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Matsumoto, Masashige, Matsuo, Akira, [Hase, Masashi](https://orcid.org/0000-0003-2717-461X), Kindo, Koichi

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[Magnetism of the antiferromagnetic spin-1/2 tetramer compound CuInVO5](https://mdr.nims.go.jp/datasets/dcc104e5-83ec-447a-9baa-2bcb23c39d2f)

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PHYSICAL REVIEW B 94, 174421 (2016)Magnetism of the antiferromagnetic spin-12 tetramer compound CuInVO5Masashi Hase,1,* Masashige Matsumoto,2 Akira Matsuo,3 and Koichi Kindo31National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan2Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan3The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan(Received 16 June 2016; revised manuscript received 26 October 2016; published 14 November 2016)We measured the temperature dependence of the magnetic susceptibility and specific heat and the magnetic-field dependence of the magnetization of CuInVO5. An antiferromagnetically ordered state appears belowTN = 2.7 K. We observed a 12 quantum magnetization plateau above 30 T at 1.3 K. We consider that the probablespin model for CuInVO5 is an interacting spin- 12 tetramer model. We evaluated values of the intratetramerinteractions as J1 = 240 ± 20 K (antiferromagnetic) and J2 = −142 ± 10 K (ferromagnetic). The ground stateof the isolated spin tetramer with the J1 and J2 values is spin singlet. The shrinkage of ordered magnetic momentsby quantum fluctuation can be expected. Detectable low-energy longitudinal-mode magnetic excitations mayexist in CuInVO5.DOI: 10.1103/PhysRevB.94.174421I. INTRODUCTIONThe interacting antiferromagnetic (AF) spin-dimer com-pounds TlCuCl3 [1–5] and KCuCl3 [6,7] show a pressure-induced or magnetic-field-induced magnetic quantum phasetransition. Experimental observations [8–11] and the the-oretical background [12,13] of massive longitudinal-modemagnetic excitations in the ordered state were reported forthese compounds. The longitudinal mode and massless trans-verse modes (Nambu-Goldstone modes) [14] are related tofluctuations in the amplitude and phase of the order parameter,respectively. The longitudinal mode is the analog of the Higgsparticle [15–17].According to the results of theoretical investigations, thelongitudinal mode can exist in interacting AF spin clustersystems that are realized in Cu2Fe2Ge4O13 and Cu2CdB2O6[18]. The spin systems in Cu2Fe2Ge4O13 [18] and Cu2CdB2O6[19–21] can be regarded as interacting AF spin tetramers(Fe-Cu-Cu-Fe and Cu-Cu-Cu-Cu tetramers, respectively).The shrinkage of ordered magnetic moments by quantumfluctuation is important for the appearance of the longitudinalmode. The ground state (GS) can be a spin-singlet state inisolated AF spin clusters. Therefore, some interacting spincluster systems are advantageous for the longitudinal mode. Anantiferromagnetically ordered state appears in Cu2Fe2Ge4O13[22] and Cu2CdB2O6 [19] in zero magnetic field under atmo-spheric pressure. The magnetic excitations in Cu2Fe2Ge4O13have been investigated by inelastic neutron-scattering (INS)experiments on single crystals [22–25]. The longitudinal modewas not confirmed because of the small INS intensities dueto the large excitation energies (> 15 meV) and because ofthe overlap of the transverse modes. The magnetic excitationsin Cu1142 Cd11B2O6 were studied by INS experiments on itspowder [21]. Although the results suggest the existence ofthe longitudinal mode, there was no conclusive evidencebecause powder was used. A single crystal suitable for themeasurements of physical properties has not been reported.*HASE.Masashi@nims.go.jpWe require further spin cluster compounds that have an an-tiferromagnetically ordered state and low-energy longitudinal-mode magnetic excitations. We focus on spin- 12 tetramersbecause of the following magnetism. The Hamiltonian of aspin tetramer is expressed asH = J1S2 · S3 + J2(S1 · S2 + S3 · S4). (1)When J1 > 0 or J2 > 0, the GS is the spin-singlet state.Therefore, the shrinkage of ordered moments can be expectedin an ordered state generated by the introduction of interte-tramer interactions. The ordered state is possible under thecondition that the value of � is comparable to or less thanthat of an effective intercluster interaction [18]. Here � isthe energy difference (spin gap) between the singlet GS andfirst-excited triplet states. The effective intercluster interactionis given by the sum of the