Phase separation and frustrated square lattice magnetism of Na1.5VOPO4F0.5
A. A. Tsirlin, R. Nath, A. M. Abakumov, Y. Furukawa, D. C. Johnston, M. Hemmida, H.-A. Krug von Nidda, A. Loidl, C. Geibel, H. Rosner
PPhase separation and frustrated square lattice magnetism of Na . VOPO F . A. A. Tsirlin, ∗ R. Nath,
1, 2, 3
A. M. Abakumov, Y. Furukawa, D. C. Johnston, M. Hemmida, H.-A. Krug von Nidda, A. Loidl, C. Geibel, and H. Rosner † Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str. 40, 01187 Dresden, Germany Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 USA School of Physics, Indian Institute of Science Education and Research, Trivandrum-695016 Kerala, India EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium Experimental Physics V, Center for Electronic Correlations and Magnetism,University of Augsburg, 86135 Augsburg, Germany
Crystal structure, electronic structure, and magnetic behavior of the spin- quantum magnetNa . VOPO F . are reported. The disorder of Na atoms leads to a sequence of structural phasetransitions revealed by synchrotron x-ray powder diffraction and electron diffraction. The high-temperature second-order α ↔ β transition at 500 K is of the order-disorder type, whereas thelow-temperature β ↔ γ + γ (cid:48) transition around 250 K is of the first order and leads to a phaseseparation toward the polymorphs with long-range ( γ ) and short-range ( γ (cid:48) ) order of Na. Despitethe complex structural changes, the magnetic behavior of Na . VOPO F . probed by magneticsusceptibility, heat capacity, and electron spin resonance measurements is well described by theregular frustrated square lattice model of the high-temperature α -polymorph. The averaged nearest-neighbor and next-nearest-neighbor couplings are ¯ J (cid:39) − . J (cid:39) . T N = 2 . . VOPO F . as a parent compound for the experimental study of tetramerized square latticesas well as frustrated square lattices with different values of spin. PACS numbers: 75.50.Ee, 75.30.Et, 75.10.Jm, 61.66.Fn
I. INTRODUCTION
Among a broad family of low-dimensional and frus-trated magnets, systems based on the frustrated squarelattice (FSL) model enjoy special attention from the-ory and experiment. The model entails competing ex-change couplings along sides ( J ) and diagonals ( J )of the square and is mostly studied for purely Heisen-berg Hamiltonian and spin- . Extensive theoreticalwork convincingly established three ordered groundstates that emerge for different values of the frustra-tion ratio α = J /J . The N´eel and columnar antiferro-magnetic (AFM) states are separated by a critical spin-liquid region around the quantum critical point at α = (Refs. 3–6). At α = − , the columnar AFM phase shouldborder the ferromagnetic (FM) phase. However, the pre-cise nature of this boundary remains controversial. Whilean earlier report proposed a nematic phase separatingthe regions of FM and columnar AFM ground states, Richter et al. demonstrated the single abrupt transitionwithout any intermediate phases around α = − .Experimentally, the FSL-type magnetic behavior hasbeen observed in V +4 compounds with [VOXO ] layerscomprising VO square pyramids and non-magnetic XO tetrahedra (the only known exception is PbVO withmagnetic layers formed by VO pyramids exclusively ). If X is a main group ( p ) cation, the leading coupling isAFM J , whereas J is usually weaker and can be ei-ther FM (X = P) or AFM (X = Si, Ge), see also Ta-ble VII. Recent efforts in crystal growth andexperimental investigation of such compounds es-tablished the columnar AFM ground state and re-vealed simple trends in thermodynamic properties, in line with theoretical predictions. A thorough compar-ison between experiment and theory, however, spots cer-tain discrepancies. For example, the sublattice magne-tization is gradually reduced from 0.6 µ B in Li VOSiO ( α (cid:39)
10) to 0.5 µ B in Pb VO(PO ) ( α (cid:39) −
2) and0.4 µ B in SrZnVO(PO ) ( α (cid:39) − By contrast,theory predicts a nearly constant sublattice magnetiza-tion of 0.6 µ B in this range of α (Ref. 8). The discrepancymay arise from the spatial anisotropy of the FSL due tothe low crystallographic symmetry of Pb VO(PO ) andSrZnVO(PO ) . In fact, none of the reported FSL-typephosphates, AA (cid:48) VO(PO ) ( AA (cid:48) = Pb , PbZn, SrZn,BaZn, BaCd) with FM J , are tetragonal, thereby theeffects of the spatial anisotropy are expected. To facilitate the experimental verification of theo-retical results for the FSL model, tetragonal systemswith a perfect square lattice of magnetic atoms arerequired. Motivated by this challenge, we exploredthe structure and properties of Na . VOPO F . . Thiscompound was prepared in 2002 by a hydrothermal a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r TABLE I. Lattice parameters, space groups, and refinement residuals for different polymorphs ofNa . VOPO F . .Polymorph Temperature a c Space group R I /R p (K) (˚A) (˚A) α
560 6.39563(1) 10.65908(2) I /mmm β
298 9.03051(2) 10.62002(3) P /mnm γ
150 12.76716(2) 10.57370(4) P /mbc γ (cid:48)
150 6.37996(4) 10.5910(1) I /mmm method. Massa et al. found a tetragonal crystal struc-ture with the three-dimensional (3D) VOPO F . frame-work formed by the FSL-type VOPO layers (Fig. 1).Sauvage et al. claimed to prepare the same compoundby a high-temperature annealing in air, and reported avery similar structure refinement. Since Na atoms inNa . VOPO F . and related systems are readily dein-tercalated and exchanged with Li, the frustrationratio α could be tuned, making Na . VOPO F . an ap-pealing FSL system. To explore this possibility, we per-formed a comprehensive study of the parent compoundNa . VOPO F . .The outline of the paper is as follows. We list experi-mental and computational procedures in Sec. II, and pro-ceed to details of the crystal structure in Sec. III. Furtheron, we perform thermodynamic and magnetic resonancemeasurements (Sec. IV) and band structure calculations(Sec. V) to evaluate the low-temperature magnetic be-havior and the microscopic magnetic model. The com-parison of Na . VOPO F . to known FSL compoundsand other structural analogs is given in Sec. VI followedby a summary and outlook. II. METHODS
Powder samples of Na . VOPO F . were prepared bya solid-state reaction of Na P O , VO , and NaF inan evacuated and sealed silica tube at 700 ◦ C for 24hours. The stoichiometric mixtures of reactants werepelletized, placed into corundum crucibles, and cov-ered with lids to avoid the reaction between silica andNaF. Na P O was obtained by the decomposition ofNa HPO in air at 400 ◦ C. The bluish-green powders ofNa . VOPO F . were single-phase, as confirmed by lab-oratory x-ray diffraction (XRD) measured with HuberG670 Guinier camera (CuK α radiation, 2 θ = 3 − ◦ angle range, image-plate detector).High-resolution XRD data for structure refinementwere collected in the 150 −
560 K temperature rangeat the ID31 beamline of European Synchrotron Radia-tion Facility (ESRF) with a constant wavelength of about0.4 ˚A. The signal was measured by eight scintillation de-tectors, each preceded by a Si (111) analyzer crystal, inthe angle range 2 θ = 1 −
40 deg. The powder samplewas contained in a thin-walled borosilicate glass capil- lary with an external diameter of 0.5 mm. The samplewas cooled below room temperature in a He-flow cryo-stat and heated above room temperature with a hot-airblower. To achieve good statistics and to avoid the effectsof the preferred orientation, the capillary was spun dur-ing the experiment. The JANA2006 program was usedfor the structure refinement. Symmetry changes at thestructural phase transitions were analyzed with ISODIS-PLACE program. The samples for an electron diffraction (ED) studywere prepared by crushing the powder in ethanol anddepositing the suspension on a holey carbon grid. EDpatterns were taken at room temperature (RT) and at100 K using a Philips CM20 microscope equipped with aGatan cooling holder.The differential scanning calorimetry (DSC) measure-ment was performed with a Perkin Elmer DSC 8500 in-strument in the temperature range 120 −
800 K in ar-gon atmosphere with a heating/cooling rate of 10 K/min.The powder sample of Na . VOPO F . was placed intoa corundum crucible.The magnetic susceptibility of Na . VOPO F . wasmeasured with an MPMS SQUID magnetometer in thetemperature range 2 −
380 K in applied fields up to 5 T.Heat capacity measurements on a pressed pellet were per-formed by relaxation technique with a Quantum DesignPPMS instrument in the temperature range 0 . −
200 Kand in fields up to 11 T. The data below 1.8 K werecollected with a He insert.The electron spin resonance (ESR) measurements wereperformed at X-band frequency ( ν = 9 .
