.1271(1) 0.1795(1)O(2) 24e 0.4391(1) 1.0163(1) 1.2271(1)O(3) 24e 0.2220(1) 0.9766(2) 0.1297(1) R p = 0.0372 , R wp = 0.0487, GoF = 2.5 Appendix A: Atomic coordinates of SrCuTe O The refined fractional atomic coordinates ofSrCuTe O were shown in Table III. The refine-ment of the powder (crushed crystalline sample) andsingle crystal data was performed using fullprof and ShelXle , respectively. Appendix B: Table of magnetic irreduciblerepresentations of SrCuTe O The magnetic structure of SrCuTe O was analyzedusing the irreducible representation theory. The anal-ysis based on the symmetry of the underlying crystalstructure (space group P 32) was carried out using basireps in the fullprof software package. Since themagnetic Bragg reflections were observed on top of thestructural reflections as shown in the inset of Fig. 5(a),the magnetic propagation vector (cid:126)k is equal to (0 , , d of magnetic Cu ions witha total of 12 spins in the unit cell as shown in Figs. 1(b)and 7(c), the decomposition of the irreducible represen-tations (IRs) can be described byΓ = 1Γ (1)1 + 2Γ (1)2 + 3Γ (2)3 + 4Γ (3)4 + 5Γ (3)5 , (B1)where the basis vectors for Γ , Γ , Γ , Γ , and Γ aregiven in Table IV. Γ and Γ are one dimensional with oneand two basis vector(s), respectively. On the other hand,Γ are of two dimensions with six basis vectors whereasΓ and Γ are of three dimensions with twelve and fif-teen basis vectors, respectively. We assume that there isonly one order parameter for the magnetic transition inSrCuTe O and hence based on the Landau theory , themagnetic structure of the low-temperature phase corre-sponds to a single IR.3 TABLE IV. Magnetic irreducible representations and their basis vectors for Cu1( x, y, z ), Cu2( − x + 1 / , − y + 1 , z + 1 / − x +1 , y − / , − z +1 / x +1 / , − y +3 / , − z +1), Cu5( z, x, y ), Cu6( z +1 / , − x +1 / , − y +1), Cu7( − z +1 / , − x +1 , y − / − z + 1 , x + 1 / , − y + 3 / y, z, x ), Cu10( − y + 1 , z + 1 / , − x + 1 / y − / , − z + 1 / , − x + 1),Cu12( − y + 3 / , − z + 1 , x + 1 / ψ (10-1) a (-10-1) (-101) (101) (-110) (-1-10) (1-10) (110) (0-11) (0-1-1) (01-1) (011)Γ ψ (101) (-101) (-10-1) (10-1) (110) (1-10) (-1-10) (-110) (011) (01-1) (0-1-1) (0-11) ψ (010) (0-10) (010) (0-10) (001) (00-1) (001) (00-1) (100) (-100) (100) (-100)Γ ψ (100) (-100) (-100) (100) (0- 0) (0 0) (0 0) (0- 0) (00- ) (00 ) (00 ) (00- )(000) (000) (000) (000) (0- √ 0) (0 √ 0) (0 √ 0) (0- √ 0) (00 √ ) (00- √ ) (00- √ ) (00 √ ) ψ (010) (0-10) (010) (0-10) (00- ) (00 ) (00- ) (00 ) (- 00) ( 00) (- 00) ( √ ) (00 √ ) (00- √ ) (00 √ ) ( √ 00) (- √ 00) ( √ 00) (- √ ψ (001) (001) (00-1) (00-1) (- 00) (- 00) ( 00) ( 00) (0- 0) (0- 0) (0 0) (0 √ 00) (- √ 00) ( √ 00) ( √ 