Magnetic properties of BaCdVO(PO_4)_2: a strongly frustrated spin-1/2 square lattice close to the quantum critical regime
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Magnetic properties of BaCdVO(PO ) : a strongly frustratedspin-1/2 square lattice close to the quantum critical regime R. Nath, ∗ A. A. Tsirlin,
1, 2, † H. Rosner, and C. Geibel Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str. 40, 01187 Dresden, Germany Department of Chemistry, Moscow State University, 119992 Moscow, Russia
We report magnetization and specific heat measurements on polycrystalline samples ofBaCdVO(PO ) and show that this compound is a S = 1 / J ) and antiferromagnetic next-nearest-neighbor ( J ) interactions. Thecoupling constants J ≃ − . J ≃ . µ H s = 4 . ) undergoes magnetic ordering at about 1 K, likely to-wards a columnar antiferromagnetic state. We find that BaCdVO(PO ) with the frustration ratio α = J /J ≃ − . PACS numbers: 75.50.-y, 75.40.Cx, 75.30.Et, 75.10.Jm
I. INTRODUCTION
Low-dimensional spin systems are one of the activelystudied subjects in solid state physics due to the pos-sibility to observe numerous quantum phenomena andto interpret these phenomena within relatively simplemodels (e.g., Ising or Heisenberg models for differentlattice types). An interesting phenomenon in the spinphysics is the formation of a spin liquid – a stronglycorrelated ground state lacking long-range magnetic or-der. This ground state is usually related to the electronicmechanism of superconductivity suggested for high- T c cuprates. Spin liquids originate from quantum fluctua-tions that are particularly strong in systems with reduceddimensionality and low spin value. The fluctuations canbe further enhanced by introducing magnetic frustrationwhich impedes long-range ordering of the system.The spin-1/2 frustrated square lattice (FSL) is one ofthe simplest models giving rise to a spin liquid groundstate. In this model (also known as the J − J model),magnetic moments on a square-lattice are subjected tonearest-neighbor interaction J along the side of thesquare and next-nearest-neighbor interaction J alongthe diagonal of the square. The FSL model is de-scribed by the frustration ratio α = J /J or, alterna-tively, by the frustration angle ϕ =tan − ( J /J ). Defin-ing the thermodynamic energy scale of exchange cou-plings as J c = p J + J , one obtains J = J c cos ϕ and J = J c sin ϕ (see Fig. 1 and Ref. 3).Extensive theoretical research on the FSL model hasbeen done in the past. Initially, the studies were fo-cused on the AFM region ( J , J > while thegeneral case (arbitrary signs for J and J ) was onlyconsidered quite recently. The phase diagramof the model (Fig. 1) includes three regions with dif-ferent ordered phases: ferromagnet [FM, wave vector Q = (0 , Q = ( π, π )], and columnar antiferromagnet [CAF, Q = ( π,
0) or(0 , π )]. Classically, first-order phase transitions shouldoccur at the NAF–CAF ( α = 0 .
5) and the CAF–FM( α = − .
5) boundaries. However, quantum fluctua-tions destroy long-range ordering, leading to the forma-tion of critical regions with disordered ground states. Ingeneral, the ground state in the critical regions is re-ferred to a quantum spin liquid (QSL) regime, but theparticular nature of the spin liquid phases is still un-der discussion. A gapless nematic state is suggestedfor α ∼ − . while different dimer phases (includingresonating-valence-bond-type ones) are claimed to exist J J Li VOSiO
Li VOGeO
PbVO VOMoO ~ 0.4~ 0.4 - ~ 0.7 - J J / = - ~ 0.7 Pb VO(PO )
BaZnVO(PO )
SrZnVO(PO )
BaCdVO(PO ) j NAFCAFFM
FIG. 1: (Color online) Phase diagram of the FSL model. Solid filling indicates the regions of long-range magnetic or-dering, while hatched filling denotes the critical regions. Po-sitions of BaCdVO(PO ) and some of the previously inves-tigated compounds are shown (see text for references). for α close to 0.5. Note also that the boundaries of thecritical regions are not known exactly: for example, an α range from 0.34 to 0.60 is reported in Ref. 5 for thespin-liquid regime, while a wider region (0 . − .