36 GHz) ona Bruker ELEXSYS E500-CW spectrometer, equippedwith a continuous He-gas flow cryostat (Oxford Instru-ments) operating in the temperature range 4 . −
300 K.The polycrystalline powder sample was fixed in a quartztube with paraffin and mounted in the center of the mi-crowave cavity. The field derivative of the microwave–absorption signal was detected as a function of the staticmagnetic field due to the lock-in technique with fieldmodulation. Resonance absorption occurs when the in-cident microwave energy matches the energy of magneticdipolar transitions between the electronic Zeeman levels.The nuclear magnetic resonance (NMR) measurementswere carried out using pulsed NMR techniques on P(nuclear spin I = and gyromagnetic ratio γ N / π =17 .
237 MHz/T) nuclei in the temperature range 1 . − a O1O2FNa ab cJ J J ^ FIG. 1. (Color online) Crystal structure of α -Na . VOPO F . : the layers in the ab plane (left)and the projection along c (right). The layers have anideal FSL geometry with the couplings J and J (left) andconnect along c via the F atoms (right). Disordered Naatoms occupy the voids of the resulting framework (right).
300 K. We did the measurements at radio frequencies of79 MHz and 8 .
62 MHz, which correspond to applied fieldsof about 4 .
583 T and 0 . K = ( H ref − H ) /H was determined by measuring the resonance field of thesample ( H ) with respect to a nonmagnetic referenceH PO (resonance field H ref ). The P spin-lattice relax-ation rate 1 /T was measured by the conventional singlesaturation pulse method.Scalar-relativistic band-structure calculations forNa . VOPO F . were performed within the frameworkof density functional theory (DFT) using the basis setof local orbitals ( FPLO9.01-35 code). We applied thelocal density approximation (LDA) with the exchange-correlation potential by Perdew and Wang and a k mesh of 216 points in the first Brillouin zone (40 pointsin the irreducible wedge). Exchange couplings wereevaluated by mapping V 3 d bands onto a multi-orbitalHubbard model (see Sec. V for further details). III. CRYSTAL STRUCTUREA. High-temperature α -polymorph According to RT studies of Refs. 26 and 27,Na . VOPO F . has a body-centered tetragonal unitcell with a sub (cid:39) .
38 ˚A and c (cid:39) .
62 ˚A. Our RT XRDpattern is largely consistent with this unit cell, although10 weak reflections remained unindexed. These reflec-tions can be assigned to the √ a sub × √ a sub × c su-percell and evidence the superstructure formation at RT.The superstructure reflections in Na . VOPO F . disap-peared upon heating. Above 500 K, the patterns could befully indexed on the a sub × a sub × c body-centered tetrag-onal unit cell that we further refer as α -modification, incontrast to β -modification at RT (Table I).The structure refinement for the α -phase (Table II) TABLE II. Atomic positions, isotropic atomic displace-ment parameters ( U iso , in 10 − ˚A ), and occupancy fac-tors ( f ) of α -Na . VOPO F . at 560 K (upper lines) and γ (cid:48) -Na . VOPO F . at 150 K (bottom lines). The f valuesare fractions of total occupancy for a given position.Atom Position x y z U iso f Na1 8 h l e d e n a conforms to the atomic positions given in Ref. 26. Wefind a 3D framework of corner-sharing VO F octahedraand PO tetrahedra (Fig. 1). Vanadium atoms form oneshort bond to the O1 atom (along the c direction), fourlonger bonds to oxygens in the ab plane (the O2 position),and one long bond to the fluorine atom (also along c ), seeTable III. The octahedra are linked via the PO tetrahe-dra in the ab plane, whereas the fluorine atoms connectthe octahedra into pairs along c . Na atoms occupy thevoids of the resulting framework.At this point, the question concerning the arrangementof O and F atoms may arise. Since atomic numbers ofthese elements differ by unity, it is hard to distinguishbetween O and F using XRD, especially in a powder ex-periment. The single-crystal refinement proposed thecomplete ordering of the O and F atoms. The orderedarrangement of these atoms is also supported by empiri-cal arguments. The O1 position corresponds to the shortV–O bond which is typical for oxygen and uncommonfor fluorine, e.g., in oxyfluorides. Further on, the O2atoms in the ab plane are parts of rigid PO tetrahe-dra and can not mix with fluorine. Therefore, fluorineis left to its 2 a position with two long bonds to vana-dium. Note also that our ESR and NMR experiments(Sec. IV) suggest unique (or very similar) positions forvanadium and phosphorous, respectively, thus indicatingthe ordered arrangement of O and F atoms below RT.While the O and F atoms in Na . VOPO F . form anordered framework, the Na atoms are disordered. Ourrefinement for the α -modification identifies two Na po-sitions, 8 h (Na1) and 16 l (Na2). After constraining thesum of occupancies to the Na . VOPO F . compositionand refining atomic displacement parameters (ADPs) to- TABLE III. Interatomic distances (in ˚A) in α -, β -, and γ -polymorphs of Na . VOPO F . at 560 K,298 K, and 150 K, respectively. α β γ V–O1 1.626(3) V–O1 1.636(2) V–O1 1.629(3)V–O2 4 × . × . × . × . × . × . × . × . × . × . gether with the occupancies, we found that the 8 h posi-tion is approximately half-filled, whereas the 16 l positionis filled by (i.e., twice less Na atoms than in 8 h ). Thisis in line with Ref. 26 that reported the occupancies of and for 8 h and 16 l , respectively (our experimentis done at 560 K compared to 300 K in Ref. 26, hencea redistribution of Na atoms is possible). By contrast,Ref. 27 assigns Na2 to a 8 j position with a negligible oc-cupancy of , i.e., 0.5 atoms per unit cell in 8 j comparedto approximately 2 atoms in 16 l in our refinement. B. Room-temperature β -polymorph The superstructure formation in the RT β -polymorphis confirmed by ED (Fig. 2). While the intense re-flections are assigned to the body-centered tetragonal a sub × a sub × c unit cell of the α -polymorph, weaker su-perstructure reflections are clearly visible in the [001],[010], and [101] patterns. XRD and ED suggest a prim-itive tetragonal √ a sub × √ a sub × c unit cell, withthe reflection condition h l , h + l = 2 n that identifiesthe P /mnm space group. This reflection conditionis present in the [010] pattern (Fig. 2). The emergenceof the forbidden reflections, such as 100 and 010 in the[001] pattern, is caused by the multiple diffraction. Fi-nally, the [110] pattern reveals sharper reflection condi-tions h + k = 2 n and h + l = 2 n corresponding to thebody-centered unit cell of the α -polymorph. The lack ofsuperstructure reflections, such as 1¯10, in the [110] EDpattern is due to their zero structure factors, in agree-ment with the complete absence of these reflections inthe XRD pattern.The superstructure formation at room temperature isrelated to the partial ordering of the Na atoms in the β -modification. The structure refinement (Table IV) showed that the VOPO F . framework remains intact(Table III), whereas Na atoms are partially ordered intwo 8 i positions. The Na1 position is completely filled,and the Na2 position is exactly half-occupied under theconstraint of the Na . VOPO F . composition. FIG. 2. ED patterns of β -Na . VOPO F . at RT. The pat-terns are indexed on a tetragonal √ a sub × √ a sub × c su-percell. Bright reflections correspond to the a sub × a sub × c subcell, whereas weak reflections indicate the supercell.TABLE IV. Atomic positions and isotropic atomic displace-ment parameters ( U iso , in 10 − ˚A ) of β -Na . VOPO F . at298 K. The Na2 position is half-filled.Atom Position x y z U iso Na1 8 i i j d
