Singlet ground state in the alternating spin- 1/2 chain compound NaVOAsO 4
U. Arjun, K. M. Ranjith, B. Koo, J. Sichelschmidt, Y. Skourski, M. Baenitz, A. A. Tsirlin, R. Nath
SSinglet ground state in the alternating spin- / chain compound NaVOAsO U. Arjun, K. M. Ranjith, B. Koo, J. Sichelschmidt, Y. Skourski, M. Baenitz, A. A. Tsirlin, and R. Nath ∗ School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram-695551, India Max Planck Institute for Chemical Physics of Solids, Nthnitzer Str. 40, 01187 Dresden, Germany Dresden High Magnetic Field Laboratory, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany Experimental Physics VI, Center for Electronic Correlations and Magnetism,Institute of Physics, University of Augsburg, 86135 Augsburg, Germany (Dated: October 16, 2018)We present the synthesis and a detailed investigation of structural and magnetic properties ofpolycrystalline NaVOAsO by means of x-ray diffraction, magnetization, electron spin resonance(ESR), and As nuclear magnetic resonance (NMR) measurements as well as density-functionalband structure calculations. Temperature-dependent magnetic susceptibility, ESR intensity, andNMR line shift could be described well using an alternating spin-1 / J/k B (cid:39)
52 K and an alternation parameter α (cid:39) .
65. From the high-field magneticisotherm measured at T = 1 . H c (cid:39)
16 T,which corresponds to the zero-field spin gap of ∆ /k B (cid:39) . is apromising compound for further experimental studies under high magnetic fields. PACS numbers: 75.50.Ee, 75.10.Pq, 75.30.Et, 71.20.Ps, 61.66.Fn
I. INTRODUCTION
For decades, there has been a flourish of interest in one-dimensional (1D) quantum antiferromagnets (AFM), asthey often provide unique opportunities to study the in-terplay between the charge, orbital, spin, and lattice de-grees of freedom [1]. In most of the 1D quantum AFMs,the frequently encountered ground state is a three di-mensional (3D) long-range order (LRO), resulting frominter-chain interactions. Therefore, any new compoundpossessing a spin-gap in the excitation spectrum is offundamental interest. Typically, the alternation of ex-change interactions along the chain direction introducesa spin gap, as in (VO) P O [2]. If this gap appearsdue to a structural distortion, which in turn causes spindimerization, it is known as the spin-Peierls transition [3].However, not every structural phase transition accompa-nied by a spin-gap formation is a spin-Peierls transition.Such a transition has only been observed in several or-ganic compounds [4, 5] and CuGeO [6]. For the mixed-valence compound NaV O , the formation of a singletground state is reported to be due to charge ordering [7],while for NaTiSi O it is due to orbital ordering [8].Spin chains with the gap in the excitation spectrumare sensitive to perturbations like inter-dimer couplingsand external magnetic field. Above a threshold limit,the spin gap is closed, and a multitude of field-inducedphases emerge. The singlet state sustains in low fields,where the magnetization remains nearly zero. But ahigher field can close the gap and trigger an AFMLRO above the critical field of the gap closing. This ∗ [email protected] fascinating field-induced phenomenon can be describedas the Bose-Einstein Condensation (BEC) of triplons,where the particle density is proportional to the ap-plied magnetic field acting as chemical potential of theBose gas [9]. Notable compounds showing the triplonBEC are BaCuSi O , Sr Cr O , TlCuCl etc [10–14].Another important field-induced phenomenon shown byspin-gap systems is the emergence of a non-Fermi liquid-type Tomonaga-Luttinger liquid (TLL) phase above thecritical field of the gap closing [15, 16]. The spin-laddercompound (C H N) CuBr presents an excellent exper-iment realization, where the TLL phase evolves from thesinglet state [17–21]. The transition between the 1D TLLand 3D BEC is successfully explained in another laddercompound (C H N) CuBr [22, 23]. Quasi-1D materi-als involving alternating spin chains may show signaturesof both TLL and BEC physics, thus giving researchers anopportunity to study the crossover effects from 1D to 3Dphysics [24, 25].In addition, spin-gap materials also exhibit vari-ous other peculiar features like Wigner crystallizationof magnons, magnetization plateaus, etc, under exter-nal magnetic field[26]. Orthogonal dimer compoundSrCu (BO ) with the Shastry-Sutherland lattice is awell-known example showing these exotic features [27].The above field-induced effects are explored only in veryfew experimental systems till date due to the shortageof model compounds. In this context, materials with thesmall spin gap are preferable, so that one can explore thefield-induced phenomena and the complete H − T phasediagram using practically accessible magnetic fields.In this work, we endeavor to elucidate the magneticbehavior of NaVOAsO by means of magnetization, elec-tron spin resonance (ESR), and As Nuclear Magnetic a r X i v : . [ c ond - m a t . s t r- e l ] O c t J' J ab c J c J a J' J
FIG. 1. Left panel: crystal structure of NaVOAsO . The black dashed lines separate the spin chains. The VO and AsO polyhedra are shown in red and green colors, respectively. Right panel: spin model showing the alternating ( J, J (cid:48) ) crossingchains and the frustrated inter-chain couplings in a different orientation.
Resonance (NMR) experiments combined with density-functional band-structure calculations. A comparison ismade with AgVOAsO in order to understand the roleof the inter-chain couplings on the size of the spin gap.AgVOAsO is reported as the alternating spin-1 / P /c ) structure with thelattice parameters a = 6 . b = 8 . c = 7 . β = 115 . ◦ , and unit cell vol-ume V cell (cid:39) . [28]. From the analysis of mag-netic susceptibility, high-field magnetization, ESR, AsNMR, and band-structure calculations, the dominant ex-change coupling, alternation parameter, zero-field spingap, and critical field of the gap closing are reported tobe
J/k B (cid:39) . α = J (cid:48) /J (cid:39) .
68, ∆ /k B (cid:39)
13 K, and H c = ∆ gµ B (cid:39)
10 T, respectively [28, 29].When Ag + (ionic radius: 1.29 ˚A) is replaced by Na + (ionic radius: 1.16 ˚A), the crystal structure remains unal-tered. The reported unit cell parameters for NaVOAsO are a = 6 . b = 8 . c = 7 . β = 115 . ◦ , and V cell (cid:39) .
