Glassiness and canted antiferromagnetism in three geometrically frustrated triangular quantum Heisenberg antiferromagnets with additional Dzyaloshinskii-Moriya interaction

PHYSICAL REVIEW B 1 FEBRUARY 2000-IIVOLUME 61, NUMBER 6

Glassiness and canted antiferromagnetism in three geometrically frustrated triangular quantumHeisenberg antiferromagnets with additional Dzyaloshinskii-Moriya interaction

Mihai A. Gırtu* and Charles M. Wynn†

Department of Physics, The Ohio State University, Columbus, Ohio 43210-1106

Wataru Fujita and Kunio AwagaDepartment of Basic Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan

Arthur J. EpsteinDepartment of Physics and Department of Chemistry, The Ohio State University, Columbus, Ohio 43210-1106

~Received 16 September 1999!

We report results of extensive magnetic studies of three triangular quantum Heisenberg antiferromagnets~TQHAF’s! with weak additional Dzyaloshinskii-Moriya interaction, Cu2(OH)3(CmH2m11COO), m57, 9,and 11. Fits of the dc susceptibility data to high-temperature series expansions are consistent with high-temperature TQHAF behavior. At low temperatures the deviations from the TQHAF predictions suggest acanted antiferromagnetic type of ordering, consistent with the strong peak in the second harmonic of thenonlinear ac susceptibility, which indicates the development of a spontaneous moment. The frequency depen-dence of the linear ac susceptibility and the irreversibility in the field-cooled/zero-field-cooled magnetizationreveal spin-glass-like behavior. Glassy behavior also is suggested by the specific heat data, which show only aweak broad feature at the transition. We propose that, instead of choosing between the resonant valence bondor noncollinear Ne´el ground states expected for the ideal TQHAF, these systems undergo, due to the additionalDzyaloshinskii-Moriya interaction, a finite temperature phase transition to a state with both Ising-like cantedantiferromagnetic and glassy characteristics. The interplay of Heisenberg exchange, causing frustration, andDzyaloshinskii-Moriya interaction, determining spin canting, leads to an unusual state in which order anddisorder appear to coexist.

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I. INTRODUCTION

In the past decade there has been increased interesystems that exhibit new types of magnetic ordering aconsequence of competing interactions.1 Frustration~occur-ring when no spin configuration can simultaneously mimize all the interactions! is generated by the competitiobetween interactions of different kind~e.g., competing ferro-and antiferromagnetic interactions or competing nearneighbor and next-nearest-neighbor interactions! or by thetopology of the lattice~e.g., triangular, kagome´, pyrochlorelattices with antiferromagnetic nearest-neighbor interactioetc.!.2

The recent interest in geometrically frustrated systemspurred by the new phenomena predicted or alreadyserved at low temperatures: noncollinear Nee`l long-range or-der ~LRO!, ‘‘order by disorder,’’ ‘‘partial order,’’ quantumdisorder, etc.1 Noncollinear Nee`l LRO occurs when frustra-tion can be released by spin configurations that are not aparallel, as in the traditional Nee`l state. Simple examples arthe triangular Heisenberg orXY antiferromagnets~AF! withclassical spins, in which cases 120° configurations minimthe total energy. Order by disorder is a phenomenonoccurs in systems with high degeneracy of the classground state~GS!, by which the system can select, by meaof thermal, quantum, or quenched fluctuations, the m‘‘flexible’’ subset of the GS manifold, the one for which thdensity of low-lying excited states is a maximum.3,4 The co-existence, below a finiteT, of partial order and partial disor

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der was called ‘‘order with disorder.’’5 Such coexistence inan equilibrium state of magnetic order and disorder occwhen one or more sublattices order below the critical teperatureTc , while at least one sublattice stays disorderedall temperatures.5,6 Quantum disorder can be manifest in thquantum spin-liquid GS associated initially with triangula7

and more recently with pyrochlore8 AF’s.Recently, increased numbers of experimental realizati

of such geometrically frustrated systems have beachieved.1,9–11Extensive experimental studies of such marials showed in some cases evidence of spin-glass-likehavior, which may not always be due to the presencedisorder, as in the case of spin glasses.12–16 These resultsraise questions such as whether the glassy behavior is insic to the pure frustrated system or even the smallest deof disorder~very difficult to detect! is sufficient to be responsible for the spin-glass-like characteristics, whetherglassiness found in the pure frustrated systems is identicthe one observed in the random frustrated systems~spinglasses!, and what is the universality class, if any.

Traditionally, spin-glass behavior required both randoness and frustration.17,18 One of the main conclusions of tha review on spin glasses17 was that although many of thpure frustrated models have nonperiodic GS’s, the statfinite temperature is always a simple periodic magnestructure, and no spin-glass phase could be found. Howethere can be long-lived nonperiodic metastable states sthat the dynamic properties of such systems are more sglass-like than the static properties. Moreover, if disorde

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added fully frustrated systems could be made into sglasses.

The theoretical possibility for disorder-free glassinessthe case of frustrated Heisenberg systems remains controsial even for the most highly frustrated systems.19 It wasargued that structural disorder could have either an ordeor a disordering effect depending on the type~site or bond!and degree of disorder, the properties of the degenemanifold, and the space dimension.4,19,20For some nominallydisorder-free geometrically frustrated compounds12–16 addi-tional interactions may be necessary to explain the sfreezing and glassiness observed.9,19 Single-ion anisotropy~SIA! has been the most common additional interaction pposed to be responsible for the glassy behavior,16,19while theDzyaloshinskii-Moriya~DM! interaction has not been extensively considered, though it can become crucial forS51/2systems, for which there is no SIA.