12 14 e j k j j f aag+g ’ bbb -phase a -phase ggg ’ g ’ Sub ce ll vo l u m e ( A ) o3 c a / r a t i o s ub FIG. 3. (Color online) Temperature dependence ofthe subcell volume (top) and c/a sub ratio (bottom) forNa . VOPO F . . Error bars are smaller than symbols. Linesare guide for the eyes, with arrowheads showing changes uponheating/cooling in the region of the β ↔ γ + γ (cid:48) transition. - -
265 K235 K D S C ( V / m g ) m FIG. 4. (Color online) DSC data showing the first-order β ↔ γ + γ (cid:48) transition at 265 K (heating, upper panel) and235 K (cooling, bottom panel). The structure refinements for α - and β -Na . VOPO F . suggest that the phase transitionat 500 K is of the second order (order-disorder type).Indeed, the temperature evolution of lattice parameters(Fig. 3) shows a smooth change in the cell volume around500 K. The structural change also conforms to sym-metry requirements for a second-order transition: the P /mnm space group can be derived from I /mmm using the X − irreducible representation. Similar tosecond-order transitions in (CuCl)LaNb O (Ref. 37),we did not observe the α ↔ β transformation by DSC,presumably, due to the small change in the entropy. TABLE V. Atomic positions and isotropic atomic displace-ment parameters ( U iso , in 10 − ˚A ) of γ -Na . VOPO F . at150 K.Atom Position x y z U iso Na1 8 h h h i − . b d
12 14 g i − . i i − . i i h − . q ag : (101): (201) 4.36 I n t e n s i t y : / II ab g FIG. 5. (Color online) XRD pattern of Na . VOPO F . collected at 150 K (black dots) and the two-phase refinement(solid line). Upper and lower ticks denote the reflections ofthe γ (cid:48) - and γ -phases. The inset shows the unit cells of α -(solid), β - (dotted), and γ - (dashed) polymorphs. C. Low-temperature phase separation
Below RT, one can expect a further ordering of theNa atoms. Indeed, we found another structural trans-formation around 250 K. This transition is of the first-order type, as shown by: i) the temperature hysteresis(about 235 K on cooling and about 265 K on heating,Fig. 4); ii) the coexistence of the low-temperature andhigh-temperature phases in a certain temperature range;iii) the abrupt change in the cell volume (Fig. 3). Thetransition is also revealed by an abrupt change in theESR linewidth (see Sec. IV C).Upon cooling the sample, new reflections appeared be-low 235 K, and the reflections of the β -phase fully dis-appeared at 210 K. Below 210 K, the pattern could beindexed on a 2 a sub × a sub × c tetragonal unit cell that we FIG. 6. ED patterns of γ -Na . VOPO F . at 100 K. Thepatterns are indexed on a tetragonal 2 a sub × a sub × c unit cell.Brighter reflections with even h , k , and l indices arise fromthe subcell, whereas weaker spots evidence the formation ofthe superstructure.FIG. 7. ED patterns of γ (cid:48) -Na . VOPO F . at 100 K.The patterns are indexed on a body-centered tetragonal a sub × a sub × c unit cell. Diffuse lines in the [110] patternevidence short-range order. further refer as γ -modification. However, a closer exam-ination showed a complex shape of the ( hkl ) reflectionswith non-zero l . In particular, the (002) reflection at2 θ = 4 . ◦ could not be fitted as a single peak (Fig. 5).This points to the coexistence of two phases with slightlydifferent lattice parameters.The low-temperature ED study confirmed the presenceof two different phases below RT. One of the phases isthe ordered γ -polymorph that shows sharp superstruc-ture reflections of the 2 a sub × a sub × c unit cell (Fig. 6).The crystallites of the second phase are less ordered andreveal diffuse satellites or diffuse intensity lines instead ofthe superstructure reflections (Fig. 7). The γ -polymorphhas a primitive tetragonal unit cell. The reflection con-ditions hhl , l = 2 n ([110] pattern) and h l , h = 2 n ([001]and [010] patterns) were confirmed by XRD, and resulted in the P /mbc space group. Similar to the [110] patternof β -Na . VOPO F . (Fig. 2), the [010] pattern of the γ -polymorph contains subcell reflections only.The second low-temperature phase, further referred as γ (cid:48) -polymorph, reveals sharp subcell reflections as well asdiffuse satellites on the [001] pattern and diffuse linesalong c ∗ on the [110] pattern (Fig. 7). The intensity of thediffuse lines is modulated in a way that the maxima co-incide with the superstructure reflections of the γ -phase.This suggests that the two low-temperatures polymorphsdevelop a similar supercell, with the long-range order in γ and the short-range order in γ (cid:48) . While the orderedstructure of γ -Na . VOPO F . can be determined fromXRD (see below), the nature of the short-range order in γ (cid:48) -Na . VOPO F . is difficult to establish because thecompound is unstable under the electron beam.In XRD, the diffuse intensity produced by the γ (cid:48) -polymorph is smeared. The remaining subcell reflec-tions are formally equivalent to that of the α -phase. The150 K pattern could be refined as a two-phase mixtureof the γ -phase with the 2 a sub × a sub × c unit cell andthe γ (cid:48) -phase with the a sub × a sub × c unit cell. The re-fined composition of the mixture yields about 40 % ofthe γ (cid:48) -phase. This ratio is nearly constant within thetemperature range under investigation.The structure refinement of γ -Na . VOPO F . (Ta-ble V) identifies three Na positions which are fully oc-cupied, as evidenced by their low ADPs. The orderingof Na has moderate effect on the framework: the V–Oand P–O distances are nearly unchanged compared tothe α - and β -phases (Table III). The refined structure of γ (cid:48) -Na . VOPO F . closely resembles the α -polymorph(Table II). Based on the XRD and ED data, we sug-gest that Na . VOPO F . develops a two-phase mixtureat the first-order transition around 250 K. Specific heat,DSC, and ESR measurements (Sec. IV C and Ref. 33)do not reveal any further structural changes below thistemperature.At low temperatures, the γ - and γ (cid:48) -phases have thesame subcell volume but a different c/a sub ratio (Fig. 3).Based on our synchrotron XRD data as well as mag-netization, NMR, and ESR measurements (Sec. IV),we conjecture that both phases contain V +4 , and theircomposition is, therefore, similar (within the availableresolution). The origin of the phase separation inNa . VOPO F . is presently unclear and should be dis-closed in future studies. For a further discussion on Naordering in Na . VOPO F . and related compounds, werefer the reader to Sec. VI A. IV. MAGNETIC BEHAVIORA. Magnetization
The temperature dependence of the magnetic suscep-tibility ( χ ) of Na . VOPO F . is typical for a low-dimensional (and possibly frustrated) antiferromagnet,see Fig. 8. At low fields ( µ H ≤ T χ max (cid:39) . T N (cid:39) . χ : the maximum shifts to lower temperatures,whereas T N slightly increases (see also Fig. 11). Above50 K, 1 /χ ( T ) shows a nearly linear behavior indicatingthe Curie-Weiss paramagnetic regime. No signatures ofthe β ↔ γ + γ (cid:48) transition around 250 K could be ob-served.Above 50 K, the fit of the data with the Curie-Weisslaw: χ = χ + CT + θ (1)yields temperature-independent susceptibility χ = − . × − emu/mol (core diamagnetism andVan Vleck paramagnetism), the Curie constant C =0 . θ = 3 . µ B conforms to the expected valueof gµ B (cid:112) S ( S + 1) = 1 . µ B for V +4 where we used S = and the powder-averaged g = 1 .