products of the absolute value ofeach intercluster interaction (|Jint,i |) and the correspondingnumber of interactions per spin (zi) as Jeff = ∑i zi |Jint,i |. Theeffective intercluster interaction is usually much smaller thanthe dominant intracluster interactions. Therefore, � should bemuch smaller than the dominant intracluster interactions forthe appearance of the ordered state.Figure 1 shows the eigenenergies of the excited statesmeasured from the GS in an isolated spin- 12 tetramer [26]. Asshown in Fig. 1(a) for J1 > 0, �/J1 can be sufficiently smallwhen J2 has negative or small positive values. Even undera small Jeff , an ordered state is expected in a spin-tetramercompound for J1 > 0 and J2 < 0. The small �/J1 is incontrast to �/J = 1 in the AF spin- 12 dimer given by JS1 · S2.As shown in Fig. 1(a), the GS and first-excited states arewell separated from the other excited states (ESs). This meansthat the low-energy physics can be described by an effectivespin-dimer (singlet-triplet) system [18]. There are compoundsthat have spin- 12 tetramers expressed as Eq. (1) and an orderedstate. Examples are Cu2CdB2O6 with J1 = 317 ± 12,J2 =−162 ± 16, and TN = 9.8 K [19–21] and SeCuO3 with J1 =225,J2 = 160, and TN = 8 K [27].We can expect spin- 12 tetramers in CuInVO5 from itscrystal structure [28]. The Cu2+ ions (3d9) have localizedspin- 12 . The positions of the Cu ions and the O ions connected2469-9950/2016/94(17)/174421(7) 174421-1 ©2016 American Physical Societyhttps://doi.org/10.1103/PhysRevB.94.174421HASE, MATSUMOTO, MATSUO, AND KINDO PHYSICAL REVIEW B 94, 174421 (2016)FIG. 1. Eigenenergies of excited states measured from the groundstate in an isolated spin- 12 tetramer expressed by Eq. (1). There aretwo ST = 0 states (|01 > and |02 >), three ST = 1 states (|11 >,|12 >, and |13 >), and one ST = 2 state (|21 >). ST is the value ofthe sum of the spin operators in the tetramer. The eigenstates |ij〉of the isolated tetramer are explicitly given in [26]. In the isolatedtetramer, the ground state is the spin-singlet |02〉 state. (a) J1 > 0. Thevertical dashed line indicates the J2/J1 value of CuInVO5 evaluatedin the present work. (b) J2 > 0.to the Cu ions are shown schematically in Fig. 2(a). Twocrystallographic Cu sites (Cu1 and Cu2) exist. Red and bluebars indicate the shortest and second-shortest Cu-Cu distances,respectively. The distances at room temperature are 3.117 and3.173 Å, respectively. The closest Cu1-Cu1 pair is bridgedby two identical Cu1-O-Cu1 paths the angle of which is89.75◦. The closest Cu1-Cu2 pair is bridged by two differentCu1-O-Cu2 paths with angles of 107.61 and 88.19◦. The otherCu-Cu distances are 4.705 Å or greater. If dominant exchangeinteractions exist in the Cu1-Cu1 and Cu1-Cu2 pairs, spintetramers given by Eq. (1) are formed. Figure 2(b) shows thearrangement of the spin tetramers. Two types of tetramers (Iand II) exist, although they are equivalent to each other as a spinsystem. In this paper, we report the magnetism of CuInVO5.An AF long-range order appears below TN = 2.7 K. We showthat the spin system consists of spin tetramers with J1 > 0 andJ2 < 0.II. EXPERIMENTAL AND CALCULATION METHODSCrystalline CuInVO5 powder was synthesized by a solid-state reaction. Starting materials are CuO, In2O3, and V2O5powder. Their purity is 99.99%. A stoichiometric mixture ofpowder was sintered at 1023 K in air for 100 h with inter-mediate grindings. We measured an x-ray powder-diffractionFIG. 2. (a) Schematic drawing of positions of Cu2+ ions havingspin- 12 and O2− ions connected to Cu2+ ions in CuInVO5 [28]. Red,blue, and white circles indicate Cu1, Cu2, and O sites, respectively.Red and blue bars represent the shortest and second-shortest Cu-Cudistances, respectively. Thin black bars represent Cu-O. We define J1and J2 as the exchange interaction parameters for the Cu1-Cu1 andCu1-Cu2 pairs, respectively. The J1 and J2 interactions form a spin- 12tetramer. (b) Schematic drawing of spin tetramers in CuInVO5. Twotypes of tetramers (I and II) exist, although they are equivalent toeach other as a spin system. (c) Interacting spin tetramer model usedto calculate magnetization using a mean-field theory based on thetetramer unit (tetramer mean-field theory).pattern at room temperature using an x-ray diffractometer(RINT-TTR III, Rigaku). The wavelength is 1.540 and 1.544 Å(Cu Kα1 and Kα2 lines, respectively). X rays of Cu Kβare excluded. We adopted a flat sample scattering geometry.We performed Rietveld refinements