83) issuggested in Ref. 6.Despite numerous theoretical investigations, experi-mental realizations of the J − J model are scarce.Layered vanadium oxides Li VOXO (X = Si, Ge) andVOMoO were the first examples of the FSL systems.Initially, these materials were ascribed to α ≃ However, later stud-ies revealed the lack of the structural distortions and es-tablished a different scenario for all three systems. Thus,Li VOXO compounds fall into the CAF region with α ≫ while VOMoO lies in the NAF regionwith α ≃ . Complex vanadium phosphates AA’VO(PO ) presentanother realization of the FSL model with ferromagnetic J and antiferromagnetic J as will be discussed in detailbelow. We should also mention two very recent proposi-tions for the FSL systems. (CuX)LaNb O (X = Cl, Br)perovskite-type compounds were claimed to realize the J − J model with ferromagnetic J and antiferromag-netic J . However, the estimates of J ’s from differentexperimental techniques are contradictory, and the valid-ity of the FSL model for these systems is still unclear. The layered perovskite PbVO also reveals an interestingsquare lattice system with α ∼ .
3, i.e., quite close to theantiferromagnetic (AFM) critical region.
However,the complicated preparation procedure strongly hampersdetailed investigation of this compound. Thus, little ex-perimental information about the critical regions of theFSL model is available, and the search for new FSL sys-tems is still challenging.Complex vanadium phosphates AA’VO(PO ) havelayered crystal structures (Fig. 2) with square lattice-likearrangement of V +4 ( S = 1 /
2) cations. [VOPO ] lay-ers are formed by VO square pyramids linked via PO tetrahedra. The tetrahedra allow for superexchange in-teractions both along the side and along the diagonal ofthe square, hence the FSL-like spin system is formed.Metal cations and additional, isolated PO tetrahedraare located between the layers. The layers are flexi-ble towards buckling, therefore metal-oxygen distancescan be tuned, and different metal cations can be accom-modated within the structure. Magnetic properties ofthe compounds with AA’ = Pb , SrZn, and BaZn havebeen recently investigated by means of thermodynamicmeasurements and neutron scattering. We foundthat all the compounds fall to the CAF region of the FSLphase diagram. They reveal ferromagnetic J and anti-ferromagnetic J with α varying from − . ,BaZn) to − . ñb aa J J FIG. 2: (Color online) Crystal structure of BaCdVO(PO ) :single [VOPO ] layer (upper panel) and stacking of the layers(bottom panel). Arrows indicate superexchange interactions J (along the side of the square) and J (along the diagonalof the square). Larger and smaller spheres denote Ba and Cdcations, respectively. tional information about the properties of the system. Yet there is also one complication. The crystal struc-tures of AA’VO(PO ) do not have tetragonal symme-try; therefore, vanadium atoms do not form a regularsquare lattice. At first glance, the distortion of the squarelattice is negligible. However, even a very slight alter-ation of the structure can lead to drastic changes in theexchange couplings as shown recently for Ag VOP O (Ref. 29). In case of the AA’VO(PO ) compounds, westudied this issue in detail using band structure calcula-tions and found considerable deviations from the squarelattice model for some of the systems. Nevertheless, thedeviation is really negligible for one of the compounds –BaCdVO(PO ) . Below, we present magnetic propertiesof this compound and interpret the results within theFSL model. A detailed examination of the appropriatespin models for other AA’VO(PO ) compounds will bepublished elsewhere. The crystal structure of BaCdVO(PO ) has been re-ported by Meyer et al. , but magnetic properties of thiscompound were not investigated. In our work, we use aset of different thermodynamic measurements to studymagnetic interactions in BaCdVO(PO ) . We show thatBaCdVO(PO ) is a FSL system with the frustration ra-tio α ≈ − .