95 from ESR(see Sec. IV C).Despite the 3D nature of the crystal structure, a low-dimensional magnetic behavior should be expected. Pre-vious studies of V +4 compounds identify the typ-ical d xy orbital ground state driven by the short V–O1bond along the z direction. Since the d xy orbital does notoverlap with the orbitals of the axial O1 and F atoms,exchange couplings along the c direction are weak. Inthe case of Na . VOPO F . , this implies the formationof VOPO magnetic layers, similar to AA (cid:48) VO(PO ) FSL-type vanadium phosphates (see the left panel ofFig. 1).
Three different phosphorus positions inthe γ -Na . VOPO F . structure lead to a number ofinequivalent exchange couplings (Sec. V and Fig. 18).However, the thermodynamic properties in the high-temperature region ( T > | J | , J ) should follow the pre-dictions for the ideal FSL model, and allow one to eval-uate ¯ J and ¯ J , the averaged nearest-neighbor (NN) andnext-nearest-neighbor (NNN) couplings on the squarelattice. Based on the above arguments, we fit the mag-netic susceptibility of Na . VOPO F . with the high-temperature series expansion (HTSE) for the FSLmodel: χ = χ + N A g µ B k B T (cid:88) n (cid:18) ¯ J k B T (cid:19) n (cid:88) m c mn (cid:18) ¯ J ¯ J (cid:19) m , (2)where χ is similar to Eq. (1), N A is Avogadro’s num-ber, g is the g -factor, and c mn with m, n ≤ J and ¯ J (see Refs. 11, 15, 24, and m H HTSE fit 100Temperature (K) c ( e m u / m o l ) - FIG. 8. (Color online) Temperature dependence of themagnetic susceptibility ( χ ) and the HTSE fit with Eq. (2): T min = 12 K, ¯ J = − . J = 6 . χ at low temperatures. The kink at2 . − . T min
12 14 16 g c J J - - - E x c h a n g e c o up li n g s ( K ) F i tt e d g c ( e m u / m o l ) - - - FIG. 9. (Color online) Details of the magnetic susceptibil-ity fit: averaged exchange couplings ¯ J and ¯ J , g -value, andtemperature-independent contribution χ evaluated for thefitting ranges with different T min . J = − . J = 6 . g = 1 . χ = − . × − emu/mol) or AFM¯ J = 5 . J = − . g = 1 . χ = − . × − emu/mol). The correct solutioncan be determined from the specific heat data, thesaturation field, or band structure calculations. Inthe case of Na . VOPO F . , band structure calculationsprovide a solid evidence for leading AFM couplings be-tween next-nearest neighbors (see Sec. V). The analysisof the saturation field also favors the solution with FM¯ J and AFM ¯ J (see below).A more subtle problem of the HTSE fit lies in thechoice of the fitting range. The convergence of theHTSE depends on the frustration ratio α , hence the lowerlimit of the fitting range ( T min ) is not universal. Fit-ting the data with different T min , we find a stable so-lution for T min = 12 −
16 K (Fig. 9). The resulting χ = − . × − emu/mol and g = 1 . χ = − . × − emu/mol, g = 1 . T min ,the high-temperature part of the HTSE fit deviates fromthe Curie-Weiss fit, thereby less accurate estimates of ¯ J and ¯ J are obtained. Using ¯ J (cid:39) − . J (cid:39) . α = ¯ J / ¯ J (cid:39) − . VO(PO ) , BaZnVO(PO ) ,and PbZnVO(PO ) , see Table VII as well as Refs. 23and 24.The magnetization curve of Na . VOPO F . (rightpanel of Fig. 11) is typical for a two-dimensional (2D)antiferromagnet. The linear increase in the magnetiza-tion ( M ) at low fields is followed by a slight curvatureand the saturation at µ H s (cid:39) . J and ¯ J against H s , we use the expressionfrom Ref. 42 for the columnar AFM state ( J > − . J )on the regular square lattice: µ H s = 2( ¯ J + 2 ¯ J ) k B / ( gµ B ) . (3)Assuming g = 1 .
95 from ESR (Sec. IV C), one finds µ H s (cid:39) . J implies theN´eel ordering (antiparallel spins on nearest neighbors),leads to µ H s (cid:39) . H s , which inthis case is proportional to 4 ¯ J instead of 2( ¯ J + 2 ¯ J ) inEq. (3). While the saturation field itself does not allowus to choose the correct solution, DFT results (Sec. V)put forward the sizable AFM ¯ J and, therefore, FM ¯ J –AFM ¯ J regime with an additional interlayer coupling.This interlayer coupling contributes to the saturationfield, and explains the underestimate of H s in the purely2D model described by Eq. (3). By contrast, the AFM ¯ J – FM ¯ J regime overestimates H s even in the 2D model,thereby any additional couplings will only exaggerate thediscrepancy between the exchange integrals and satura-tion field. This way, we choose the FM ¯ J – AFM ¯ J solution, which is confirmed by the similarity to otherFSL-type vanadium phosphates. B. Heat capacity
The measured specific heat ( C p ) of Na . VOPO F . resembles that of BaCdVO(PO ) and other FSLcompounds. The sharp anomaly at T N (cid:39) . C p above 8 K (Fig. 10). The maximum is a signature ofthe magnetic contribution C mag , whereas the data above T N = 2.6 K 6 9 12Temperature (K)04 Sp ec i f i c h e a t:( J m o l K ) C p --
20 1 3 4 T (K ) C T p / ( J m o l K ) -- FIG. 10. (Color online) Zero-field specific heat ( C p ) ofNa . VOPO F . with a transition anomaly at T N = 2 . C p /T against T . Solid line indicates the idealized T behavior. γ - and γ (cid:48) -polymorphs having different phononspectra.The specific heat maximum around 4.0 K suggestsweaker exchange couplings in Na . VOPO F . com-pared to Pb VO(PO ) and PbZnVO(PO ) that reveal¯ J = 9 −
10 K and the maxima of C mag at a highertemperature of 4 . − . By contrast, Li VOSiO with a similar J = 6 . C mag at 3.6 K. Since the phonon contribution at 4 K issmall, the measured value of C p is a reasonable esti-mate for the maximum of C mag . This way, we find C maxmag /R (cid:39) .
43 that compares well to 0.44, as expectedfor the unfrustrated or moderately frustrated squarelattice, and experimentally observed in a number of FSLcompounds. The low-temperature evolution of C p resembles clas-sical antiferromagnets. However, the data do not fol-low a simple T behavior (see the inset of Fig. 10).The origin of this deviation is presently unclear. Notethat other FSL-type compounds also show complexlow-temperature features in the specific heat. Thedata for Pb VO(PO ) and Li VOGeO can be de-scribed by a combination of cubic and linear terms. InNa . VOPO F . , the linear term is vanishingly small,but the cubic term itself poorly fits the data below 2 K.Heat capacity measurements in magnetic field es-tablish the temperature-vs-field phase diagram ofNa . VOPO F . (Fig. 11). With increasing magneticfield, the specific heat maximum is suppressed, whereasthe transition anomaly is growing and T N increases upto 2.8 K at 5 T. Above 5 T, both the anomaly and T N are gradually suppressed. The overall shape of thephase boundary is typical for 2D antiferromagnets. T paramagnetAFM order H s T = 1.4 K0.01.07Temperature (K) Temperature (K) Magnetization ( /f.u.) m B F i e l d ( T )
04 12840166 Sp ec i f i c h e a t:( J m o l K ) C p -- FIG. 11. (Color online) Left: specific heat of Na . VOPO F . measured in different applied fields. Right: the field-vs-temperature phase diagram, based on the heat capacity and NMR measurements, as well as the magnetization isotherm at1.4 K (Ref. 19). The initial increase in T N is due to the field-inducedanisotropy that weakens quantum fluctuations. Uponfurther increase in the field, the tendency towards thefully polarized state competes with AFM correlations andimpedes the AFM long-range ordering. C. Electron spin resonance
Typical ESR spectra of Na . VOPO F . are shownin Fig. 12. The compound reveals a single exchange-narrowed resonance line which is perfectly described bya Lorentzian curve in the whole temperature range underinvestigation. The temperature dependence of the corre-sponding fit parameters, i.e., the resonance field H res ,intensity I ESR , and half-width at half-maximum ∆ H ,is illustrated in Fig. 13. At room temperature, the g -factor, derived from the resonance field H res = 3422 Oevia hν = gµ B H res is found to be g = 1 .