72 ˚A [30]. The mainobjective of this work is to tune the spin gap and hencethe critical field of the gap closing by chemical substi-tution, which may facilitate the study of field-inducedeffects using commonly accessible laboratory fields.In NaVOAsO , there are one As atom, one Na atom,one V atom, and five nonequivalent O atoms. Similarto AgVOAsO , each AsO tetrahedron is coupled withfour VO octahedra to form alternating spin chains thatrun nearly perpendicular to each other in the ab -plane,as shown in Fig. 1. There are also weak inter-chain ex- change interactions, which make a frustrated network be-tween the chains. The directions of all the exchange in-teractions are clearly depicted in the right panel of Fig. 1.The As atom is strongly coupled to the two neighbouringmagnetic V ions along the direction of the spin chain.On the other hand, the Na atom is located in betweenthe chains and seems to be very weakly coupled to themagnetic V ions. To the best of our knowledge, noinformation about the magnetic properties of this com-pound has been reported till date. II. METHODS
Polycrystalline sample of NaVOAsO was synthesizedby the conventional solid-state reaction technique by an-nealing the stoichiometric mixture of Na O (99.99%),As O (99.99%), and VO (99.99%) (all from Sigma-Aldrich) in an evacuated silica tube at 500 ◦ C for 48 hwith one intermediate grinding and pelletization. Toavoid the hydration, the reactants were handled in an Arfilled glove box. The resulting sample was green in color.Its phase purity was confirmed by powder x-ray diffrac-tion (XRD, PANalytical powder diffractometer with Cu K α radiation, λ ave = 1 . T ) dependent powder XRD measurementswere performed in the T -range 15 K - 300 K using thelow- T attachment (Oxford Phenix) to the x-ray diffrac-tometer. Rietveld refinement of the observed XRD pat-terns was performed using the FullProf package [31],taking the initial parameters from Ref. 30.DC magnetization ( M ) was measured as a functionof T and applied magnetic field H using the vibratingsample magnetometer (VSM) attachment to the Phys-ical Property Measurement System [PPMS, QuantumDesign]. High-field magnetization isotherm ( M vs H )was measured at T = 1 . P absorbed by the sample from a transverse magneticmicrowave field (X-band, ν (cid:39) . B . The lock-in technique wasused to improve the signal-to-noise ratio which yields thederivative of the resonance signal dP/dB . The g -factorwas determined by the resonance condition g = hνµ B B res ,where h is the Planck’s constant, µ B is the Bohr magne-ton, ν is the resonance frequency, and B res is the corre-sponding resonance field.The NMR experiments on As nucleus (nuclear spin I = 3 /
2, gyromagnetic ratio γ/ π = 7 .
291 MHz/T) werecarried out using pulsed NMR technique at a radio fre-quency ν (= γ /2 πH ) of 50.44 MHz, which correspondsto the magnetic field of H ≈ . As NMR spectrum was obtained by sweeping themagnetic field while keeping the frequency fixed. TheNMR shift K ( T ) = [ H ref − H ( T )] /H ( T ) was determinedfrom the resonance field H ( T ) of the sample with re-spect to the resonance field H ref of the reference GaAssample. The spin-lattice relaxation rate (1 /T ) measure-ments were done using the standard inversion recoverymethod.Exchange couplings of the Heisenberg spin Hamilto-nian ˆ H = (cid:88) (cid:104) ij (cid:105) J ij S i S j , (1)where S = and the summation is over the latticebonds (cid:104) ij (cid:105) , were obtained by a mapping procedure us-ing density-functional (DFT) band-structure calculationsperformed in the FPLO code [33] within the general-ized gradient approximation (GGA) for the exchange-correlation potential [34]. The mean-field DFT+ U cor-rection for correlation effects in the V 3 d shell wasused. Additionally, we analyzed the uncorrelated GGAband structure and extracted the hopping parameters,which were further used to obtain antiferromagnetic ex-change couplings on the level of second-order perturba-tion theory, as further explained in Sec. III E. The typ-ical k -mesh included 128 points in the irreducible partof the first Brillouin zone. In the DFT+ U calculations,we applied U d = 4 eV and J d = 1 eV for the on-siteCoulomb repulsion and Hund’s exchange, respectively,as well as the around-mean-field correction for double-counting [28, 35, 36]. The structural parameters fromRef. 30 were used.Magnetization curves were simulated using the dirloop_sse [38] algorithm of the ALPS simulation pack- age [39] on two-dimensional finite lattices with periodicboundary conditions and up to 256 sites.
III. RESULTS AND DISCUSSIONA. X-Ray diffraction
Figure 2 shows the powder XRD pattern of NaVOAsO at two different temperatures, T = 15 K and 300 K,along with the refinement. The obtained best fit pa-rameters at 300 K and 15 K are [ a = 6 . b = 8 . c = 7 . β = 115 . ◦ , V cell (cid:39) .
89 ˚A , and the goodness-of-fit χ (cid:39) . a = 6 . b = 8 . c = 7 . β = 115 . ◦ , V cell (cid:39) .