Triangular Heisenberg AF’s recently have been intensstudied both theoretically and experimentally.11 Most of theexperimental realizations have been systems with staclattices. Various noncollinear states~and in particular the120° GS! predicted theoretically were actually observehowever the ordering in these systemsthree-dimensional.11 It was determined that interlayer inteactions and SIA play an important role in the magneticdering, while site disorder did not affect significantly thordered GS in such systems.11

Regarding the triangular spin-1/2~quantum! Heisenbergantiferromagnet~TQHAF!, the suggestion that this is thsimplest system to have a resonating valence bond~RVB!ground state,7 as opposed to the noncollinear semiclassiNeel state, has resulted in much debate and controveOther RVB-type variational wave functions have beproposed,21 supporting the disordered spin liquid GS, blower energy variational states preserving some of the lorange 120° noncollinear Ne´el-type order also have beefound.22 Exact diagonalization of small clusters23 suggestedno long range order, while two more recent calculatioreached opposite conclusions regarding the existencemagnetic LRO.24,25Spin-wave theory,26 high-temperature series expansions,27 and renormalization group effective fieltheories28 support various degrees of long-range noncollinNeel ordering, however a true consensus is yet to be reac

The previously studied magnetic realization of TQHAFhas not provided conclusive results regarding the naturthe GS. The difficulties in the preparation of NaTiO2 haveimpeded extensive studies, though preliminary resultsconsistent with a disordered low-temperature phase.29 Theinterest in TQHAF’s has recently increased due to new toretical and experimental studies on nonmagneanalogues.30–34However, these reports did not settle the cotroversy concerning the GS of TQHAF’s, and new systeare needed to address it.

In this paper we present extensive magnetic studies ofrecently reported35 hybrid organic/inorganic TQHAF’s withweak additional Dzyaloshinskii-Moriya interactionCu2(OH)3(CmH2m11COO), m57, 9, and 11. These hybridnanocomposites are examples of molecule-based magn36

with a two-dimensional~2D! magnetic lattice, the most important interactions being Heisenberg and DM exchange35

Based on dc magnetization and magnetic irreversibility st

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ies, and on linear and nonlinear ac susceptibility datashowed35 that these compounds have an unusual magnbehavior, presenting, due to the additional DzyaloshinsMoriya ~DM! interaction, neither the RVB nor the noncolinear Neel GS. Instead, we proposed35 that the interplaybetween the Heisenberg antiferromagnetic exchange, caufrustration, and the DM interaction, leading to spin cantinallow these systems to evolve, into a new, unusual stateboth 2D Ising-like canted antiferromagnetic and spin-glalike characteristics. Here we expand those studies, presenextensive data for all three compounds. Based on dc matization and magnetic irreversibility studies, linear and nolinear ac susceptibility data, and specific heat data we shthat these three compounds have a similar but unusual mnetic behavior. Moreover, static scaling analyses of the mnetization and of the susceptibility as well as dynamic scing and Cole-Cole analyses strengthen our previconclusions.

The paper is organized as follows: In Sec. II we introduthe materials, discussing their structure as well as the minteractions likely to be responsible for their magnetic bhavior. We follow with a description of the experimentapparatus and the measurement techniques used. Sectipresents the experimental data together with the resultstatic and dynamic scaling analyses. In Sec. IV we discthe correlation between structure and magnetic behavior,role of additional interactions, structural disorder, and qutum fluctuations. We speculate on the possibility of partorder in these compounds. Section V is devoted to consions.

II. EXPERIMENT

A. Materials

The samples studied are compounds obtained by intelation of saturated organic chains between inorganic layercopper hydroxides.37,38 The copper hydroxy saltsCu2(OH)3CmH2m11COO, m>0, exhibit a botallackite-typestructure,39 Fig. 1~a!, in which two-crystallographically dis-tinct copper atoms lie in slightly different octahedrenvironments.37 Extended x-ray absorption fine structumeasurements showed40 that the powder samples of the intercalation compounds maintain the basic framework ofcrystalline inorganic layer of the parent compouCu2(OH)3NO3, with only minor distortions of the local environment. The x-ray powder diffraction studies revealedlayered structure with interlayer distances of 24.1, 29.4,34.4 Å for m57, 9, and 11, respectively.37 Based on thewidth of the diffraction peaks we estimate the size of tcrystallites to ;300 Å, consistent with values obtainefrom TEM studies. The TEM photographs revealed interfence patterns usually observed only in structurally ordematerials.41

The spin-carrying units areS51/2 Cu21 ions with nosingle-ion anisotropy, located on a planar lattice.37,40 Themost important interaction consistent with the structure psented in Fig. 1~a! is the isotropic Heisenberg~symmetric!exchange: HH52(2Ji j Si•Sj . The ~super-! exchangeinteraction42 is mediated by two types of bridging oxygeatoms, one from the OH groups, the other from COO grouAs there are unique Cu-O-Cu angles between each pair o

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PRB 61 4119GLASSINESS AND CANTED ANTIFERROMAGNETISM IN . . .

ions, the strength of the exchange interactions also va40

causing the magnetic lattice to consist of nonequilateralangles.