based on the space groupP 21/c (No. 14) as in the literature [28] using the FULLPROFSUITE program package [29] with its internal tables forscattering lengths. All observed reflections can be indexedon the basis of the published structure data of CuInVO5.We confirmed that our sample was a nearly single phaseof CuInVO5. The lattice constants are a = 8.776(1), b =6.158(1), c = 15.268(1) Å, and β = 106.48(1)◦. These arealmost the same as the values reported in the literature [28][a = 8.793(2), b = 6.1542(6), c = 15.262(2) Å, and β =106.69(2)◦]. The atomic positions in our results are closeto those in the literature. We measured the specific heatusing a physical property measurement system (QuantumDesign). We measured the magnetization in magnetic fieldsof up to 5 T using a superconducting quantum interference174421-2MAGNETISM OF THE ANTIFERROMAGNETIC SPIN- . . . PHYSICAL REVIEW B 94, 174421 (2016)device magnetometer magnetic property measurement system(Quantum Design). High-field magnetization measurementswere conducted using an induction method with a multilayerpulsed field magnet installed at the Institute for Solid StatePhysics, the University of Tokyo.We obtained the eigenenergies and eigenstates of isolatedspin- 12 tetramers using an exact diagonalization method [26].We calculated the temperature T dependence of the magneticsusceptibility and the specific heat and the magnetic-field Hdependence of the magnetization M(H ) using the eigenen-ergies and eigenstates. We calculated M(H ) for the modelshown in Fig. 2(c) using a mean-field theory based on thetetramer unit (tetramer mean-field theory). Finite magneticmoments were initially assumed on the Cu sites in the tetramer.The mean-field Hamiltonian was then expressed by a 16 × 16matrix form under consideration of the external magneticfield and the molecular field from the nearest-neighbor sites.The eigenstates of the mean-field Hamiltonian were used tocalculate the expectation value of the ordered moments onthe Cu sites. We continued this procedure until the valuesof the magnetic moments converged. We finally obtained aself-consistently determined solution for M(H ).III. RESULTS AND DISCUSSIONThe red circles in Figs. 3 and 4 show the T dependenceof the specific heat C(T ) of CuInVO5 in zero magnetic fieldand the magnetic susceptibility χ (T ) in a magnetic field ofH = 0.01 T, respectively. We can observe a peak in C(T ) at2.7 K and a clear decrease in χ (T ) below this temperature,indicating the occurrence of an AF long-range order. A broadmaximum can be seen around 8 K in C(T ) and around 11 Kin χ (T ), indicating that the origin of the broad maximum inFIG. 3. Temperature T dependence of the specific heat C(T )of CuInVO5 in zero magnetic field. A green line indicates C(T )calculated for an isolated spin- 12 tetramer. The J1 and J2 values arelisted in Table I.FIG. 4. Temperature T dependence of the magnetic susceptibilityχ (T ) of CuInVO5 (circles) in a magnetic field of H = 0.01 T. Green,red, and blue lines indicate χ (T ) calculated for the total, Cu1, andCu2 spins, respectively, in an isolated spin- 12 tetramer. The J1 and J2values are listed in Table I.C(T ) is magnetic. As T is increased, χ (T ) decreases rapidlyup to T = 40 K then decreases slowly at higher temperatures.Other phase transitions were not observed in C(T ) and χ (T )below 300 K.The thick red lines in Figs. 5(a) and 5(b) show the Hdependence of the magnetization M(H ) of CuInVO5 measuredat 1.3 and 30 K, respectively. We can observe a 12 quantummagnetization plateau above 30 T at 1.3 K. The g value wasevaluated to be 2.09 ± 0.02 from the magnetization of theplateau. The magnetization plateau is smeared at 30 K.We compare χ (T ), C(T ), and M(H ) for CuInVO5 withthose calculated for isolated spin tetramers. We calculatedχ (T ) for various sets of exchange parameters. The set inwhich J1 = 240 and J2 = −142 K is the best. The green line inFig. 