06 ˚A , and χ (cid:39) . compared to AgVOAsO islikely due to the fact that Na + (ionic radius: 1.16 ˚A) hasa smaller ionic radius than Ag + (ionic radius: 1.29 ˚A).The atomic positions obtained from the structural re-finement at room temperature ( T = 300 K) are listed inTable I.The powder XRD pattern at different temperatureswere also analyzed by the Rietveld refinement to un-derstand the temperature variation of the crystal struc-ture. Going from 300 K to 15 K, the overall XRD pat-tern remains intact, thus ruling out any structural phasetransitions or lattice distortions that take place in otherspin-gap compounds like CuGeO [40], NaV O [41], andNaTiSi O [8]. The temperature dependence of the lat-tice parameters obtained from the refinement is plottedin Fig. 3. The lattice constants a , b , and c , and the angle β are found to decrease upon cooling. As a result, V cell also shows a gradual decrease with decreasing tempera-ture from 300 K to 15 K.The temperature variation of V cell was modeled follow-ing the Gr¨uneisen approximation for the zero-pressurestate, where the effects of thermal expansion are consid-ered to be equivalent to the elastic strain [42]. V cell ( T )can be written as [43] V cell ( T ) = γU ( T ) /K + V , (2)where V is the unit cell volume at T = 0 K, K is thebulk modulus, γ is the Gr¨uneisen parameter, and U ( T ) isthe internal energy. The U ( T ) can be expressed in termsof the Debye approximation as U ( T ) = 9 N k B T (cid:18) Tθ D (cid:19) (cid:90) θ D T x ( e x − dx, (3)where θ D is the characteristic Debye temperature, N isthe number of atoms per unit cell, and k B is the Boltz-mann constant. The fit is shown as the solid line in thebottom panel of Fig. 3. The obtained best fit parametersare θ D (cid:39)
375 K and V (cid:39)
376 ˚A .
10 20 30 40 50 60 70 80 = 3.65 I obs I cal I obs - I cal Bragg positions I n t e n s it y ( a r b . un it s )
2 (degree)T = 300 K = 3.87T = 15 K
FIG. 2. (Color online) Powder x-ray diffraction pattern (opencircles) for NaVOAsO at two different temperatures ( T =15 K and 300 K). The solid line represents the full-profilerefinement, with the vertical bars showing the expected Braggpeak positions, and the lower solid blue line representing thedifference between the observed and calculated intensities.TABLE I. Atomic coordinates and isotropic atomic displace-ment parameters ( U iso , in units of 10 − ˚A ) of NaVOAsO ob-tained from the Rietveld refinement of the powder XRD dataat room temperature (300 K). The crystal structure is mon-oclinic with the space group P /c (No. 14). The obtainedlattice parameters are a = 6 . b = 8 . c = 7 . β = 115 . ◦ , and V cell (cid:39) .
89 ˚A . The U iso values for oxygen were constrained. The error bars arefrom the Rietveld refinement.Atom Wyckoff x y z U iso positionNa 4 e e e e e e e e ca () c ( ¯ ) b ( ¯ ) V cell b a ( ¯ ) T (K) V ce ll ( ¯ ) FIG. 3. The lattice constants ( a , b , c ), monoclinic angle ( β ),and unit cell volume ( V cell ) are plotted as a function of tem-perature. The solid line in the bottom panel represents thefit using Eq. (2). B. Magnetization
The temperature-dependent magnetic susceptibility χ ( T ) ( ≡ M/H ) measured in an applied field of H = 0 . χ ( T )increases in the Curie-Weiss manner as expected in theparamagnetic regime and then shows a broad maximumat around T max χ (cid:39)
29 K, indicative of a short-range mag-netic order, as expected for low-dimensional spin systems.Below T max χ , it shows a rapid decrease and then an up-turn. This low temperature upturn is likely due to theextrinsic paramagnetic impurities and/or defects presentin the sample [44]. There is no clear indication of anymagnetic LRO down to 2 K.To extract the magnetic parameters, χ ( T ) at high tem-peratures was fitted by the following expression χ ( T ) = χ + CT − θ CW , (4)where χ is the temperature independent contributionconsisting of the diamagnetic susceptibility ( χ core ) of coreelectron shells and Van-Vleck paramagnetic susceptibil-ity ( χ VV ) of the open shells of the V ions. The secondterm in Eq. (4) is the Curie-Weiss (CW) law with the CWtemperature θ CW and Curie constant C = N A µ / k B .
10 1000.0000.0020.0040.006
Eq. (5) imp spin
Eq. (6) ( c m / m o l ) T (K)H = 0.5 T
1/ Eq. (4) ( c m / m o l ) - T (K)
FIG. 4. χ ( T ) measured at H = 0 . χ imp , and the dashed line shows the fit on the χ spin data usingEq. (6). Inset: 1 /χ ( T ) data along with the fit using Eq. (4)(solid line). Here, N A is Avogadro’s number, µ eff = g (cid:112) S ( S + 1) µ B isthe effective magnetic moment, g is the Land´e g -factor, µ B is the Bohr magneton, and S is the spin quantumnumber. Our fit in the temperature range 150 K to 380 K(inset of Fig. 4) yields χ (cid:39) − . × − cm /mol, C (cid:39) .
39 cm K/mol, and θ CW (cid:39) − . C , the effective moment was calculated to be µ eff (cid:39) . µ B in good agreement with the expected spin-only value of 1.73 µ B for S = 1 /
2, assuming g = 2. Thenegative value of θ CW indicates that the dominant ex-change couplings between V ions are antiferromagneticin nature.In order to estimate the exchange couplings, the χ ( T )data were fitted by χ ( T ) = χ + C imp T − θ imp + χ spin . (5)Here, C imp represents the Curie constant correspondingto the impurity spins and θ imp quantifies the effectiveinteraction between them. χ spin is the expression forthe spin susceptibility of the alternating spin-1 / ≤ α (= J (cid:48) /J ) ≤ k B T /J ≥ . J (cid:48) and J are the exchange couplings along thechain.[45] As shown in Fig. 4, Eq. (5) fits very well to the χ ( T ) data over the whole temperature range, yielding J/k B (cid:39) . α (cid:39) . χ (cid:39) − . × − cm /mol, C imp (cid:39) . K/mol, and θ imp (cid:39) − . g was fixed to g (cid:39) . C imp corresponds to the concentra-tion of nearly 4.1 %, assuming the impurity spins to be S = 1 /
2. Using the value of
J/k B and α , the spin gap is T = 1.5 K M ( a r b . un it s ) H (T) H C = 16 T d M / d H H (T)H C FIG. 5. Magnetization ( M ) vs H measured at T = 1 . H c (cid:39)
16 T ismarked with an arrow. Inset: dM/dH vs H highlights theposition of H c . estimated to be ∆ /k B = J/k B [(1 − α ) / (1+ α ) / (cid:39) .