According to the Goodenough-Kanamori42 rules the ex-change coupling constant depends on the angles betwmagnetic ion–ligand–magnetic ion, angles of 180°~strongorbital overlap! and 90° ~orthogonal orbitals! leading tostrong antiferromagnetic and ferromagnetic interactions,spectively. For intermediate angles, even small deviatifrom 90° and/or the presence of side groups coupled toligands may be sufficient for the interactions to becoantiferromagnetic.43 In the Cu2(OH)3(CmH2m11COO), m57, 9, and 11 compounds, given the Cu-O-Cu anglesthe presence of the side groups bound to the oxyligands,43 the intralayer exchange interaction was found toantiferromagnetic.40,44

When the local environments of adjacent spin siare different the DM ~antisymmetric! exchange,45 HDM5(Di j •Si3Sj , caused by the spin-orbit interaction, can ato the usual Heisenberg exchange. This additional interacis always weaker than the Heisenberg exchange and faperpendicular spin alignments. The direction of the DM axvectorDi j could be determined based on the symmetrieslating adjacent spin sites. In particular, if an inversion cenis located half way between the two sitesDi j is zero, while if

FIG. 1. ~a! Proposed intralayer structure oCu2(OH)3(CmH2m11COO), m57, 9, and 11 based on the struture of the parent compound Cu2(OH)3NO3 ~Ref. 43!. Arrowsshow the directions of slight anisotropy.~b! Suggested magnetilattice of Cu2(OH)3(CmH2m11COO), m57, 9, and 11. The differ-ent symbols for the lines connecting the spin sites suggest diffestrengths of the Heisenberg exchange coupling, while the arrshow probable orientations of the various DM vectorsDi j . Theinteractions are distributed periodically throughout the lattice.

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there is a mirror plane including the two sitesDi j is perpen-dicular to that plane.46 For the compounds reported here toctahedral symmetry around the Cu ions is slightly alteby the inequivalency of the six oxygen ligands, some bepart of an OH group others of an COO group.37 This smallanisotropy varies from site to site leading to a variety of Dvectors.

Figure 1~b! shows the suggested simplified magnetic ltice of Cu2(OH)3(CmH2m11COO), m57, 9, and 11. Thedifferent symbols for the lines connecting the spin sites sgest different strengths of the Heisenberg exchange coupThe arrows show probable directions of the DM vectorsDi j~not necessarily in the plane of the lattice!, the different sym-bols suggesting different interaction strengths. The intertions are distributed periodically but not uniformly througout the lattice. Though this suggested magnetic latticeclearly oversimplified~especially in predicting the directionof the DM vectors! it represents the principal magnetic inteactions and could constitute a starting point for theoretistudies.

The interlayer interactions are expected to be very smlikely negligible. The direct exchange between unpairelectrons located in adjacent layers is very small becausthe negligible overlap of the orbitals involved, due to tlarge interlayer separation. The exchange through spin poization is also negligible as the organic chains have osaturated carbon atoms. Also, the interlayer dipole-dipoleteractions are likely negligible (;1024 K) due to the largeinterlayer distances, which are about tenfold larger thanintralayer spin-spin distances. The dipolar interaction47

probably play an important role only at low temperaturesis, therefore, very likely that these compounds are goodalizations of 2D systems.

B. Measurement techniques

The powder samples, with masses of;20 mg, weresealed at room temperature in quartz tubes with known mnetic background signal. The measurements of the lineamagnetic susceptibility and its harmonics were made witLake Shore 7225 AC Susceptometer/DC Magnetometethe temperature range 5<T<30 K, on warming. Both thein-phase (x18) and out-of-phase (x19) linear susceptibilities,x15x181 ix19 , were measured under an ac fieldHac

5h0 sin(2pft) with h051 Oe~in zero dc applied field! anda wide range of frequencies (5< f <10000 Hz). The secondand third harmonics of the magnetic susceptibilityx2 andx3were obtained by reading the 2f and 3f lock-in responses,respectively, to an ac field with frequencyf. The harmonicswere measured on warming, in zero applied magnetic fiat fixed ac field amplitude~1.3 Oe! and frequencies betwee10 and 3330 Hz. The linear ac magnetic susceptibility walso measured in various dc applied fields 0<Hdc<50 kOe and in the temperature range 5<T<40 K, atconstant field on warming.

The magnetization was measured with a Quantum DesMPMS 5 SQUID magnetometer. The temperature depdence of the static susceptibility was determined basedmagnetization data collected on cooling between 350 K5 K in a dc applied field of 5000 Oe. Hysteresis curves wobtained at 5 K for applied fields of 255000<Hdc

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<55000 Oe, after cooling with no applied field through ttransition. The remanent magnetization was measuredtaking precautions for properly zeroing the applied dc fieto within 0.05 Oe. The sequence for the measurementssisted of: cooling the system in a small dc field~5 Oe at first!from 50 K ~a temperature at least twice theTc) to 5 K, wellbelow the transition; turning the applied field to zero at 5taking data on warming in zero applied dc field;0.2 K/min; repeating the sequence for higher fields, up50 Oe.~Fields higher than 50 Oe were avoided becausethe danger of trapping flux in the superconducting magnwhich would than compromise the zeroing of the dc fieFields lower than 5 Oe lead to noisy data.! Field-cooled andzero-field-cooled magnetization data were collectedwarming in the range 5<T<30 K in various applied dcfields. The sequence for the measurements consistedcooling the system in zero applied dc field from 50 K to 5turning the applied field on at 5 K; taking zero-field-cool~ZFC! data on warming at;0.5 K/min in the applied dcfield; cooling the system in the same applied dc field fromK to 5 K; taking FC data on warming in the applied dc fielrepeating the sequence for higher fields (5<Hdc<200 Oe).

The specific heat data were collected with a QuantDesign PPMS. The samples were cut from pressed pe~masses between 7.5 and and 9.8 mg! and were mounted onthe sample holder using a small amount of Apiezongrease. To separate the contribution of the sample heapacity from the addenda, the heat capacity of the samholder with the grease was measured over the full tempture range (1.9<T<300 K) prior to mounting each sampleAll measurements were done in the absence of an appmagnetic field. Each data point corresponds to a single reation measurement, where the sample holder is heateconstant power for half the characteristic time-constant ofcalorimeter, followed by a cooling period. The heat-capacvalues were extracted from the temperature response cuby fitting to the solution of a dual time-constant thermmodel,48 thus minimizing thermal contact artifacts.