9, i.e., lying very close the critical region ofthe FSL.The paper is organized as follows: Sec. II deals with theexperimental details. In Sec. III, we present our resultson BaCd(VO)(PO ) . Section IV contains a detailed dis-cussion of the results in the light of the J − J modelfollowed by our conclusions. II. EXPERIMENTAL DETAILS
Polycrystalline samples of BaCd(VO)(PO ) were pre-pared by solid state reaction technique using BaCO ,CdO, V O , V O , and (NH ) HPO as starting mate-rials (all the chemicals had at least 99.9% purity grade).The process involved two steps. First, the intermedi-ate compound BaCdP O was prepared by firing thestoichiometric mixture of BaCO and (NH ) HPO at950 ◦ C in air for 48 h with one intermediate grinding. Inthe second step, stoichiometric amounts of BaCdP O ,V O , and V O were grinded, pelletized and annealedin dynamic vacuum (10 − mbar) or evacuated and sealedquartz tube (10 − mbar) at 800 ◦ C for 30 h.The phase composition of the prepared samples waschecked by x-ray diffraction (XRD) (Huber G670f cam-era, CuK α radiation, ImagePlate detector). The sam-ples contained BaCd(VO)(PO ) and minor amount( ∼ O . One then expects anequal amount of unreacted VO to be present in thesamples. However, a minor impurity of VO can notbe resolved by XRD due to the overlap of the strongestreflection of VO with that of BaCdVO(PO ) . Instead,a minor amount of the VO impurity is evidenced by asmall kink in the magnetic susceptibility data at 340 K(see Sec. III).We tried to improve the quality of the samples by vary-ing temperature and duration of the annealing. Unfortu-nately, BaCdVO(PO ) is rather unstable, and the for-mation of an unknown impurity phase was observed af-ter long annealings at 800 ◦ C or any annealings above800 ◦ C. The annealing at 900 ◦ C resulted in melting andcomplete decomposition (as seen by powder XRD) of thecompound towards unknown phases. Regarding thesedifficulties, we did not attempt to grow single crystalsof BaCdVO(PO ) , because the data measured on poly-crystalline samples are sufficient to determine the param-eters of the FSL model and the location of the systemwithin the phase diagram.Magnetization ( M ) data were measured as a functionof temperature using a SQUID magnetometer (QuantumDesign MPMS). Specific heat C p ( T ) was measured on apressed pellet with a standard relaxation technique us-ing a Quantum Design PPMS. All the measurementswere carried out over a wide temperature range (0 . ≤ T ≤