95, which is atypical value for V (3 d ) ions coordinated by oxygenatoms in a square pyramid or an octahedron. On ap-proaching T N , the g -value decreases only slightly. Theintegrated ESR intensity nicely matches the static sus-ceptibility χ obtained from SQUID measurements, andexhibits a maximum around 7 K. This indicates that allvanadium spins are observed by ESR.The linewidth is ∆ H (cid:39)
110 Oe at room temperature,and exhibits two anomalies upon cooling. One anomalyis close to T N , probably due to short-range order effects,while the other one matches the structural β ↔ γ + γ (cid:48) transition at T str (cid:39)
250 K. Similar to the structuralphase transition in the quasi one-dimensional magnetCuSb O , the change in the ESR linewidth for T < T str is well described in terms of a thermally activated behav-ior: ∆ H ( T ) = ∆ H + A · exp ( − ∆( T ) T ) (4)with the BCS-like mean-field gap ∆( T ) = 1 . · (1 − T /T str ) . . We obtain ∆ = 560(13) K, T str = 257(1) K, H (kOe) 150 K d P / d H ( a r b . u . ) = 9.36 GHz 4 KNa VPO F FIG. 12. (Color online) ESR spectra of Na . VOPO F . measured at X-band frequency for selected temperatures inthe paramagnetic regime. Solid lines indicate fits with thefield derivative of a Lorentzian. A = 13(1) Oe, and the residual linewidth ∆ H =93(1) Oe. It is interesting to consider the relation be-tween the energy gap ∆ and the transition tempera-ture T str , which yields 2∆ /T str = 4 .
36. This is enhancedcompared to the value of 3.52 predicted by mean-field0 H ( k O e ) T(K)ESR I ES R , ( e m u / m o l ) SQUID (H = 5 kOe) Na VPO F g = 1.94 H r e s ( k O e ) g = 1.95 V ( 3d ) = 9.36 GHz FIG. 13. (Color online) Temperature dependence of the ESRparameters: resonance field H res (upper frame), ESR intensitycompared to the magnetic susceptibility (middle frame), andlinewidth ∆ H (lower frame) in Na . VOPO F . . The solidred line indicates the fit using a mean-field BCS-like ansatz[Eq. (4)]. theory for a second-order phase transition. Indeed, thetransition at T str is of the first order, as shown by thetemperature evolution of unit cell parameters (Fig. 3). D. Nuclear magnetic resonance
For an I = nucleus, one expects a single spectralline that is indeed observed in the NMR spectraof Na . VOPO F . in the 1 . −
300 K temperature range(Fig. 14). Although our structural study (Sec. III) iden-tifies two inequivalent phosphorous positions at 300 K( β -phase) and at least three inequivalent phosphorouspositions in the γ -phase below 200 −
250 K, this will notnecessarily lead to separate lines in the spectra. Theline shift for an individual P site is determined by its lo-cal environment and the hyperfine couplings to electronicspins on V +4 . Since the inequivalent P sites have verysimilar coordination and similar connections to the foursurrounding vanadium atoms (Table III and Fig. 18), webelieve that the line splitting is too small to be observedin the present experiment. While this does not allow torefine our model of the low-temperature crystal structureand phase separation, the single spectral line facilitates n m H = 4.583 T I n t e n s i t y : / II m ax FIG. 14. (Color online) Fourier-transformed P NMRspectra at different temperatures T ( T > T N ) forNa . VOPO F . .
400 10 100 nm = 79 MHz( = 4.58 T) H Temperature (K)08001200 L i n e s h i ft( pp m ) K
500 0.8 1.6 K ( pp m ) c (10 emu/mol) - FIG. 15. (Color online) Temperature dependence of the PNMR line shift K measured at 79 MHz. The inset shows K plotted against the magnetic susceptibility χ with tempera-ture as an implicit parameter. The solid line represents thelinear fit with Eq. (5). the study of the magnetic properties by NMR.Since our measurements are done on randomly-oriented polycrystalline samples, the asymmetric shapeof the spectra corresponds to a powder pattern dueto an asymmetric hyperfine coupling constant and ananisotropic susceptibility. The line position was foundto shift with temperature, similar to the magnetic sus-ceptibility (Fig. 15). To establish the relation between K and χ , we use the expression: K ( T ) = K + A hf N A χ spin ( T ) , (5)where K is the temperature-independent chemicalshift, A hf is the hyperfine coupling constant, and χ spin /N A is the intrinsic (spin) magnetic susceptibility ofNa . VOPO F . in units of µ B /Oe per electronic spin(i.e., per formula unit). In order to calculate A hf , weplotted K vs. χ with T as an implicit parameter (the1 m H
79 MHz ( = 4.583 T) m H Temperature (K) / ( s ) T - FIG. 16. (Color online) Temperature dependence of the spin-lattice relaxation rate 1 /T measured at two different frequen-cies of 79 and 8.62 MHz. n = 8.62 MHz I n t e n s i t y : / II m ax FIG. 17. (Color online) Temperature-dependent P NMRspectra measured at 8.62 MHz. inset of Fig. 15). The linear fit with Eq. (5) over thewhole temperature range yields A hf = (212 ±
20) Oe/ µ B that is comparable to hyperfine couplings found in otherFSL-type vanadium phosphates. For an I = nucleus, the recovery of the longitudinalmagnetization is expected to follow a single-exponentialbehavior. In Na . VOPO F . , the recovery of the nu-clear magnetization after a saturation pulse was indeedfitted well by the exponential function1 − M ( t ) M = Ae − t/T , (6)where M ( t ) is the nuclear magnetization at a time t afterthe saturation pulse, and M is the equilibrium magneti-zation. The temperature dependence of 1 /T is depictedin Fig. 16. Above 10 K, 1 /T is essentially temperature-independent, which is typical for paramagnetic momentsfluctuating fast and at random. Below 10 K, 1 /T de-creases slowly, shows a peak around 2.8 K, and furtherdecreases at low temperatures. A similar behavior hasbeen observed in Pb VO(PO ) from P NMR (Ref. 23), SrZnVO(PO ) from P NMR (Ref. 20), VOMoO from Mo NMR (Ref. 10), and Cu(HCO ) · O from HNMR (Ref. 51). The decrease in 1 /T can be explainedby the cancellation of antiferromagnetic spin fluctuationsat the probed nuclei. The sharp peak in 1 /T identifies the onset of the long-range magnetic ordering. The peak position depends onthe NMR frequency and reveals the field dependence of T N . In Fig. 11, we plot the two transition temperaturesderived from our NMR experiments, and find perfectagreement with the phase boundary obtained from theheat capacity measurements.Below T N , the P line measured at 79 MHz broad-ens abruptly. In order to check whether any extra fea-tures could be resolved, we remeasured the spectra at alower frequency of 8.62 MHz (Fig. 17). Two symmetricalshoulder-like features develop on both sides of the cen-tral line and move systematically away from each otherwith decrease in temperature. The symmetric positionand the systematic evolution of the shoulders in the low-frequency low-temperature spectra is an indication of acommensurate magnetic ordering. Our low-temperaturespectral shape can be compared to the Li-NMR spectraof Li VOSiO reported by Melzi et al. In the columnarAFM phase, they observed a central line and two sym-metrical satellites from the single-crystal measurement.In our case, such satellites appear as broad shoulders dueto the random distribution of the internal field for themeasurements on polycrystalline samples. Nevertheless,the overall spectral shape is consistent with Li-NMR re-sults on Li VOSiO and hence points to the columnarAFM ordering in Na . VOPO F . . V. MICROSCOPIC MAGNETIC MODEL