III. EXPERIMENTAL RESULTS AND ANALYSIS

A. Canted antiferromagnetic behavior

The temperature dependence of the static susceptibxdc for Cu2(OH)3(CmH2m11COO), m57, 9, and 11, waspresented elsewhere asxdcT vs T.35 It was shown thatupon decreasingT from room temperature thexdcT productfirst decreases, indicating antiferromagnetic correlationsthen, below 50 K, increases, with a peak at;20 K. ThexdcT curves form57, 9, and 11 normalized by their Curiconstants overlay each other, demonstrating independencinterlayer distances and suggesting true 2D behavior.

The fit to the high-T series expansions for TQHAF27 isconsistent with such behavior for 120<T<350 K. The val-ues of the two free parameters, the average exchangepling constant, and the Curie constant~or, equivalently, theLande g factor!, were found35 similar for the three com-pounds: 22J562,56,54 K and C50.53,0.54,0.58 emuK/mol (g52.37,2.40,2.48), form57, 9, and 11, respectively. The large increase in interlayer separation correla

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with a small decrease in the exchange and a slight increasthe g factor of the copper spins.

The upward trend of thexdcT product below 50 K indi-cated 35 ferromagnetic correlations between the spins,sharp peak suggesting a transition with a critical temperaTc'20 K for all three compounds. This behavior is likedue to the spin canting caused by the additional DMchange, whose strength is estimated46 at Di j ;@(g22)/g#Ji j '5 K.

The fit to the Curie-Weiss lawxdc215(T1QCW)/C for T

>200 K ~Fig. 2! gives mean fieldQCW of 140, 130, and 120K, for m57, 9, and 11, respectively. This quantity allowthe calculation of the ratioQCW /Tc , which may correlatewith the degree of frustration.9 We obtain values of 7, 6.5and 6, respectively, suggesting moderate strenfrustration.9 A better estimate of the degree of frustrationf R5QCW /TN which relies on the use of the Nee`l tempera-ture, corresponding to an antiferromagnetic transition~in thiscase in the absence of the DM interaction, perhaps due tointeractions, etc.!. Thus, we estimate even stronger frustrtion, asTN would very likely be much lower than 20 K.

Hysteresis curves at 5 K~Fig. 3! give saturation magnetization values of;1440, 1480, and 1550 emu Oe/mol-C~below the value of 5585 emu Oe/mol expected for a sp1/2 g52 ferromagnet!, and coercive fields of 600, 900, an1000 Oe form57, 9, and 11, respectively. The saturatiomagnetizations correlate with theg values determined fromthe high-temperature static susceptibility. It is noted that bg and the saturation magnetization increase monotonicwith m. If instead of the spin-only value for theg factor g52 we use the values obtained above from the fit to hitemperature series expansions (g52.37,2.40,2.48) we calculate expected ferromagnetic saturation magnetizations vaof 6620, 6700, and 6925 emu Oe/mol. Therefore, the expmental saturation magnetizations are significantly sma

FIG. 2. xdc21 vs T for Cu2(OH)3(CmH2m11COO), m57, 9, and

11 at Hdc55000 Oe, and fits to Curie-Weiss mean field predtions.

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(;4.5 fold! than the ones expected for a ferromagnetic mterial, consistent with spin canting and/or noncollinear sconfiguration pictures.

The lowest frequency~5 Hz! x18 , Fig. 4, data demon-strates the independence of the peak temperaturesTp519.1, 19.3, and 19.0 K, form57, 9, and 11, respectively!on the interlayer distances. Thus the magnetic behaviogoverned by the intralayer interactions and, therefore, thcompounds are true 2D systems.

The sharp peaks~variations of almost three orders omagnitude over less than three degrees! suggest divergencieof the susceptibility and, therefore, true phase transitio

FIG. 3. Hysteresis curves for Cu2(OH)3(CmH2m11COO), m57, 9, and 11 atT55 K. Inset: details of the low-field regionshowing the coercive fields.

FIG. 4. x18 of Cu2(OH)3(CmH2m11COO), m57, 9, and 11, inh051 Oe ~zero applied dc field! at f 55 Hz.

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The divergence of the magnetic susceptibility can be studusing Kouvel-Fisher scaling analysis,49 which makes use ofthe scaling law:

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Kouvel-Fisher scaling analyses of the linear susceptibidata~Fig. 4! are presented in Fig. 5 for all three compoundThe parameters obtained through such analyses areTc519.0, 19.7, and 19.5 K andg51.81, 1.75, and 1.78 form57, 9, and 11, respectively. The error in the determinatof theg values is estimated to about610–15 %, mainly dueto the various choices for the temperature ranges wherewere fitted to a straight line. The final choices reflect the bfits in ranges of reduced temperature of about 0.01<(T2Tc)/Tc<0.15.

All three compounds have similar critical exponentsg,with values close to that of a 2D-Ising system,g51.75,significantly larger than 1.24 for 3D Ising or 1.39 for 3DHeisenberg systems.50 Although the error in these fits is relatively large, it is clear that the lower bound for theg expo-nent would still be far above the 3D ones, suggesting tthese systems have Ising-like behavior, despite the absof single-ion anisotropy. Such a behavior is likely due to tanisotropy caused by the additional DM interaction.

The field dependence of the remanent magnetizationthe normalized remanent magnetization—obtained dividthe actual value at eachT by the value at the lowest temperature measured 5 K,M (T)/M (5K)—for all three compoundsare shown in Fig. 6. The low-field normalized remanendata are consistently above the higher-field values and ssharper temperature variations, the transition being bprobed at the lowest applied field of 5 Oe. These data wused for static scaling analyses, to findTc and the criticalexponentb:

M}~Tc2T!b. ~4!