00) (0 √ 0) (0 √ 0) (0- √ 0) (0- √ ψ (00 ) (00 ) (00- ) (00- ) ( 00) ( 00) (- 00) (- 00) (0-10) (0-10) (010) (010)(00- √ ) (00- √ ) (00 √ ) (00 √ ) ( √ 00) ( √ 00) (- √ 00) (- √ 00) (000) (000) (000) (000) ψ (0 0) (0- 0) (0 0) (0- 0) (00 ) (00- ) (00 ) (00- ) (-100) (100) (-100) (100)(0- √ 0) (0 √ 0) (0- √ 0) (0 √ 0) (00 √ ) (00- √ ) (00 √ ) (00- √ ) (000) (000) (000) (000) ψ ( 00) (- 00) (- 00) ( 00) (0 0) (0- 0) (0- 0) (0 0) (00-1) (001) (001) (00-1)(- √ 00) ( √ 00) ( √ 00) (- √ 00) (0 √ 0) (0- √ 0) (0- √ 0) (0 √ 0) (000) (000) (000) (000)Γ ψ (100) (-100) (100) (-100) (000) (000) (000) (000) (0-10) (010) (0-10) (010) ψ (010) (0-10) (0-10) (010) (000) (000) (000) (000) (-100) (-100) (100) (100) ψ (001) (001) (001) (001) (000) (000) (000) (000) (00-1) (00-1) (00-1) (00-1) ψ (000) (000) (000) (000) (-110) (110) (1-10) (-1-10) (000) (000) (000) (000) ψ (00-1) (001) (0-1) (001) (010) (0-10) (010) (0-10) (000) (000) (000) (000) ψ (0-10) (0-10) (010) (010) (001) (00-1) (00-1) (001) (000) (000) (000) (000) ψ (-100) (-100) (-100) (-100) (100) (100) (100) (100) (000) (000) (000) (000) ψ (000) (000) (000) (000) (000) (000) (000) (000) (0-11) (011) (01-1) (0-1-1) ψ (000) (000) (000) (000) (-100) (100) (-100) (100) (001) (00-1) (001) (00-1) ψ (000) (000) (000) (000) (00-1) (00-1) (001) (001) (100) (-100) (-100) (100) ψ (000) (000) (000) (000) (0-10) (0-10) (0-10) (0-10) (010) (010) (010) (010) ψ (10-1) (101) (-101) (-10-1) (000) (000) (000) (000) (000) (000) (000) (000)Γ ψ (100) (-100) (100) (-100) (000) (000) (000) (000) (010) (0-10) (010) (0-10) ψ (010) (0-10) (0-10) (010) (000) (000) (000) (000) (100) (100) (-100) (-100) ψ (001) (001) (001) (001) (000) (000) (000) (000) (001) (001) (001) (001) ψ (000) (000) (000) (000) (110) (-110) (-1-10) (1-10) (000) (000) (000) (000) ψ (000) (000) (000) (000) (001) (001) (001) (001) (000) (000) (000) (000) ψ (001) (00-1) (001) (00-1) (010) (0-10) (010) (0-10) (000) (000) (000) (000) ψ (010) (010) (0-10) (0-10) (001) (00-1) (00-1) (001) (000) (000) (000) (000) ψ (100) (100) (100) (100) (100) (100) (100) (100) (000) (000) (000) (000) ψ (000) (000) (000) (000) (000) (000) (000) (000) (011) (0-11) (0-1-1) (01-1) ψ (000) (000) (000) (000) (000) (000) (000) (000) (100) (100) (100) (100) ψ (000) (000) (000) (000) (100) (-100) (100) (-100) (001) (00-1) (001) (00-1) ψ (000) (000) (000) (000) (001) (001) (00-1) (00-1) (100) (-100) (-100) (100) ψ (000) (000) (000) (000) (010) (010) (010) (010) (010) (010) (010) (010) ψ (101) (10-1) (-10-1) (-101) (000) (000) (000) (000) (000) (000) (000) (000) ψ (010) (010) (010) (010) (000) (000) (000) (000) (000) (000) (000) (000) a A parenthesis represents ( m x m y m z ). b The basis vectors for Γ3