5K [45]. The critical field for closing a gap of ∆ /k B (cid:39) . H c = ∆ / ( gµ B ) (cid:39) . χ imp = χ + C imp T − θ imp ) was sub-tracted from χ ( T ) and the obtained χ spin ( T ) is plot-ted in the same Fig. 4. It shows a clear exponentialdecrease towards zero demonstrating the singlet groundstate. χ spin ( T ) below the broad maximum is fitted us-ing the expression for the spin susceptibility of a gappedspin-1 / χ ( T ) ∝ (cid:115) ∆ χ k B T e − ∆ χ /k B T . (6)The fit in the low- T region ( T ≤
15 K) returns ∆ χ /k B (cid:39) . H c (cid:39) . T =1 . M is very small and remains almost constant up to about16 T, suggesting that the critical field of the gap clos-ing is H c (cid:39)
16 T. It is slightly higher than H c (cid:39)
10 Tin AgVOAsO but is still in the accessible range. Thisvalue of H c corresponds to the zero-field spin gap of∆ /k B (cid:39) . χ ( T ) analysis but consistent with thatobtained from the fit using Eq. (6). Typically, in thespin-gap compounds, the magnetization remains zero upto H c . However, in our compound, a finite value of themoment was obtained below H c which is likely due tothe saturation of paramagnetic impurities in the powdersample. The data shown in Fig. 5 are corrected for thisparamagnetc contributions. For H ≥
16 T, M increases
10 1000.00.51.01.5
320 330 340 350 360
Eq. (7) I E S R ( a r b . un it ) T (K) data fit d P / d B ( a r b . un it ) B (mT)T = 300 K
T (K) / I E S R ( a r b . un it ) T (K) g g E S R g
300 K28 K I E S R ( a r b . un it ) (cm /mol) FIG. 6. Upper panel: Temperature-dependent ESR inten-sity, I ESR ( T ), obtained by integrating the ESR spectra ofthe polycrystalline NaVOAsO sample. The solid line rep-resents the fit as described in the text. Inset: A typical spec-trum (symbols) at T = 300 K together with the fit using thepowder-averaged Lorentzian shape for the uniaxial g -factoranisotropy. Lower panel: 1 /I ESR vs T along with the fit (seethe text). Top inset: Temperature-dependent g -factor ( g (cid:107) and g ⊥ ) obtained from the Lorentzian fit. Bottom inset: I ESR vs χ with temperature as an implicit parameter. almost linearly with H and then shows a pronounced cur-vature. It does not saturate even at 60 T. A pronouncedcurvature above H c reflects strong quantum fluctuations,as expected for 1D spin chains. The lack of saturationeven at 60 T suggests a large value of the exchange cou-pling. C. ESR
ESR results on the NaVOAsO powder sample arepresented in Fig. 6. The inset of the upper panel ofFig. 6 depicts a typical ESR powder spectrum at roomtemperature. We tried to fit the spectra using thepowder-averaged Lorentzian line for the uniaxial g -factoranisotropy. The fit reproduces the spectral shape verywell at T = 300 K, yielding the anisotropic g -tensorcomponents g (cid:107) (cid:39) .
963 (parallel) and g ⊥ (cid:39) .
998 (per-pendicular). The isotropic g (cid:104) = (cid:113) ( g (cid:107) + 2 g ⊥ ) / (cid:105) value iscalculated to be ∼ .
99. Such a value of g is typicallyobserved for V compounds [28, 47]. As shown in thetop inset of the lower panel of Fig. 6, both g (cid:107) and g ⊥ are found to be temperature-independent down to about10 K. Only below 10 K, an anomalous behaviour was ob-served, which could be due to extrinsic foreign phases.The ESR intensity ( I ESR ) as a function of temperatureshows a pronounced broad maximum at ∼
30 K, similarto the bulk χ ( T ) data. In the bottom inset of the lowerpanel of Fig. 6, I ESR is plotted as a function of χ withtemperature as an implicit parameter. It shows linearityover the temperature range of 28 K to 300 K, implyingthat I ESR traces χ ( T ) nicely.In order to estimate the exchange couplings, I ESR ( T )data were fitted by the equation I ESR = A + Bχ spin ( T ) , (7)where A and B are arbitrary constants. The fit in therange 9 K to 300 K (upper panel of Fig. 6) with thefixed g (cid:39) .
99 yields
J/k B (cid:39) . α (cid:39) .
64. Thisvalue of
J/k B is close to the one obtained from the χ ( T )analysis. In the lower panel of Fig. 6, the inverse ESR in-tensity ( 1 /I ESR ) is plotted as a function of temperature.At high temperatures, the linear behavior resembles the χ − data. We fitted 1 /I ESR ( T ) by an expression1 /I ESR = (cid:20) M + NT − θ CW (cid:21) − , (8)where M and N are arbitrary constants. The fit in thehigh-temperature region yields θ CW (cid:39) −
45 K, which islittle higher in magnitude than the value obtained fromthe χ − analysis. D. As NMR
NMR is a powerful local technique to study the staticand dynamic properties of a spin system. In NaVOAsO ,the As nuclei are strongly hyperfine-coupled to themagnetic V ions along the spin chains. Therefore,the low-lying excitations of the V spins can be probedby means of the As NMR spectra, NMR shift, andspin-lattice relaxation time measurements. As discussedearlier, the χ ( T ) data do not show an exponential de-crease as anticipated for a spin gap system since the low-temperature impurity contribution masks the spin-gapbehavior. Secondly, due to several fitting parameters,the χ ( T ) analysis often doesn’t provide reliable magneticparameters. In this context, the NMR shift has an ad-vantage over the bulk χ ( T ). The bulk χ ( T ) data includeadditional contributions coming from impurities and de-fects present in the sample, whereas the NMR shift di-rectly measures the intrinsic spin susceptibility and iscompletely free from the extrinsic contributions. There-fore, one can precisely estimate the magnetic parametersby analyzing the temperature-dependent NMR shift in-stead of the bulk χ ( T ), and underpin the singlet groundstate.