The fits to the power law behavior belowTc , Fig. 6, giveTc519.9, 19.8, and 19.5 K and unusually large values ob50.81, 0.76, and 0.80 form57, 9, and 11, respectively. Therrors in the determination of these values is estimatedabout 610–15 %, mainly due to the various possibchoices for the temperature ranges where data were fiThe final choices were made such that the region fittedbeyond saturation and within the critical regime~ranges ofreduced temperature of roughly 0.005<(T2Tc)/Tc<0.1)and such that the best fit could be obtained.

All three compounds have similar critical exponentsb,with values significantly larger than the 0.125~0.3 experi-mentally! for 2D Ising, 0.32 for 3D Ising, or 0.36 for 3DHeisenberg systems.50 Even though the errors of these fi

4122 PRB 61GIRTU, WYNN, FUJITA, AWAGA, AND EPSTEIN

FIG. 5. Kouvel-Fisher scaling analysis ofx18data of Fig. 6 for Cu2(OH)3(CmH2m11COO), ~a!m57, ~b! m59, and~c! m511.

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are relatively large, the lower limit of the values for thcritical exponents are still much larger than any of the knob exponents. These large values deserve further study.

The nonlinear susceptibility can provide important infomation regarding the existence of a phase transition, theture of the transition, and the dynamics near the transitFor a ferromagnet~FM! the magnetizationM can be ex-panded in a power series with respect to the magnetic fiH, as:

MFM5M01x1H1x2H21x3H31•••, ~5!

where M0 is the spontaneous magnetization,x1 the linearmagnetic susceptibility, andx2 ,x3 , etc. are the nonlineacomponents of the susceptibility. The even nonlinear comnents can be observed because of the lack of inversion smetry with respect to the applied field.51 For a SG there is nospontaneous magnetization andM can be expressed in termof only the odd powers ofH:52

MSG5x1H1x3H31•••. ~6!

In the case of a small amplitude ac field and no appliedfield the linear and nonlinear susceptibilities are relatedthe harmonics measured experimentally by:53

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Both second and third harmonics (2f and 3f responses,respectively!, for the m59 compound~again them57 and11 compounds have almost identical behavior35!, Fig. 7,have relatively sharp peaks~variations of almost three orderof magnitude over a temperature range less than one deg!suggesting divergencies of both these quantities and, thfore, true magnetic transitions. The peak in the secondmonic indicates that a spontaneous moment is formed attransition, while the frequency dependence of both comnents of the nonlinear susceptibility suggests slow relaxaprocesses and glassiness.

We note that the fluctuations surveyed by both the linand nonlinear ac susceptibilities start above the transiand peak~in the low-frequency limit! at the critical point.Similarly, the remanent magnetization measured in theexperiment has nonzero values even above the transi

PRB 61 4123GLASSINESS AND CANTED ANTIFERROMAGNETISM IN . . .

FIG. 6. Normalized remanent magnetization for Cu2(OH)3(CmH2m11COO), ~a! m57, ~b! m59, and~c! m511 at 5<Hdc<50 Oe withpower law fits. Inset: the unnormalized data.

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These effects may be caused by impurities, the finite sizthe system, the time of measurement, etc., but mostly byinstrumental resolution for measuring the applied fields athe response of the system.54,55

B. Spin-glasslike behavior

The frequency dependence of the real and imaginaryof the linear susceptibility is shown in Fig. 8. The peak teperature ofx18 increases while the peak height decreastrongly with increasing frequency indicative of slow relaation processes that characterize glassy behavior.18,56 Thevalues of the relative variation of the peak temperaturedecade of frequency, (DTp /Tp)/D(log10 f )50.003, 0.008,

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and 0.008, form57, 9, and 11, respectively,35 place thesecompounds within the range of canonical spin glasses.18

Given the glassy behavior observed in the ac susceptity data, we attempted to perform dynamic scaling analyfor all three compounds, using the linear scaling procedur57

This procedure allows the determination ofTc , and the spin-glass critical exponents separately and independently, aving the usual log-log plot that often conceals departures frscaling. Attempts to achieve data collapse using the sglass scaling expressions were not successful, likely duthe complex behavior of these systems~apparent divergencein the linear susceptibility and the presence of the secharmonic of the susceptibility to be discussed below!. Thusthese materials are not typical spin glasses.

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4124 PRB 61GIRTU, WYNN, FUJITA, AWAGA, AND EPSTEIN

The frequency dependence of the ac susceptibility psented in Fig. 8 indicates long relaxation times. A detaianalysis of these relaxation times and their dependencetemperature can be made using the phenomenologicascription of Cole and Cole,58 which involves a modeling ofthe dynamics at a given temperature onto a distributionrelaxation times that is symmetric on the logarithmic timscale. The Cole-Cole formalism introduces a paramea (0,a,1), which determines the width of the distributioof relaxation times,g(ln t), around the median relaxatiotime, tc :59

g~ ln t!51

2p

sin~ap!

cosh@~12a!ln~t/tc!#2cos~ap!. ~10!

This distribution is determined based on the Cole-Cole eqtion for the complex linear susceptibility:

x1,CC5xS1x02xS

11~ ivtc!12a

, ~11!

wherex0 and xS are the isothermal (v→0) and adiabatic(v→`) susceptibilities, respectively.