Despite the tetragonal crystallographic symme-try, the complex low-temperature superstructure of γ -Na . VOPO F . gives rise to a number of inequivalentsuperexchange pathways and the overall distortion of theFSL. Using the atomic positions for γ -Na . VOPO F . (Table V), we arrive at six different couplings in the ab plane: J , J (cid:48) , and J (cid:48)(cid:48) between nearest neighbors as wellas J , J (cid:48) , and J (cid:48)(cid:48) between next-nearest neighbors, seeFig. 18. The spin lattice entails regular J − J (cid:48)(cid:48) pla-quettes alternated with less regular plaquette-like unitshaving one type of the NNN couplings (either J or J (cid:48) )yet different NN couplings.While it is hardly possible to evaluate all the six afore-mentioned parameters experimentally, DFT calculationsare capable of providing microscopic insight into indi-vidual superexchange pathways and establishing realisticspin models for complex crystal structures. In thefollowing, we present DFT results for the ordered struc-ture of γ -Na . VOPO F . . The second low-temperaturephase, the γ (cid:48) -polymorph with the short-range order ofNa atoms (Sec. III), is more difficult to model due tothe unresolved short-range order, although we may ex-2 J J J ’ J ’ J ’’ J ’’ O1 O2 F ab FIG. 18. (Color online) Magnetic layers in the crystal struc-ture of γ -Na . VOPO F . (left) and the respective spin lat-tice with six inequivalent couplings (right). The J − J (cid:48)(cid:48) pla-quette units are shaded. pect a similar scenario with FM couplings between near-est neighbors and AFM couplings between next-nearestneighbors.The LDA density of states (DOS) for γ -Na . VOPO F . identifies oxygen and fluorine2 p valence bands below − d bandsat the Fermi level (Fig. 19). The vanadium statesshow a crystal-field splitting into the t g (below 0.7 eV)and e g (above 1.3 eV) sublevels, as expected for theoctahedral coordination of a transition metal. Furtheron, the short bond to the axial oxygen atom (O1) splitsthe t g bands into the half-filled d xy states lying atthe Fermi level and unoccupied d yz , d xz states lyingbetween 0.2 eV and 0.7 eV (here, z aligns with thecrystallographic c axis). The positions of the crystal-fieldlevels justify our empirical conclusions on the symmetryof the magnetic orbital that induces leading couplingsin the ab plane (Sec. IV A). The crystal-field splittingin γ -Na . VOPO F . closely resembles the positions ofthe 3 d sublevels in Pb V O (Ref. 53), Ca(VO) (PO ) (Ref. 39), and other V +4 oxides.The gapless (metallic) nature of the energy spectrumis a typical problem of LDA due to the underestimateof correlation effects in the V 3 d shell. While the corre-lation effects can be introduced in a mean-field way viathe LSDA+ U method, the huge size of the unit cell (144atoms) tempts us to avoid this procedure and apply anLDA-based model approach instead. The latter is re-markably efficient for the evaluation of weak exchangecouplings in V +4 compounds. Using Wannierfunctions with proper orbital characters, we fit 80 vana-dium 3 d bands with a tight-binding model, and map theresulting hopping parameters ( t ) onto the multi-orbitalHubbard model with the effective on-site Coulomb repul-sion U eff and on-site Hund’s coupling J eff . The notablylow t values (below 5 meV) compared to U eff (cid:39) t (cid:28) U eff so that theexpression of the Kugel-Khomskii model for the exchange - - D O S ( e V ) - FIG. 19. (Color online) LDA density of states for γ -Na . VOPO F . . The Fermi level is at zero energy. couplings can be applied: J = 4 t xy U eff − (cid:88) α t xy → α J eff ( U eff + ∆ α )( U eff + ∆ α − J eff ) . (7)Here, α denotes unoccupied d orbitals, t xy are hoppingsbetween the xy orbitals, t xy → α are hoppings from the xy (half-filled) to α (empty) orbitals, and ∆ α are the crystal-field splittings between the xy and α orbitals. The firstterm is AFM superexchange, whereas the second termis FM and arises from the Hund’s coupling in the 3 d shell. Using U eff = 4 eV and J eff = 1 eV, wearrive at individual exchange couplings listed in Table VI.The weak exchange couplings in Na . VOPO F . are asevere challenge for computational methods. In contrastto PbZnVO(PO ) and Pb V O (Refs. 25 and 53), thecalculated J ’s are exclusively AFM, despite the fits of themagnetic susceptibility (Sec. IV A) reveal the presence ofFM exchange. This problem is likely unavoidable andrestricts the DFT results to qualitative conclusions thatare nevertheless essential for understanding the system.The sizable AFM NNN couplings, compared to weakerNN couplings, support the FM ¯ J – AFM ¯ J scenario.The weakest NNN exchange J (cid:48)(cid:48) is about 70 % of J and J (cid:48) so that a spatial anisotropy of the diagonal couplingsis moderate, as in Pb VO(PO ) and PbZnVO(PO ) (Refs. 14 and 25). By contrast, the difference between J (cid:48)(cid:48) and J (cid:48) ( J ) is about 5 K, which is larger than theexperimental | ¯ J | (cid:39) . J ,¯ J ) and individual exchanges is more intricate than in AA (cid:48) VO(PO ) . Since there are two J i bonds yet sin-gle J (cid:48) i and J (cid:48)(cid:48) i bonds per lattice site, one writes ¯ J i =(2 J i + J (cid:48) i + J (cid:48)(cid:48) i ) /
4. Applying this expression to the DFTestimates in Table VI, we arrive at ¯ J = 13 . J (cid:39) . AA (cid:48) VO(PO ) phos-phates: compare ¯ J = 18 . J = 10 . ) (Ref. 25). This3 TABLE VI. FM ( J FM ), AFM ( J AFM ), and total ( J ) exchangeintegrals calculated with Eq. (7). The intralayer couplings aredepicted in Fig. 18, whereas J ⊥ couples the neighboring layers(see Fig. 1).Distance J AFM J FM J (˚A) (K) (K) (K) J − . J (cid:48) − . J (cid:48)(cid:48) J J (cid:48) J (cid:48)(cid:48) J ⊥ gives a strong support to the DFT results, and ascribestheir apparent inaccuracy to systematic errors that donot alter qualitative trends.The notable reduction in ¯ J (6.5 K in Na . VOPO F . vs. 9 −
10 K in PbZnVO(PO ) and similar phosphates)can be traced back to longer bonds comprising the su-perexchange pathways. The relevant parameters areV–O and O–O distances. The latter are edges of thePO tetrahedra and stay nearly constant, whereas theformer are flexible. In Na . VOPO F . , the V–O dis-tances exceed 2.0 ˚A (Table III) and contrast with shorter(1 . − . AA (cid:48) VO(PO ) (Table IV in Ref. 14). Na . VOPO F . resembles Li VOSiO in terms of the diagonal couplings( ¯ J = 6 . anion is smaller than SiO .While the nature of the diagonal couplings inNa . VOPO F . is readily comprehended, the evalua-tion of the NN couplings remains a challenge becauseof the computational inaccuracies of DFT. To makethings even more complicated, the effects of the spa-tial anisotropy cannot be resolved by the available ex-perimental data on Na . VOPO F . . The orthorhom-bic distortion of the FSL favors a certain directionfor the spin columns, stabilizes the columnar AFM stateand, therefore, increases the saturation field H s . Whenthe experimental H s exceeds the prediction of the regu-lar FSL model, their difference approximates the magni-tude of the orthorhombic distortion for NN couplings. The spin lattice of Na . VOPO F . , however, lacks anyorthorhombic distortion and rather shows a complextetragonal superstructure with three inequivalent cou-plings ( J , J (cid:48) , and J (cid:48)(cid:48) ). The couplings of each type runalong both dimensions of the square lattice (Fig. 18) anddo not stabilize the columnar AFM state, thereby no ef-fect on H s should be expected. Presently, we are un-able to give an experimental-based estimate of the spatialanisotropy in Na . VOPO F . . As long as the DFT re-sults are considered, appreciable anisotropy effects shouldbe expected (Table VI).The last important feature of Na . VOPO F . is the X M Z R A
G G - - E n er gy ( e V ) FIG. 20. (Color online) Vanadium d xy bands at the Fermilevel (zero energy): LDA band structure (thin light lines) andthe fit of the tight-binding model (thick dark lines). sizable interlayer coupling J ⊥ . It is related to dimersof octahedra sharing the fluorine atom (right panel ofFig. 1), and couples each lattice site to one of the neigh-boring layers only. Since the magnetic saturation requiresall spins to be parallel, the presence of the AFM inter-layer coupling increases the saturation field. Eq. (3) canbe rewritten as µ H s = (2 ¯ J + 4 ¯ J + J ⊥ ) k B / ( gµ B ) (8)to yield J ⊥ = 1 . J = − . J =6 . µ H s = 15 . g = 1 .