1. NMR Spectra As is a quadrupole nucleus with the nuclear spin I = 3 / I = 3 / Q and the surrounding electric-field gradient(EFG). The nuclear spin Hamiltonian can be written as asum of the Zeeman and quadrupolar interactions [48, 49], H = − γ (cid:126) ˆ I z H (1+ K )+ hν Q I z (cid:48) − ˆ I )+ η ( ˆ I x (cid:48) − ˆ I y (cid:48) )] . (9)Here, the nuclear quadrupole resonance (NQR) frequencyis defined as ν Q = ν z (cid:112) η / e qQ I (2 I − h (cid:112) η / e is the electron charge, (cid:126) (= h/ π ) is the Planck’s con-stant, H is the applied field along ˆ z , K is the magneticshift due to hyperfine field at the nuclear site, V αβ are thecomponents of the EFG tensor, eq = V z (cid:48) z (cid:48) is the largesteigenvalue of the EFG, and η (= | V x (cid:48) x (cid:48) − V y (cid:48) y (cid:48) | /V z (cid:48) z (cid:48) )is the asymmetry parameter. The principal axes { x (cid:48) , y (cid:48) , z (cid:48) } of the EFG tensor are defined by the local symmetryof the crystal structure. Therefore, resonance frequencycorresponding to any nuclear transition is strongly de-pendent on the direction of the applied field relative tothe crystallographic axes. The parameters ν z , η , and ˆ z (cid:48) can fully characterize the EFG, where, ˆ z (cid:48) is the unit vec-tor in the direction of the principal axis of the EFG withthe largest eigenvalue.When the Zeeman term dominates over the quadrupoleterm, first-order perturbation theory is enough for de-scribing the system. In such a scenario, for a I = 3 / I z = ± / ←→± /
2) should appear on either side of the central peak( I z = +1 / ←→ − / ν Q . On the otherhand, when the quadrupole effects are large enough,second-order perturbation theory is required, and and thepeak positions depend strongly on the angle θ betweenthe applied field H (along ˆ z ) and ˆ z (cid:48) . The expression forthe resonance frequency for the central transition can be written as ν ( ± ) = ν + ν Q ν (1 − cos θ ) (cid:20) (54cos θ ) (cid:18) η cos2 φ (cid:19) − (cid:18) − η cos2 φ (cid:19)(cid:21) + η ν Q ν (cid:20) − − θ −
272 cos φ (cos θ − (cid:21) (10)where ν is the Larmor frequency and φ is the azimuthalangle. For a polycrystalline sample, the NMR spectraare typically broad due to random distribution of theinternal field. They use to show the central transition( I z = +1 / ←→ − /
2) split into two horns, which cor-respond to crystallites with θ (cid:39) . ◦ (lower frequencypeak) and to θ = 90 ◦ (upper frequency peak) [49, 50].The As NMR spectra measured at different tempera-tures are shown in Fig. 7. Their asymmetric double-hornline shape can be described well by the second-order nu-clear quadrupolar interaction. The obtained fitting pa-rameters corresponding to the spectrum at T = 115 Kare: the isotropic shift K iso (cid:39) . K axial (cid:39)− .
031 %, anisotropic shift K aniso (cid:39) ν Q (cid:39) .
12 MHz, asymmetry parameter η (cid:39) . (cid:39) . octahedra and/or lattice distortion.This is in sharp contrast to what is observed in the spin-Peierls compound CuGeO where the lattice distortionleads to the spin dimerization [6] and in α (cid:48) − Na x V O where the singlet ground state is driven by charge order-ing [51] at low temperatures. This is indeed consistentwith our temperature dependent powder XRD data.
2. NMR Shift
From Fig. 7 we can clearly see that the line posi-tion is shifting with temperature. The temperature-dependent NMR shift K ( T ) was extracted by taking thezero quadrupole shift with respect to the reference field ofGaAs and is presented in Fig. 8. Similar to χ ( T ), K ( T )also passes through a broad maximum at around 30 K,which indicates low-dimensional short-range order. Atlow temperatures, K ( T ) decreases rapidly towards zero,which clearly signifies the reduction of the spin suscepti-bility of V and the opening of a spin gap between thesinglet ( S = 0) ground state and triplet ( S = 1) excitedstates. It also implies that the low-temperature upturnobserved in χ ( T ) is purely extrinsic in nature and couldbe due to a small amount of extrinsic impurities, defects,and/or finite crystallite size. In powder samples, the de-fects often break the spin chains and the unpaired spinsat the end of finite chains give rise to the staggered mag-netization, which also appears as low-temperature Curietail in χ ( T ). I n t e n s it y ( a r b . un it s ) H(T)
190 K As NMR50.44 MHz H r e f FIG. 7. Temperature-dependent field-sweep As NMR spec-tra of NaVOAs measured at ν = 50 .
44 MHz. The solid linesare the fits to the spectra at T = 70 K and 115 K. The verticaldashed line corresponds to the As resonance position of thereference GaAs sample.
In general, one can write K ( T ) in terms of χ spin as K ( T ) = K + A hf N A µ B χ spin , (11)where K is the temperature-independent chemical shiftand A hf is the total hyperfine coupling between the Asnuclei and V spins. A hf includes contributions fromtransferred hyperfine coupling and the nuclear dipolarcoupling, both of which are temperature-independent.The magnitude of the nuclear dipolar coupling is usu-ally negligible compared to the transferred hyperfine cou-pling. From Eq. (11), A hf can be calculated by taking theslope of the linear K vs χ plot (inset of Fig. 8) with tem-perature as an implicit parameter. The χ ( T ) data used inthe inset of Fig. 8 were measured at 7 T, which is close tothe field at which our NMR experiments were performed.The data for T ≥
30 K were fitted well to a linear func-tion, and the slope of the fit yields A hf (cid:39) .
75 T/ µ B .Nearly the same value of A hf is reported for the isostruc-tural compound AgVOAsO [29]. However, it is an or-
10 1000123 As NMR50.44 MHz K ( % ) T (K)
K(T) Eq. (11)
210 K 30 K K ( % ) (cm /mol-V )
1D fit ( K - K ) T / -1 )
3D fit ( K - K ) T - / FIG. 8. Upper panel: As NMR shift as a function of tem-perature. The solid line is the fit of K ( T ) by Eq. (11). In-set: K vs χ (measured at 7 T) with temperature as an im-plicit parameter. The solid line is the linear fit. Lower panel:( K − K ) T / vs 1 /T and ( K − K ) T − / vs 1 /T along theleft and right y -axes, respectively. The solid lines are the fitsusing activation laws ( K − K ) T / ∝ e − ∆ K /k B T (1D model)with ∆ K /k B (cid:39) .