Based on the Cole-Cole equation one can determineexpression forx19(x18), which allows the fit of the experi-mental data.59 Such fits~with a,tc , and the isothermal susceptibility as parameters! are presented in the Argand plot oFig. 9, where the phenomenological Cole-Cole model is

FIG. 7. Nonlinear susceptibilitiesux2,exph0u and u 34 x3,exph0

2u ofCu2(OH)3rm(C9H19COO) measured at 2f and 3f , respectively, inh051.3 Oe~zero dc field! at frequencies 10< f <3330 Hz.

-done-

f

er

a-

an

-

scribed by circular arcs of size (12a)p cutting thex18 axisat x0 andxS with a maximum atvtc51. Noteworthy is theshift of the data points from a nearly isothermal susceptiity above 19.5 K to a nearly adiabatic susceptibility belo18.7 K. As it will be seen below, this behavior corresponto a large increase of the median relaxation time whencreasing the temperature a few degrees through the tration.

The parametersa andtc determined from the Cole-Coleanalysis at eachT allow the construction of the distribution

FIG. 8. x18 and x19 of Cu2(OH)3(C9H19COO), in h051 Oe~zero applied dc field! at various frequencies 5< f <10000 Hz.

FIG. 9. Argand plots,x19(x18), and Cole-Cole analysis of datshown in Fig. 9, for Cu2(OH)3(C9H19COO) at 18.7<T<19.5 K.

edem

achays

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PRB 61 4125GLASSINESS AND CANTED ANTIFERROMAGNETISM IN . . .

of relaxation times~Fig. 10!. As the temperature is decreasthrough the transition the median relaxation time increasindicating the growth of the correlation length of the systeof spins, while the parametera, describing the width of thedistribution of relaxation times, increases, reflecting the fthat the distribution of cluster sizes broadens. Such a beior was seen in all three compounds, though only the analfor the m59 sample~based on the data of Fig. 8! is shownhere.

The temperature dependence of the median relaxatime determined by Cole-Cole analysis for all three copounds, presented in Fig. 11, shows thattc varies almost sixdecades over less than three degrees. The divergence orelaxation time can be studied using scaling analysis baon the scaling law:

tc}~T2Tc!2zn. ~12!

The parameters obtained through such analyses arTc518.7, 18.6, and 18.3 K andzn56.1, 7.7, and 5.6 form57, 9, and 11, respectively. The error in the determinatof these values is estimated to about 15–20 %, again madue to the various choices for the temperature ranges wdata were fitted to a straight line. The final choices reflectbest fits in ranges of reduced temperature of about 0<(T2Tc)/Tc<0.25. These values are much larger ththose found in pure systems without frustration, wherez;2 and n;0.6, but comparable to those found in spglasses, wherezn is larger than 5, in some cases even largthan 10.50,56

The field-cooled~FC! and zero-field-cooled~ZFC! mag-netization curves for them59 compound (m57 and 11compounds have almost identical behavior35!, Fig. 12, show,with decreasingT, a rapid rise just above 20 K. At lowerTthe ZFC magnetization deviates below the FC magnetizaindicating history dependence of the magnetization procein the T range wherex18 shows frequency dependence. Tfield dependence of the bifurcation pointTb between the FC

FIG. 10. Distribution of relaxation times g(t) forCu2(OH)3(C9H19COO) obtained through Cole-Cole analysis of tdata shown in Fig. 9, at 17<T<20 K.

s,

tv-is

n-

theed

nlyree2

r

nes

and ZFC curves (Tb decreases with increasing the appliedfield! reinforces the glassy behavior description.60,61

Specific heat data is plotted as the ratioC/T vs T in Fig.13~a!. In the absence of a nonmagnetic sample with a simstructure isolation of the magnetic component of the specheat is not obvious. However, a weak broad feature intotal specific heat is observed just above the temperaturethe magnetic transition, at about 21 K, where there is a mest change in slope, which we attribute to the magnetic scific heat. Weak features in the magnetic specific heatvery common in spin glasses and spin-glass-like system17

because of the gradual freezing of the magnetic degreefreedom starting well above the spin-glass transition.

The log-log plot of the specific heat shown in Fig. 13~b!reveals a roughly quadratic power law behavior. More pcisely, the exponents are about 1.9,n,2.2, depending onthe region of the fit, being closer ton;2 below the transi-tion. This result is expected for both phonons and antifermagnetic spin waves on a 2D lattice. It is, therefore, likethat the layered crystalline structure causes aT2 behavior ofthe specific heat in the temperature range probed. The crover toT3, expected for any 3D solid, probably takes plaat temperatures below 1 K.

FIG. 11. Temperature dependence of the mean relaxationobtained through Cole-Cole analysis of thexac data forCu2(OH)3(CmH2m11COO), ~a! m57, ~b! m59, and ~c! m511with power-law fits.

p-e ofand

iderthe

f

-at

4126 PRB 61GIRTU, WYNN, FUJITA, AWAGA, AND EPSTEIN

FIG. 12. FC~filled symbols! and ZFC~empty symbols! magne-tization of Cu2(OH)3(C9H19COO) in dc applied fields of 5<Hdc

<200 Oe.

FIG. 13. Specific heat of Cu2(OH)3(CmH2m11COO), m57, 9, and 11, plotted as the ratioC/T vs T, ~a!, and as a log-logplot of C vs T, ~b!.

The low-temperature dynamic susceptibility in zero aplied dc field and various frequencies and in the presencvarious constant dc magnetic fields are shown in Fig. 14Fig. 15, respectively for the compound withm511. Figs. 14and 15 display the data on logarithmic scales and in a wrange of temperatures, which allows the examination ofmagnetic behavior especially below the'20 K transition.