95. This experimental estimate of J ⊥ compares well to the calculated value of1.7 K (Table VI) and provides additional justification forthe reliability of our DFT results.The interlayer coupling in Na . VOPO F . amountsto 20 % of ¯ J and is larger than in any of the knownFSL compounds (see Table VII). J ⊥ arises from theshort V–V distance of 4.21 ˚A, which is in fact the short-est V–V separation in the structure. There are compa-rable V–V separations of 4 . − .
55 ˚A in Li VOSiO and Li VOGeO , but the lack of the bridging fluorineatom reduces the interlayer coupling to J ⊥ (cid:39) . The AA (cid:48) VO(PO ) phosphates show even weaker inter-layer couplings J ⊥ (cid:28) . owing to the V–V separa-tions of 8 . − . . VOPO F . is furtherdiscussed in Sec. VI B. VI. DISCUSSIONA. Crystal structure
The crystal structure of Na . VOPO F . , as reportedin Refs. 26 and 27, is deceptive in its simplicity. It dif-fers from our findings for Na . VOPO F . prepared byhigh-temperature solid-state method. The α -type struc-ture, previously found at RT, is observed above 500 K4 F filled Na positions partially filled Na positions vacancies a -Na VOPO F b -Na VOPO F g -Na VOPO F ab FIG. 21. (Color online) The arrangement of Na atoms in different polymorphs of Na . VOPO F . . Dark spheres denote filledNa positions, light spheres are partially filled Na positions, and open circles represent vacancies. only. The actual RT structure is marginally different,owing to the partial ordering of the Na atoms in the β -polymorph. Low-temperature studies reveal the forma-tion of the complex superstructure in the γ -polymorphand the β ↔ γ + γ (cid:48) phase separation. The apparentdifferences in the RT structure are probably related to avariety of preparation procedures that could lead to slightdeviations in the sample composition, especially the O/Fratio (see Sec. I). In our case, magnetization measure-ments combined with ESR and NMR local probes con-firm the oxidation state of +4 for the vanadium atomsand ensure the ordered arrangement of O and F. Sinceour samples are synthesized from stoichiometric mixturesof the reactants, the Na . VOPO F . composition issafely established.The course of the structural transformationsin Na . VOPO F . is reminiscent of isotypicNa . M PO F . fluorophosphates with M +3 = Fe,Cr, V, Al, and Ga. These compounds accommodatetrivalent M cations by substituting F for O1. Thisway, the M cation attains a nearly regular octahedralcoordination with fluorines in the axial positions.High-temperature structures are always of the α -type,but low-temperature features are variable. When M =Fe and V +3 , the RT structure is of the β -type, with a = √ a sub , P /mnm space group, and partiallyordered Na atoms. The complete ordering of Na takesplace in the orthorhombic γ -phase ( a (cid:39) b (cid:39) a sub ) belowRT. By contrast, the fluorophosphates with M = Cr, Al,and Ga drop directly to a γ -type RT polymorph having a = 2 a sub , P /mbc space group, and still disorderedNa atoms. Below RT, these compounds presumablyundergo another phase transition, but its nature hasnot been established. Na . VOPO F . complementsthis welter of polymorphs and transitions by the orderedtetragonal γ -phase and the β ↔ γ + γ (cid:48) phase separationbelow RT.Although the origin of the structural transforma-tions in Na . M PO F . is far from being clear, elec-tronic effects should likely be ruled out. Indeed, mag- netic (Cr) and non-magnetic (Al, Ga) trivalent cationsgive rise to the same sequence of the transitions. InNa . VOPO F . , the effect is even more tangible. Weakexchange couplings manifest themselves below 10 −
20 Kwhich is far below the first-order structural transitionand phase separation around 250 K. In Fig. 21, we plotthe arrangement of Na atoms in different polymorphs ofNa . VOPO F . . The partial filling of the two multiple-site positions in the α -polymorph is an indication ofNa mobility. β -Na . VOPO F . reveals clusters of or-dered (Na1) and partially disordered (Na2) atoms. In γ -Na . VOPO F . , we find a peculiar ordering patternreminiscent of a -depleted square lattice. A similarpattern is observed in γ -Na . FePO F . , the only ex-ample of the complete cation ordering in this structuretype. The structural transformations/phase separationare possibly caused by electrostatic interactions betweenNa and F atoms together with an apparent difficulty offinding a symmetric configuration for the -filled “squarelattice” of Na.A notable feature of the phase separation inNa . VOPO F . is the lack of any visible compositionalchanges. When electronic degrees of freedom are inac-tive, phase separation is typically driven by a size mis-match and results in different compositions of the twophases. In Na . VOPO F . , the fully ordered struc-ture of the γ -phase implies the ideal Na . VOPO F . composition and imposes the same composition for the α -phase, as further confirmed by the unique oxidationstate of +4 for vanadium. Neither ESR nor NMR re-veal any differences between the two phases coexisting atlow temperatures. Another surprising observation is themacroscopic scale of the separated phases: both γ - and γ (cid:48) -polymorphs show narrow XRD reflections correspond-ing to well-defined crystallites rather than nanoscale do-mains. These unusual features suggest that a furtherstudy of the phase separation in Na . VOPO F . couldbe insightful.5 B. Magnetic properties
Our detailed study of the magnetic propertiesand electronic structure confirms the assignment ofNa . VOPO F . to the quasi-2D FSL-type spin model.The ultimate goal of finding an ideal FSL with FM J is, however, defeated by the complexity of structuraltransformations and low-temperature phase separation.The structural changes are not caused by the magneticfrustration (Sec. VI A), while the magnetism, in turn, ismarginally influenced by the spin lattice distortion. Sim-ilar to AA (cid:48) VO(PO ) , thermodynamic properties con-form to the regular FSL model. A closer and micro-scopic look, based on DFT, detects sizable deviationsfrom the regular model, although experimental data givelittle information on such effects. In contrast to our ear-lier conjecture in Ref. 19, the underestimate of the satu-ration field should be ascribed to the interlayer coupling(Sec. V).Despite seeming to be overcomplicated at first glance,the spin lattice comprising six inequivalent intralayercouplings (right panel of Fig. 18) expands the family ofFSL-type models realized in inorganic compounds. The AA (cid:48) VO(PO ) phosphates reveal a somewhat more sim-ple, orthorhombic distortion of the FSL. By contrast,the tetragonal superstructure in γ -Na . VOPO F . splits the square lattice into plaquettes (Fig. 18) andtherefore relates to a different branch of theoretical re-sults. The models of tetramerized square lattices, alsoknown as square lattices with plaquette structure, ex-hibit a quantum phase transition between the long-range-ordered and gapped ground states but so far lack any pro-totype materials. A cation substitution/intercalation inNa . VOPO F . could be a way to obtain further com-pounds of this type, although the possible problem of thephase separation should not be overlooked.Apart from the spatial anisotropy, Na . VOPO F . features an unusually strong interlayer coupling. In Ta-ble VII, we list some basic parameters of various FSLcompounds with columnar AFM ground state: experi-mental values of ¯ J and ¯ J , thermodynamic energy scale J c = (cid:112) ¯ J + ¯ J , DFT estimates of J ⊥ , and experimen-tal temperatures of the magnetic ordering ( T N ). To ac-count for the single interlayer coupling per lattice site inNa . VOPO F . , we take one half of the experimental J ⊥ . The comparison identifies the effects of the frustra-tion and interlayer couplings on T N . Based on T N /J c , thecompounds can be divided into three groups: stronglyfrustrated ( α (cid:39) − T N /J c (cid:39) . α (cid:39) − T N /J c (cid:39) . α > T N /J c = 0 . − .