400 K) and a field up to 7 T. The low- temperature measurements were done partly using anadditional He setup.
III. RESULTS
Figure 3 shows the magnetic susceptibility χ = M/H of BaCdVO(PO ) as a function of temperature. Athigh temperature (above 20 K), the data behave ina Curie-Weiss (CW) manner. At lower temperatureand low fields, the curve exhibits a broad maximumat T χ max ≃ . ∼ χ value at the maximum ( χ max ), and an increaseof χ at the lowest investigated temperatures. This be-havior is quite similar to that observed in the other FSLsystems and rather typical for low-dimensional and/orfrustrated antiferromagnetic systems.Careful examination reveals a small kink in 1 /χ vs. T curve at 340 K. This kink is indicative of the VO im-purity, since VO undergoes metal-insulator transitionat 340 K. We emphasize that this impurity does notaffect any of the results reported below, as the paramag-netic contribution of VO below 300 K is temperature-independent, while the energy scale of the exchange in-teractions in BaCdVO(PO ) is lower by two orders ofmagnitude. However, the impurity prevents us from us-ing the data above 300 K in further analysis.High-temperature (20 −
300 K) susceptibility data canbe fitted with a CW law corrected for the temperature-independent contribution χ that accounts for diamag-netism of core shells and Van Vleck paramagnetism: χ ( T ) = χ + CT + θ CW (1)The fitting resulted in χ = − . × − emu/mol, C = 0 . θ CW = 0 . C value corresponds to an effective magnetic moment µ eff =1 . µ B in perfect agreement with the expected spin-only value for V +4 . Note that θ CW is slightly dependenton the lower limit of the data used due to the curvaturein 1 /χ related to the maximum in χ ( T ). However, valuesof θ CW below 1 K are obtained for any fit with a lowerlimit above 20 K.Thus, we find a very low θ CW value, despite the sus-ceptibility maximum at T χ max ≃ . FIG. 3: (Color online) Temperature dependence of the sus-ceptibility χ ( T ) of BaCd(VO)(PO ) measured at differentapplied fields. Insets show inverse susceptibility (open cir-cles) along with the HTSE [Eq. (2), upper inset] and theCurie-Weiss [Eq. (1), lower inset] fits. In the upper inset,dashed and solid lines denote solutions (a) and (b), respec-tively. In the lower inset, solid line is the Curie-Weiss fit,while the arrow indicates the kink at 340 K due to the VO impurity. expansion (HTSE) for the FSL model: χ ( T ) = χ + N A g µ B k B T X n (cid:18) J k B T (cid:19) n X m c m,n (cid:18) J J (cid:19) m , (2)where χ is temperature-independent contribution, g isLande g -factor, and c m,n are the coefficients listed in Ta-ble I of Ref. 17. We find χ = − . × − emu/mol, g = 1 . J = − . J = 3 . solu-tion (a)], or, alternatively, χ = − . × − emu/mol, g = 1 . J = 2 . J = − . solu-tion (b)]. The fit with the solution (b) looks somewhatbetter and extends to lower temperatures as comparedto the fit with the solution (a) (see the upper inset ofFig. 3). Therefore, one may speculate that the solution(b) is the correct one. However, the difference betweenthe two fits is caused by the lower exchange couplingsin the solution (b), since the HTSE is valid at T ≥ J i .In the region used for the fitting (above 5 K), both thesolutions produce the fits of similar quality, hence theyare indistinguishable. The solution (a) locates the sys-tem in the CAF region close to the FM critical region(i.e., the region on the CAF–FM boundary), while thesolution (b) corresponds to a position deep into the sta-ble NAF region. The presence of two solutions in fitting χ ( T ) with the HTSE for the FSL is a well-known prob-lem. The ambiguity has to be resolved by the use of otherexperimental data.To discriminate the valid set of J ’s, we turn to mag-netization data and analyze the value of the saturation FIG. 4: Magnetization ( M ) as a function of the applied field( H ) measured at 0 . T N . The arrow marksthe saturation field µ H s . field H s . According to theoretical results by Schmidt etal. , the saturation field can be calculated as µ H s = J c k B zSgµ B (cid:2)(cid:0) − (cos Q x + cos Q y ) (cid:1) cos ϕ +(1 − cos Q x cos Q y ) sin ϕ ] , (3)where z = 4 (magnetic coordination number), S = 1 / ϕ and J c are defined in Section I, and( Q x , Q y ) is the wave vector of the ordered state. Usingthe appropriate wave vectors for the CAF and NAF re-gions, one finds µ H s = 2( J + 2 J ) k B / ( gµ B ) for CAFand 4 J k B / ( gµ B ) for NAF phases. The first set of J ’s[solution (a)] results in µ H s = 4 . µ H s = 6 .