The overall shape of the temperature dependence ox18remains qualitatively the same regardless of frequency~Fig.14! or dc fields ~Fig. 15!. The in-phase susceptibility increases with decreasingT, has a peak and then levels offlow temperatures. The magnitude ofx18 ~which describes the

FIG. 14. x18 of Cu2(OH)3(C11H23COO), inh051 Oe~zero ap-plied dc field! at various frequencies 5< f <10000 Hz.

FIG. 15. x18 of Cu2(OH)3(C11H23COO), in h051 Oe at f51000 Hz and various dc applied fields 0<Hdc<20000 Oe. In-set:Hdc at the peak inx18 vs peak temperatureTp .

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PRB 61 4127GLASSINESS AND CANTED ANTIFERROMAGNETISM IN . . .

fluctuations in magnetization! decreases monotonically witincreasingHdc ~Fig. 15!, as expected, indicating that as thapplied field is raised more and more spins are locked,ented by the dc field, and do not participate in fluctuatioAlso, the peak in the in-phase susceptibility shifts towahigherT asHdc is increased~inset of Fig. 15!. The presenceof the field compensates for the disordering effect of thermfluctuations, allowing the ‘‘ordering’’ to occur at higherT.Such behavior has been seen in Monte Carlo simulation2D Ising magnets.62

It is important to note the leveling off ofx18 at low tem-peratures. While the usual 2D-Ising magnet has vanishsusceptibility asT decreases to zero,62 the constant lowT x18seen in Fig. 15 indicates that fluctuations do not disappfor these compounds, consistent with the zero field dc datFig. 12. Therefore, some spin degrees of freedom areserved at lowT, which is in accord with frustration playingcrucial role in these systems. Given the nonuniform butriodic magnetic lattice, it is possible that one or more subtices order below the critical temperature of'20 K, whileat least one sublattice remains disordered even below 51

IV. DISCUSSION

All three systems show very similar magnetic propertdespite the large differences between the interlayer distanThe magnetic behavior must, therefore, be determined byintralayer interactions, which are expected to be very simibased on the intralayer structural similarities betweenthree compounds. The absence of single-ion anisotropy aspin carrying sites and the presence of oxygen mediatedperexchange pathways lead to strong isotropic Heisenantiferromagnetic exchange, which on a triangular lattcauses spin frustration. The slight differences in the envirment of adjacent Cu sites causes a weak DM exchawhich favors spin canting leading to anisotropies.

The antiferromagnetic correlations revealed by the dc sceptibility data show that the Heisenberg antiferromagnexchange is indeed the dominant interaction. Moreover,fits to high-temperature series expansions indicate thathigh-temperature behavior of these compounds is consiswith that of a TQHAF. At low temperatures the deviatiofrom the TQHAF predictions, together with the strong peain both the linear ac susceptibility and its harmonics~thesecond harmonic indicating the developing of a spontanemoment! suggest a canted antiferromagnetic type of ording, consistent with the presence of weak DM interaction

The nature of the low-temperature phase is very unushowever. The sharpness of the peaks in the linear andlinear susceptibilities~orders of magnitude variations oveonly a few degrees! suggests divergencies and, thereforetrue phase transition. The critical exponents obtainedKouvel-Fisher scaling analysis of the linear susceptibisuggest a 2D Ising-like low-temperature phase. Howethis simple picture is complicated by the strong frequendependence of the linear ac susceptibility and the irrevibilities present in the FC/ZFC magnetization data~with fielddependence of the bifurcation temperature! which indicatespin-glass-like behavior. Moreover, the dynamic critical eponents obtained through Cole-Cole analysis of the dynasusceptibility, followed by fitting the temperature depe

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dence of the median relaxation times to power laws, hlarge values, reminescent of glassy dynamics. The speheat data has only a weak broad feature at the transitconsistent with the spin-glass-like behavior. The small vaof the 5 K saturation magnetization, which is only a smfraction of the value expected for a spin-1/2 system, is csistent with spin canting and/or noncollinear spin configutions as well as with partial order. Furthermore, the leveloff below 10 K of the linear ac susceptibility with or withoua superimposed dc magnetic field suggests that somedegrees of freedom are preserved below the transitionappears that the interplay of Heisenberg and DM exchanleads to an unusual state in which order and disorder seecoexist.

We should note that, although very similar, these thcompounds do not have completely identical magnetic prerties. For instance, there are some slight differences inexchange constants andg values, in the saturation magnetzation and coercive fields, etc. These differences are likrelated to the slight structural differences within the layebetween the three systems.40

Also, although there is a general consistence of somethe results of the scaling analyses, some puzzling questpersist. One is related to the unusually large values ofcritical exponentb. A second question is related to the diferent values of the critical temperature obtained throughvarious scaling analyses. The relatively large error bars inTc~estimated generally to;10%) show that the different values are within the accuracy of our data and of our fittiprocedures. The values that we believe best describe thesition in these systems are the ones obtained throughKouvel-Fisher analysis. In that case the estimated erwere at a minimum, despite the fact that the fitting procedrequired taking the derivative of the susceptibility data.

It has been known, that in real compounds, there usuexist either symmetry-reducing lattice distortions63 or addi-tional interactions,9 which relieve frustration and allow thesystem to order at a temperature determined by the domiinteraction strength. In the case of our compounds we ppose, based on both structural information and magndata, that the additional DM interaction~whose strength wasestimated to 5 K, about ten times smaller than the Heisberg exchange! is the cause of the anisotropy leading at lotemperatures to 2D Ising-like behavior.