84 K and ( K − K ) T − / ∝ e − ∆ K /k B T (3Dmodel) with ∆ K /k B (cid:39) . der of magnitude larger than the one observed for P in1D spin-1 / ) andK CuP O with similar interaction pathways [52, 53].Such a large hyperfine coupling suggests a strong overlapbetween the p orbitals of As and d orbitals of V ionsvia the 2 p orbitals of O − . This also explains why the su-perexchange interaction between V ions is stronger viathe V-O-As-O-V pathway than via the structural chainswith the shorter V-O-V path (see Ref. 28 and Sec. III Ebelow).In order to extract the exchange couplings, the K ( T )data were fitted using Eq. (11) over the whole tempera-ture range, taking χ spin for the spin-1 / et. al. [45]. To minimize thenumber of fitting parameters, A hf and g were fixed tothe values obtained from the K vs χ analysis and ESRexperiments, respectively. Our fit in the whole temper-ature range yields K (cid:39) .
006 %,
J/k B (cid:39) .
14 K, and α (cid:39) .
66. The quality of the fit was very good (seeupper panel of Fig. 8) and the obtained values of
J/k B and α are close to those obtained from the χ ( T ) and I ESR ( T ) analysis. Using the above values of J/k B and α , the estimated spin gap is ∼ . et. al. are ap-plicable for the estimation of J/k B and ∆ /k B from the χ ( T ) data in the zero-field limit but our NMR experi-ments were carried out at a high field of 6.8 T, where thegap is expected to decrease to about ∼ . /k B (cid:39) . /k B = 0 at H c [54].One can also estimate the spin gap by analyzing thelow-temperature part of the K ( T ) data. We fitted thelow-temperature ( T ≤
11 K) K ( T ) data by K ( T ) = K + A χ ( T ), where K and A are arbitrary constantsand χ is given by Eq. (6) with a change that ∆ χ /k B is replaced by ∆ K /k B . The obtained fitting parametersare K (cid:39) . A (cid:39) .
02 %, and ∆ K /k B (cid:39) . K − K ) T / vs 1 /T . The y -axis is shownin the log scale in order to highlight the linear behaviorin the gapped region. This value of ∆ K /k B is still higherthan the expected value of ∼ . H = 6 . K ( T )is obtained in the low-field limit [46, 55] while our ex-perimental data are taken in the field of 6.8 T, which iscomparable to the thermal energy below ∼ d -dimensional spin system,the low-temperature ( k B T << ∆) susceptibility can beexpressed as χ d ∝ T ( d/ − × e − ∆ /k B T . (12)Assuming that our low-temperature K ( T ) data are dom-inated by the 3D ( d = 3) correlations, the above ex-pression is reduced to χ = cT / × e − ∆ /k B T , where c is a constant. In the lower panel of Fig. 8 (right y -axis), we have plotted ( K − K ) T − / versus 1 /T , whichshows a linear regime for T ≤
11 K. Our fit in thisregime returns K (cid:39) − .
023 %, c (cid:39) . − / , and∆ K /k B (cid:39) . K /k B extracted fromthe K ( T ) data using the 3D model is very close to theexpected value of ∼
3. Spin-lattice relaxation rate, /T Spin-lattice relaxation rate, 1 /T , is an important pa-rameter for understanding the dynamic properties of aspin system. It gives direct access to the low-energy spinexcitations by probing the nearly zero-energy limit (in As NMR50.44 MHz / T ( m s - ) T (K) 1/T fit / T ( m s - ) -1 ) t (ms) [ - M ( t ) / M ()] / FIG. 9. Upper panel: 1 /T is plotted as a function of tem-perature. Inset: Recovery of longitudinal magnetization asa function of waiting time t at three different temperatures.Solid lines are the fits using Eq. (13). Lower panel: 1 /T is plotted against 1 /T . The solid line is the fit in the low-temperature region using Eq. (16). the momentum space) of the local spin-spin correlationfunction [57]. The 1 /T measurements for the As nu-clei were done at a field corresponding to the center ofthe spectra and using relatively narrow pulses. Recoveryof the longitudinal magnetization at different tempera-tures, measured using the inversion recovery sequence, π − τ − π/
2, was fitted using the double exponential func-tion [58]12 (cid:20) − M ( t ) M ( ∞ ) (cid:21) = 0 . − t/T ) + 0 . − t/T ) , (13)which corresponds to the I = 3 / As nuclei. Here, M ( t ) and M ( ∞ ) are the nuclear magnetizations at time t and at equilibrium ( t → ∞ ), respectively. The insetof Fig. 9 depicts the recovery curves at three differenttemperatures. The 1 /T values obtained from the fit areplotted as a function of temperature in Fig. 9. At hightemperatures, 1 /T remains constant, a typical behaviorexpected in the paramagnetic regime [57]. Below about12 K, 1 /T decreases rapidly towards zero because of the0opening of a spin gap in the excitation spectrum.Generally, T T can be expressed in terms of the dy-namical susceptibility χ ( (cid:126)q, ω ) per mole of electronic spinsas [59, 60] 1 T T = 2 γ N k B N (cid:88) (cid:126)q | A ( (cid:126)q ) | χ (cid:48)(cid:48) ( (cid:126)q, ω ) ω . (14)Here, the summation is over all the wave vectors (cid:126)q in thefirst Brillouin zone. A ( (cid:126)q ) is the form factor of the hy-perfine interactions as a function of (cid:126)q and χ (cid:48)(cid:48) M ( (cid:126)q, ω ) isthe imaginary part of the dynamic susceptibility at thenuclear Larmor frequency ω . When q = 0 and ω = 0,the real component χ (cid:48) M ( (cid:126)q, ω ) corresponds to the uni-form static susceptibility χ . In the paramagnetic regimewhere spins are uncorrelated, 1 /T is dominated by theuniform q = 0 fluctuations and hence 1 /T remains inde-pendent of T [57]. In that case, 1 /χT T is expected tobe a constant. In our compound, the temperature inde-pendent 1 /T behavior at high temperatures is consistentwith this expectation.For the estimation of spin gap, we tried to fit the 1 /T data by the 1D expression [46, 55]1 /T ∝ ∆ T √ T e − T / k B T . (15)From the fit below 11 K, the value of the spin gap isestimated to be ∆ T /k B (cid:39)
18 K, which is higher than ∼ . /T data using the expression1 /T ∝ T α exp (cid:20) gµ B ( H − H c )2 k B T (cid:21) , (16)which accounts for the 3D magnon excitations over thegapped region ( H < H c ). Here, the exponent α dependson the effective dimension of the magnon dispersion rela-tion as selected by thermal fluctuations k B T [61]. On in-creasing T , α gradually varies from 2 (for the 3D regime, k B T < J ) to 0 (for the 1D regime, J (cid:28) k B T < J ).The lower panel Fig. 9 shows 1 /T vs 1 /T plot alongwith the fit using Eq. (16), where the y -axis is shownin the log scale in order to highlight the activated be-havior at low temperatures. Our fit in the low- T region( T ≤