55 T.Figure 4 presents the magnetization curve measuredat 0.5 K, i.e., well below the ordering temperature T N .At low fields, the M ( H ) dependence is linear, while apositive curvature is observed above 2 T leading to amarked kink at the saturation field µ H s ≃ . Thesaturation magnetization M s ≃ . µ B /V +4 is slightlyless than the expected value of 1 µ B . The reduction of M s is likely caused by the weight error due to the pres-ence of non-magnetic impurity (BaCdP O ) in the sam-ples under investigation (see Sec. II). The curvatureabove 2 T is also somewhat puzzling. At first glance, onemay interpret it as a spin-flop transition. However, themagnetic anisotropy of V +4 is weak. Thus, a spin-floptransition in Pb VO(PO ) is observed at 1 T, anda strong increase of the anisotropy in the isostructuralBaCdVO(PO ) compound is unlikely. In the next sec-tion, we will suggest an alternative and plausible expla-nation for the curvature of the M ( H ) dependence above2 T.The experimentally determined saturation fieldmatches exactly the value calculated for the solution FIG. 5: (Color online) Temperature dependence of the specificheat C p ( T ) of BaCd(VO)(PO ) measured at zero appliedfield. Open circles are the raw data, the solid line showsthe phonon contribution [according to the fit with Eq. (4)],and the dashed line indicates the magnetic contribution C mag .The anomaly at T N ≃ S mag ) is plotted as a function oftemperature, dashed lines show the entropy of 0 . R ln 2 andthe respective temperature. (a) of the susceptibility fit but is far below the valueexpected for the solution (b). This clearly demonstratesthat the solution (a), with a ferromagnetic J andan antiferromagnetic J is the correct one. Thus,BaCdVO(PO ) belongs to the frustrated ferromagneticsquare lattice system as all the other AA ′ VO(PO ) structural homologs.Specific heat ( C p ) measurement at zero field is shownin Fig. 5. At high temperatures, C p is completely dom-inated by phonon excitations. Below 5 K, a decreaseof temperature is accompanied by an increase of C p indi-cating that the magnetic contribution to the specific heatbecomes prominent. At low temperatures, C p ( T ) showsa broad maximum at T C max ≃ . T N ≃ T N , C p ( T ) drops rapidly.To extract the magnetic contribution to the specificheat C mag , we subtract an estimated phonon contributionfrom the total measured specific heat C p . For this pur-pose, the experimental data at high temperatures (15 K ≤ T ≤
200 K) were fitted with C p ( T ) = AT + 9 R n =4 X n =1 c n Tθ ( n ) D ! θ ( n ) D /T Z x e x ( e x − dx, (4)where A/T accounts for the magnetic contribution athigh temperatures, R = 8 .
314 J/mol K is the gas con- stant, and the sum of Debye functions accounts for thephonon contribution. The use of several Debye functionswith distinct characteristic temperatures θ ( n ) D is necessarydue to the large difference in atomic masses of the ele-ments forming the BaCdVO(PO ) compound. In gen-eral, the procedure is similar to that reported in Refs. 26,37, and 38 for the related vanadium phosphates. The re-sulting C mag ( T ) curve is shown in Fig. 5. In the inset,we plot the temperature dependence of the magnetic en-tropy S mag ( T ) as obtained by integrating C mag ( T ) /T . Athigh temperatures, S mag ( T ) converges towards the value S ∞ ≃ . R ln 2 = 5 .