The spin-glass-like behavior seen in these compoundsfeatures similar to typical spin glasses@the relative variationof the Tp in x18 per decade of frequency(DTp /Tp)/D(log10f ), as well as the dynamical critical exponents# but the evidence for canted antiferromagnetic ordershows that these systems cannot be simple spin glasses

In the case of triangular Heisenberg AF’s with Ising-likanisotropy it was suggested that site dilution produces adom anisotropy field that leads to low-temperature propersimilar to a 2D-Ising spin glass.64 Numerical studies considering the appropriate amount of impurities lead to goagreement with experimental specific heat data65 confirmingthe initial suggestion that the glassiness in that systemdue to site disorder.66 A spin wave analysis of the triangulaquantum Heisenberg AF with vacancies found that frustion remains the dominant influence even in the presencdefects and, also, that the magnetic properties depend on

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4128 PRB 61GIRTU, WYNN, FUJITA, AWAGA, AND EPSTEIN

relative position of the vacancies and not simply on thdistance of separation.67

It is questionable that the glassy behavior in our copounds could be attributed solely to the existence of frustion, especially as the degeneracy of the ground stateclassical THAF is not macroscopic. It is unlikely that sudegeneracy can lead to a disordered state. More likely,the interplay between the Heisenberg and DM exchan~the former causing frustration, the latter causing anisotrin the spin orientation! that might lead to a rough free-energlandscape responsible for the glassiness observed in tsystems. If the frustration caused by the Heisenbergchange could be released, in the absence of other intetions, by a 120° noncollinear spin configuration, the scanting favored by the additional DM interaction may opose such tendencies and lead to spin freezing along anropy directions. The influence of DM interaction in metallspin-glasses and re-entrant spin-glasses has been discusthe past and it was shown that the DM exchange seemplay a role in the observed macroscopic irreversibilitysuch systems.68

It is difficult to address the role of structural disorderthe magnetic behavior of our compounds because of theited structural information available at this time.37,40 Thex-ray powder diffraction revealed the layered structure athe presence of crystallites, while the TEM micrograpshowed interference patterns usually seen in structurallydered systems.41 It is possible to have some site defects adislocations in the lattice, which, along with possible surfaeffects, might also play some role in causing some ofglassy behavior. However, given the strength of the intertions involved (2J;60 K andD;5 K) we speculate thaany effects due to possible structural disorder are inhidden~as it was proposed for some nominally disorder-frgeometrically frustrated compounds19! by the more impor-tant ones caused by the two major interactions.

Quantum fluctuations are very important at low tempetures, even though it is still debated whether they couldstrong enough to prevent the pure TQHAF from achievzero temperature noncollinear magnetic LRO.27 Given therelatively high temperature where the transition occ(;20 K) and the strength of the additional DM interactioit is unlikely that quantum fluctuations play the crucial roeven though their contribution cannot be ruled out.

The interlayer interaction~superexchange or dipole-dipointeraction!, expected to be negligible based on the structudata, do not affect the magnetic behavior in the rangetemperatures probed, as all three compounds have very slar behavior despite the large differences in interlayer seration. If 3D behavior could be seen in these compoundshould be at temperatures much lower than 20 K and pably less than 1 K~based on specific heat data!.

Given the mixture of canted antiferromagnetism and spglass-like characteristics in the low temperature behaviothese compounds we are lead to two possible explanatiOne is based on the coexistence of antiferromagnetic sh

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range order and spin-glass long-range order predicted fordiluted fcc AF,69 while the other is ‘‘partial order’’ proposedfor the bcc Heisenberg AF.6 In the absence of clear evidencfor structural disorder the first possibility seems less likeMoreover, given the nonuniform but periodic magnetic latice it is conceivable that some of the spins will be interaing more strongly than others,~in particular the absence oDM interaction between spins on particular sites might gextra freedom! such that ordered and disordered sublatticmight be formed. Therefore we are lead to speculate thatlow temperature state might be partially ordered. The pobility of partial order in these compounds is, however,open question and deserves further consideration. A deunderstanding of this apparent coexistence of canted antromagnetic and spin-glass-like characteristics awaits theoical studies of systems with both Heisenberg and DM intactions.

V. CONCLUSION

We reported extensive magnetic studies of three trianlar quantum Heisenberg antiferromagnets~TQHAF’s! withweak additional Dzyaloshinskii-Moriya interactionCu2(OH)3(CmH2m11COO), m57, 9, and 11.

We proposed that the unusual behavior of these thTQHAF systems is determined by the interplay betweenHeisenberg exchange, causing geometrical frustration,the DM exchange, leading to spin canting. Instead of choing between the resonant valence bond or noncollinear N´elground states, we proposed that these systems evolve, dthe additional DM interaction, toward a finite temperatu2D Ising-like canted antiferromagnetic state. Geometrifrustration, together with the anisotropy caused by the adtional DM interaction, proves to be strong enough to causome kind of spin freezing, slow relaxation processes,glassiness. The interplay of Heisenberg and DM exchanleads to an unusual state in which order and disorder apto coexist.

We also speculated that the effects of structural disordquantum fluctuations and interlayer interactions are likelybe hidden at the relatively high temperature of the transitdue to the strength of the main two interactions. We sgested that glassiness in our systems is different fromfound in typical spin glasses. Based on structural informatwe proposed that these systems are candidates for studpartial order as the interactions are nonuniform but periodwith possibilities for making distinctions between the variosublattices.

ACKNOWLEDGMENTS

We thank Randall C. Black and Jost Diederichs froQuantum Design for providing the specific heat data. Wacknowledge useful discussions with H.T. Diep, EShender, D.L. Huber, and M.J.P. Gingras. The support ofDepartment of Energy Division of Materials Science~DE-FG02-86BR45271! is gratefully acknowledged.

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PRB 61 4129GLASSINESS AND CANTED ANTIFERROMAGNETISM IN . . .

*Present address: Department of Physics, Ovidius UniversityConstant¸a, Constant¸a, 8700, Romania.

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