11 K) with the fixed g = 1 .
99 (from ESR), and H = 6 . α = 2 (forthe 3D regime) yields the critical field of the gap clos-ing H c (cid:39) .
08 T. The corresponding value of the zero-field spin gap is ∆ T /k B (cid:39) .
84 K. This value is invery good agreement with our previous estimation fromthe high-field magnetization data. Moreover, the valueof the exponent α = 2 suggests that at low tempera-tures the spin-lattice is dominated by 3D correlations,as corroborated by the χ ( T ) analysis and band-structurecalculations. TABLE II. The V–V distances (in ˚A), hopping parameters t i (in meV) obtained on the GGA level, and exchange couplings J i (in K) calculated via the DFT+ U mapping procedure forboth NaVOAsO and AgVOAsO . See Fig. 1 for the notationof individual exchange pathways. Note that the DFT+ U re-sults for AgVOAsO differ from those in Ref. 28, because theGGA functional has been used.NaVOAsO AgVOAsO d V − V t i J i d V − V t i J i J J (cid:48) J a J c − − E. Microscopic magnetic model
Two complementary techniques can be used to esti-mate exchange couplings in S = materials. On onehand, total-energy DFT+ U calculations allow a directparametrization of the spin Hamiltonian through theso-called mapping procedure [62], although its resultsmay become ambiguous and depend on computationaldetails, such as the double-counting correction of theDFT+ U method [63]. On the other hand, hopping pa-rameters t i of the uncorrelated (GGA) band structurecan be directly introduced into an effective one-orbitalHubbard model, which in the half-filling regime mapsonto a spin Hamiltonian for low-energy excitations. Con-sequently, the AFM part of the exchange is obtained as J AFM i = 4 t i /U eff , where U eff (cid:39) d band [64].This way, relative strengths of the exchange couplings canbe directly linked to the hopping parameters t i , which arenot biased by the choice of the Hubbard U and by detailsof its DFT+ U implementation.Table II presents the comparative DFT results forthe exchange couplings in NaVOAsO and AgVOAsO .Both compounds feature spin chains formed by J and J (cid:48) . These spin chains do not coincide with the structuralchains, as shown in Fig. 1. The values of t and t (cid:48) suggestthat the intrachain couplings in NaVOAsO are enhancedcompared to the Ag analog, yet the alternation ratiogiven by α = ( t (cid:48) /t ) (cid:39) .
67 remains nearly constant.The interchain couplings are ferromagnetic (FM) J c andAFM J a , and they both become slightly weaker upon thereplacement of Ag by Na. This way, NaVOAsO is some-what closer to the 1D alternating-chain regime than theAg compound.On a more quantitative level DFT+ U overestimatesthe alternation ratio α and renders J nearly equal to J (cid:48) , the result not supported by the hopping parametersand also contradicting the experiment. A similar prob-lem occurs in the case of AgVOAsO [28] and may berelated to the intrinsic difficulties of evaluating weak ex-change couplings (or small differences between the ex-1 FIG. 10. Comparison of magnetization isotherms forAgVOAsO and NaVOAsO measured at 1.5 K (paramag-netic background subtracted). Dashed lines are simulatedcurves for alternating spin chains uniformly coupled in 2D byan effective interchain coupling J ⊥ . The fit parameters are J = 34 K, α = 0 .
60, and J ⊥ /J = − .
15 for AgVOAsO and J = 51 K, α = 0 .
63, and J ⊥ /J = − .
05 for NaVOAsO . change couplings) within DFT+ U [65]. All qualitativetrends are, nevertheless, consistent with the experimentalfindings. The susceptibility fit with the alternating-chainmodel yields stronger exchange couplings in NaVOAsO ( J (cid:39) . J (cid:48) (cid:39) . ( J (cid:39) . J (cid:48) (cid:39) . α (cid:39) .
65 (Na) and α (cid:39) . saturatesat 48.5 T, whereas the saturation of NaVOAsO is notreached even at 59 T, the highest field of our measure-ment. With the 23% increase in J and J (cid:48) , one ex-pects a similar increase in H c that amounts to 10 Tin AgVOAsO and should then be around 12 T inNaVOAsO , much lower than the experimental value of16 T. This discrepancy confirms that not only the intra-chain couplings increase but also the interchain couplingsdecrease upon the replacement of Ag by Na. Direct sim-ulation of the magnetization curves (Fig. 10) shows thebest agreement for J ⊥ /J = − .
15 in AgVOAsO vs. J ⊥ /J = − .