76 J/mol K taking into accountthe uncertainty in the estimated phonon contribution.The result indicates that C mag reflects the intrinsic con-tribution of BaCdVO(PO ) .At the lowest limit ( T = 15 K) of the specific heat fitto Eq. (4), the magnetic contribution A/T amounts toa small part of the total specific heat only. Therefore,the parameter A is obtained with a large uncertainty, A = 100 ±
30 J K/mol. The
A/T term corresponds tothe lowest order in the HTSE for the specific heat andmay be expressed as A = 0 . J c = 0 . R ( J + J ).We find A = 72 J K/mol and A = 29 J K/mol using the J i values of the solutions (a) and (b) of the susceptibilityfit, respectively. Both theoretical values are below theexperimental one, but the result for the solution (a) isstill within the error bar, while that for the solution (b)is far below. Thus, this analysis also favors the solution(a) and the CAF scenario for BaCdVO(PO ) .A further rough estimate of the thermodynamic en-ergy scale J c can be obtained from the T -dependenceof the magnetic entropy. In a general approximation, J c is approximately twice the temperature at which theentropy reaches half of its high-temperature limit. InBaCdVO(PO ) , S ( T ) = 0 . R ln 2 at T = 2 . J c = 5 K, in perfect agree-ment with the solution (a) ( J c = 4 . J c = 3 . . C p at T N is enhanced. The maximumenhancement of the peak value is observed at 2 T wherethe transition anomaly is most pronounced. With fur-ther increase of the field, the peak value decreases againand the anomaly broadens slightly. However, between 3 . T N increases very slightly from 0 to 2 T up to a maximum T N (2 T) = 1 . H − T phase diagram of the system (inset of Fig. 6). Thefield dependence of the specific heat is similar to thatof Pb VO(PO ) , but the smaller energy scale of theexchange couplings facilitates experimental access to thelarger part of the H vs. T phase diagram.The field dependence may be understood as follows. FIG. 6: Specific heat of BaCdVO(PO ) measured in differentmagnetic fields. The inset shows the H − T phase diagram ofthe system. The broad maximum at 1 . T N . As we havementioned in the introduction, long-range magnetic or-dering in low-dimensional and/or frustrated spin systemsis suppressed by quantum fluctuations. Magnetic fieldsuppresses these fluctuations, therefore T N is slightly en-hanced at low fields. However, above 2 T the field isstrong enough to overcome the antiferromagnetic order-ing, hence T N is reduced and finally suppressed below0.4 K at 4 T. IV. DISCUSSION
Our experimental results and our analysis demonstratethat BaCdVO(PO ) is a frustrated square lattice witha ferromagnetic exchange J along the side of the squareand an antiferromagmetic exchange J along the diago-nal. The first evidence for the presence of antiferromag-netic and ferromagnetic exchange of similar size is givenby the very small value of the Weiss constant θ CW < T χ max ≃ . χ ( T ). The latter onereflects the onset of antiferromagnetic correlations. Ina non-frustrated S = 1 / T χ max = 0 . J (Refs.39 and 40), suggesting some antiferromagnetic exchangewith J & ) . Then a much smaller θ CW , which is the sum of all the exchange interactionsin the system under investigation, implies an additionalexchange of the opposite sign and thus ferromagnetic.A fit of the susceptibility in the T range 5 K ≤ T ≤
300 K with the HTSE for the FSL resulted intwo sets of exchange coupling parameters: solution (a)with J ≃ − . J ≃ . J ≃ . J ≃ − . µ H s = 4 . µ H s = 6 .
55 T despite both J and J are weaker than in the solution (a). Magne-tization measurements at low temperatures show a welldefined saturation at the saturation field µ H s = 4 . µ H s expected for thesolution (a) but is well below the µ H s expected for thesolution (b). Further on, an analysis of the magnetic spe-cific heat at high temperatures ( >
15 K) as well as of the T dependence of the magnetic entropy at low tempera-tures ( < J c in reasonable agreement withthe solution (a) but in clear disagreement with the solu-tion (b). Therefore, the solution (a) is the appropriateone to describe the spin system of BaCdVO(PO ) .The solution (a) with J ≃ − . J ≃ . J /J = − . ϕ/π = 0 . ) is quite close to the border of the FMcritical regime ( J /J = − . ) ( J /J = − .