50) systems. An order of mag-nitude difference in J ⊥ /J c between Na . VOPO F . andPbZnVO(PO ) has no appreciable effect on T N .Despite the weak influence on T N , J ⊥ likely affectsother features related to the magnetic ordering. InFig. 22, we compare specific-heat transition anomalies forLi VOSiO , Na . VOPO F . , and Pb VO(PO ) . Thearea under the C p /T curve is a measure of the transi-
22 4
Li VOSiO
Na VOP F
O 4.0 0.90.61.2 C T p / ( J m o l K ) -- FIG. 22. (Color online) Zero-field specific heat at the mag-netic ordering transition in Na . VOPO F . (powder sam-ple), Li VOSiO (powder sample), and Pb VO(PO ) (singlecrystal). The data for the last two compounds are fromRef. 15. tion entropy S . Pb VO(PO ) shows the smallest S ,below 0.01 J mol − K − . A similar frustration ratio of α (cid:39) − S (cid:39) .
065 J mol − K − in Na . VOPO F . , while Li VOSiO reveals an evenbroader anomaly with S (cid:39) .
11 J mol − K − . Althoughnumerical estimates of S are tentative, the qualitativetrend is robust and identifies the strong dependence of S on J ⊥ . The reduction in J ⊥ shifts the entropy to thebroad maximum above T N and reduces the transitionentropy, which determines the magnitude of the tran-sition anomaly. At weak J ⊥ , the anomaly ultimatelytransforms into a kink, especially for powder samples,see the data in Refs. 15, 24, and 25. A similar trend forthe transition entropy depending on J ⊥ has been pro-posed in a theoretical study of a simplified spin modelwith non-frustrated square lattices coupled by J ⊥ . Wemention that the 3D derivative of the FSL model hasbeen addressed in recent theoretical studies focused onground-state properties. Our results call for a furtherinvestigation of finite-temperature behavior: in particu-lar, the evaluation of N´eel temperature and temperaturedependence of the specific heat.Finally, Na . VOPO F . can be a starting point to ex-plore FSL-type systems with different values of the spin.Experimental work has been mostly restricted to the caseof spin- , although theoretical results propose pecu-liar features of systems with larger spin. The family ofNa . M PO F . fluorophosphates should retain the ba-sic FSL-type geometry with FM couplings between near-est neighbors and AFM couplings between next-nearestneighbors in the ab plane. Trivalent M cations representspin-1 (V +3 ), spin- (Cr +3 ), and even spin- (Fe +3 ) sothat the effect of the spin value on the FSL model couldbe explored experimentally.In summary, we have presented a detailed study ofthe crystal structure, magnetic behavior, and electronicstructure of Na . VOPO F . , a spin- FSL-type com-pound. We observed essential structural changes, includ-6
TABLE VII. Comparison of FSL compounds with the columnar AFM ground state: exchangecouplings ¯ J and ¯ J (in K); thermodynamic energy scale J c = (cid:112) ¯ J + ¯ J (in K); the frustrationratio α = ¯ J / ¯ J ; the interlayer coupling J ⊥ ; and the magnetic ordering temperature T N .Compound ¯ J ¯ J α J c J ⊥ /J c T N /J c Ref.BaCdVO(PO ) − . − . < .
01 0.21 24SrZnVO(PO ) − . − . < .
01 0.22 15BaZnVO(PO ) − . − . < .
01 0.36 15Pb VO(PO ) − . − . < .
01 0.33 11PbZnVO(PO ) − . − . < .
01 0.35 25Na . VOPO F . − . − . VOSiO VOGeO ing the low-temperature phase separation, that preventNa . VOPO F . from being a simple model system. Acomplete experimental evaluation of the spin model forthis compound is a formidable challenge. Despite thisfact, Na . VOPO F . is the only experimental exampleof the tetramerized (plaquette-structure) square latticeand bears a relation to a group of spin models which so farlacked any prototype materials. Our experimental dataconform to the regular FSL model supplied with a siz-able interlayer coupling J ⊥ of about 10 % of the effectiveintralayer exchange. This interlayer coupling marginallyaffects the magnetic ordering temperature, yet the transi-tion anomaly in the specific heat is notably increased. Ingeneral, Na . VOPO F . and isostructural compoundsestablish several promising connections to FSL-type spinmodels. This should outweigh the apparent structuralcomplexity of these materials and stimulate further ex- perimental investigation. ACKNOWLEDGMENTS
We are grateful to ESRF for providing the beam timeat ID31 and specifically acknowledge Andrew Fitch forhis efficient support during the data collection. Wewould also like to thank Yurii Prots, Horst Borrmann,and Roman Shpanchenko for laboratory XRD measure-ments and Stefan Hoffmann for the DSC study. A.T.was supported by Alexander von Humboldt Founda-tion. This work was partially supported by the DeutscheForschungsgemeinschaft (DFG) within the TransregionalCollaborative Research Center TRR 80 (Augsburg, Mu-nich). Work at the Ames Laboratory was supported bythe Department of Energy-Basic Energy Sciences undercontract No. DE-AC02-07CH11358. ∗ [email protected] † [email protected] G. Misguich and C. Lhuillier, in
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Structure refinement of α -Na . VOPO F . at 560 K: experimental (symbols),calculated (green line), and difference (black line) patterns. Ticks show reflection positions. q I n t e n s i t y : / II I = 30000 T = 298 K FIG. S2.
Structure refinement of β -Na . VOPO F . at RT: experimental (symbols), calcu-lated (green line), and difference (black line) patterns. Ticks show reflection positions.0 I n t e n s i t y : / II q I = 81000 T = 150 K FIG. S3.
Structure refinement at 150 K: experimental (symbols), calculated (green line), anddifference (black line) patterns. Upper and lower sets of ticks denote reflections of the γ (cid:48) -and γ -polymorphs, respectively.
100 50 100 150 2000 Temperature (K)0200 Sp ec i f i c h e a t ( J m o l K ) - - FIG. S4.
Specific heat of Na . VOPO F .5