05 in NaVOAsO , thus suggesting a signifi-cant reduction in the interchain exchange [66].Finally, we note that individual exchange couplings inNaVOAsO follow the general microscopic scenario forthe V compounds. The distortion of the VO octahe-dra puts the magnetic orbital in the plane perpendicularto the structural chains [28]. This facilitates magnetic in-teractions between the structural chains ( J and J (cid:48) ) andsuppresses the coupling J c along these chains. The inter-action J a through a single AsO tetrahedron is of similar strength as J c but of the opposite sign, which leads to acompeting exchange scenario and the frustration of the3D spin lattice, where the four-fold loops with one FMand three AFM couplings are present (Fig. 1). The frus-trated nature of the interchain couplings was arguablythe reason for inconsistencies in the fitting parametersfor AgVOAsO [28]: compare J (cid:39) . J (cid:39)
34 K from the fit of the magneti-zation curve. This discrepancy is basically remedied inNaVOAsO , where we find a consistent fit of both suscep-tibility and high-field magnetization with J (cid:39)
51 K and α = 0 . − .
65. Therefore, the frustration seems to playminor role in NaVOAsO compared to its Ag analogue. IV. SUMMARY
We have studied in detail the magnetic behavior ofthe quantum magnet NaVOAsO and establish it as anew alternating spin-1 / ions are coupled antiferromagnetically and form al-ternating chains running in two crossing directions in the ab -plane. Compared to its Ag-analogue, the lattice pa-rameters and unit cell volume are smaller in magnitudefor NaVOAsO . As a result, the strength of the exchangeinteractions are found to be stronger in NaVOAsO .The χ ( T ) data analysis with the alternating spin-1 / J/k B (cid:39) . J (cid:48) /k B (cid:39) . /k B (cid:39) . H c (cid:39)
16 T, where the magnetization de-parts from zero. The relatively large H c compared to H c (cid:39)
10 T in AgVOAsO is not fully accounted by theincreased intrachain couplings and also reflects the reduc-tion in the interchange couplings. Therefore, NaVOAsO is closer to the 1D regime.The As nuclei are strongly coupled to the magneticV spins with a hyperfine coupling A hf (cid:39) .
75 T/ µ B .The NMR shift K ( T ) and the spin-lattice relaxation rate1 /T ( T ) show the low-temperature activated behavior,which unambiguously demonstrates the singlet groundstate in this compound. The spin gap ∆ K3D /k B (cid:39) . K ( T ) data using the 3D model ismuch closer to the expected value of ∼ . /k B (cid:39) .
84 from the 1 /T analysis, accounting forthe 3D magnon dispersion, is consistent with that ob-tained from the magnetization isotherm data, suggestingthat at low temperatures NaVOAsO acts as a 3D mag-net despite its relatively weak interchain couplings. Thegapped nature of the spectrum and the availability ofthe microscopic parameters render NaVOAsO a modelcompound for high-field studies especially for exploringfield-induced effects.2 ACKNOWLEDGMENTS
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16 T, and the corre-sponding zero-field spin gap is ∆ /k B (cid:39) . As NMR measurements were carried out in the fieldof 6.8 T, the spin gap will be reduced. The reductionis estimated as ∆ (cid:48) /k B = Hgµ B k B (cid:39) . As NMR at 6.8 T one should observe a spin gap of∆ /k B − ∆ (cid:48) /k B (cid:39) . /k B .[55] K. Damle and S. Sachdev, “Spin dynamics and transportin gapped one-dimensional Heisenberg antiferromagnetsat nonzero temperatures,” Phys. Rev. B , 8307–8339(1998).[56] S. Taniguchi, T. Nishikawa, Y. Yasui, Y. Kobayashi,M. Sato, T. Nishioka, M. Kontani, and K. Sano, “Spingap behavior of s = 1 / O ,” J. Phys. Soc. Jpn. , 2758–2761 (1995). [57] Tru Moriya, “Nuclear magnetic relaxation in antiferro-magnetics,” Progr. Theor. Phys. , 23–44 (1956).[58] M.I. Gordon and M.J.R. Hoch, “Quadrupolar spin-latticerelaxation in solids,” J. Phys. C: Solid State Phys. ,783 (1978); W. W. Simmons, W. J. O’Sullivan, andW. A. Robinson, “Nuclear spin-lattice relaxation in di-lute paramagnetic sapphire,” Phys. Rev. , 1168–1178(1962).[59] R. Nath, Y. Furukawa, F. Borsa, E. E. Kaul, M. Baenitz,C. Geibel, and D. C. Johnston, “Single-crystal PNMR studies of the frustrated square-lattice compoundPb VO(PO ) ,” Phys. Rev. B , 214430 (2009).[60] K. M. Ranjith, M. Majumder, M. Baenitz, A. A. Tsir-lin, and R. Nath, “Frustrated three-dimensional antifer-romagnet Li CuW O : Li NMR and the effect of non-magnetic dilution,” Phys. Rev. B , 024422 (2015).[61] S. Mukhopadhyay, M. Klanjˇsek, M. S. Grbi´c, R. Blin-der, H. Mayaffre, C. Berthier, M. Horvati´c, M. A. Con-tinentino, A. Paduan-Filho, B. Chiari, and O. Pi-ovesana, “Quantum-critical spin dynamics in quasi-one- dimensional antiferromagnets,” Phys. Rev. Lett. ,177206 (2012).[62] H. J. Xiang, E. J. Kan, S.-H. Wei, M.-H. Whangbo, andX. G. Gong, “Predicting the spin-lattice order of frus-trated systems from first principles,” Phys. Rev. B ,224429 (2011).[63] A. A. Tsirlin, O. Janson, and H. Rosner, “Unusual fer-romagnetic superexchange in CdVO : The role of Cd,”Phys. Rev. B , 144429 (2011).[64] A. A. Tsirlin, R. Nath, C. Geibel, and H. Rosner, “Mag-netic properties of Ag VOP O : An unexpected spindimer system,” Phys. Rev. B , 104436 (2008).[65] A. A. Tsirlin and H. Rosner, “Extension of the spin-1/2frustrated square lattice model: The case of layered vana-dium phosphates,” Phys. Rev. B , 214417 (2009).[66] Similar to Ref. 28, we perform simulations for a 2D arrayof alternating spin chains uniformly coupled by an effec-tive interchain coupling J ⊥ . The actual 3D spin lattice ofNaVOAsO4