1) and Pb VO(PO ) or BaZnVO(PO ) ( J /J = − . ) is given by the position of thespecific heat maximum ( T C max ) and the maximum valueof the magnetic specific heat ( C max ). The specific heatof the FSL at low temperatures ( T < J c ) is not knownprecisely, since there is presently no established way tocompute it. However, for an unfrustrated square lat-tice, the value at the maximum seems nowadays to beestablished within a few percent, C max ≃ . R = 3 . T C max /J = 0 . It is further establishedthat tuning the FSL towards a critical region leads toa broadening of this maximum, a reduction of the C mag value at the maximum, and a shift of the maximum to-wards lower temperatures. Such a trend was alreadyobserved in the previously studied FSL systems, where C max /R and T C max /J c are decreased from 0 .
46 and 0 . VOSiO (far away from any critical region) towards0 .
44 and 0 .
40 in Pb VO(PO ) and finally to 0 .
40 and0 .
29 in SrZnVO(PO ) (closer to the critical region). In BaCdVO(PO ) , C max /R reaches only 0 .
33, wellbelow the value found in the other FSL systems, while T C max /J c = 0 .
31. The very small value of C max cannot beattributed to a scaling problem due to a large amount of aforeign phase, since the magnetic entropy at high temper-atures is close to R ln 2 (the phonon contribution at T C max amounts to less than 1% of the total specific heat and cantherefore safely be neglected). Further on, a comparisonof C mag ( T ) near its maximum for different FSL systemsusing a reduced T /J c temperature scale indicates thatthe maximum in C mag ( T ) of BaCdVO(PO ) is broader,and the decrease of C mag ( T ) above T C max is much weakerthan in the other known FSL systems. Thus, this broadmaximum with a small C mag value at the maximum isalso a direct experimental evidence for the strong frus-tration.A further (and new) manifestation of the frustrationis likely observed in our magnetization curve (Fig. 4).Above 2 T, the slope of M ( H ) increases with magneticfield and steeps up just before saturation is reached. Sucha behavior has been recently predicted by theoretical cal-culations for the FSL. This steeping up just below H s is suggested to be more pronounced for systems that areclose to the critical regions. However, further magnetiza-tion measurements on other FSL systems are needed toconfirm this trend.Using exact diagonalization of finite-size clusters,Shannon et al. calculated the dependence of a numberof characteristic properties of the FSL (such as T χ max , T C max , χ max , and C max ) as a function of ϕ . Compar-ison of these predictions with the experimental obser-vation for BaCdVO(PO ) and the related compoundsindicates that while the overall ϕ dependence seems tobe reproduced, the calculated absolute values are slightlytoo large. This difference is likely a consequence of the fi-nite cluster size that leads to a gapped energy excitationspectra and thus an exponential vanishing of C mag ( T )and χ ( T ) at very low T in contrast to the real behavior.The missing entropy has then to be recovered at highertemperatures, thus enhancing C mag ( T ) at its maximumand shifting the maximum to higher T .A last comment on the antiferromagnetic order ob-served at T N ≃ . χ ( T ) and C p ( T ) are much smaller than those expected for a classicalthree-dimensional magnetic system but very similar tothose observed in the other FSL systems, e.g., Li VOSiO (Refs. 14 and 26) or Pb VO(PO ) (Refs. 25 and 26).Since in some of these systems long-range magnetic or-der was directly confirmed by NMR or neutron scatteringexperiments, the similarity in the behavior allowsus to safely claim that these small anomalies in χ ( T )and C p ( T ) correspond to the onset of (columnar) anti-ferromagnetic order in BaCdVO(PO ) too. The ratio R = T N /T χ max is smaller in BaCdVO(PO ) ( R = 0 . . < R < . ) shows thatthe magnetic properties of this compound are well un-derstood within the FSL model. Magnetic susceptibility,magnetization, and specific heat data consistently sug-gest J ≃ − . J ≃ . α ≃ − .
9, lo-cating the compound into the CAF region of the FSLphase diagram. BaCdVO(PO ) lies closer to the criti-cal region of the FSL than any of the previously reportedcompounds. This conclusion is supported by a stronglyreduced maximum of the magnetic specific heat and apositive curvature of the magnetization curve consistentwith the recent theoretical predictions. Acknowledgments
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