4(b) indicates χ (T ) calculated for an isolated spin tetramerwith J1 = 240 and J2 = −142 K. The J1 and J2 values arelisted in Table I. The agreement between the experimentaland calculated χ (T ) is nearly perfect above 30 K, whereas adiscrepancy is seen below 30 K. The green line in Fig. 3(a)indicates C(T ) calculated for the isolated spin tetramer withthe same J1 and J2 values. The positions of the broad maximumin the experimental and calculated C(T ) are close to eachother. However, the specific heat around the broad maximumis larger in the calculated result. Note that the experimentalC(T ) contains not only the magnetic specific heat but alsothe lattice specific heat [30]. We calculated C(T ) for isolatedspin tetramers with several sets of exchange parameters. Thetemperature of the maximum depends on the J1 and J2 values,whereas the height of the maximum is independent of thevalues. Similar results were obtained in other spin systemssuch as the AF uniform spin- 12 chain [31]. Therefore, we didnot estimate the J1 and J2 values in the specific-heat data.174421-3HASE, MATSUMOTO, MATSUO, AND KINDO PHYSICAL REVIEW B 94, 174421 (2016)FIG. 5. Magnetic-field dependence of the magnetization ofCuInVO5 (thick red lines). Green, red, and blue lines indicatethe magnetization calculated for the total, Cu1, and Cu2 spins,respectively, in the interacting spin- 12 tetramer model in Fig. 2(c).Black lines indicate the magnetization calculated for an isolatedspin- 12 tetramer. The values of the exchange interactions are listedin Table I. (a) Magnetization at 1.3 K. (b) Magnetization at 30 K.The black lines in Fig. 5 indicate M(H ) calculated for anisolated spin tetramer with the same J1 and J2 values. Thecalculated M(H ) is similar to the experimental M(H ) at 30 K,whereas the isolated spin tetramer model fails to reproducethe experimental M(H ) at 1.3 K. We could not find a set ofexchange parameters that reproduced the experimental M(H )at 1.3 K on the basis of an isolated tetramer.The agreement between the experimental and calculatedresults in the susceptibility above 30 K indicates that the spinsystem in CuInVO5 consists of spin tetramers with J1 = 240and J2 = −142 K. To stabilize the ordered state, intertetramerinteractions must exist in CuInVO5. Intertetramer interactionshave a greater effect on the magnetization at lower T . There-fore, the discrepancy between the experimental results andthose calculated for the isolated spin tetramer appears at low T .The magnetic structure of CuInVO5 has not yet been reported.It is difficult to determine which intertetramer interactions areeffective. Therefore, we assumed the simple model shown inFig. 2(c) and calculated M(H ) using the tetramer mean-fieldTABLE I. Values of exchange interaction parameters and g value.We used the central values for the calculations of the magneticsusceptibility in Fig. 4, the magnetization in Fig. 5, and theeigenenergies in Fig. 6.J1 (K) J2 (K) Jeff (K) g240 ± 20 −142 ± 10 30 ± 4 2.09 ± 0.02theory. Since multiple intertetramer interactions are expectedin CuInVO5, Jeff is the effective interaction between tetramers.As described below, the magnetic moment on Cu1 sites issmall in the spin tetramer with J1 = 240 and J2 = −142 K.Therefore, we assumed intertetramer interactions between Cu2spins. The green lines in Fig. 5 indicate M(H ) calculated forthe interacting spin tetramer with J1 = 240, J2 = −142, andJeff = 30 K. The experimental and calculated magnetizationsare in agreement with each other at both 1.3 and 30 K.We were not able to explain the experimental magneticsusceptibility below 30 K using the simple model shown inFig. 2(c) and the tetramer mean-field theory because of thefollowing reason. We evaluated TN = 8.7 K for the simplemodel with the exchange interaction values in Table I using thetetramer mean-field theory. Mean-field theories become lessvalid for calculation of susceptibility when the temperatureapproaches TN owing to strong fluctuations. A Monte Carlosimulation is one of the applicable theories near the transitiontemperature. It requires a set of realistic intertetramer interac-tions. As described, we do not know them. Therefore, we focuson the magnetization curve at low temperatures, where theordered moment becomes substantial and the fluctuations aresuppressed. The mean-field approximation becomes reliable.Therefore, we can reproduce M(H ) at 1.3 K with the effectiveintertetramer interaction Jeff = 30 K. We consider that ourpresent model is idealized and not fully appropriate to explainthe magnetism of CuInVO5. As a future study, we haveto determine an interacting spin- 12 tetramer model that canexplain quantitatively the experimental susceptibility andmagnetizations. We will mention this point later.Figure 6 shows the eigenenergies of the excited statesmeasured from the GS (|02〉 state) in the isolated spin tetramerwith J1 = 240 and J2 = −142 K. The first excited states arethe spin-triplet |13〉 states located at � = 17 K. The conditionfor the appearance of the ordered state (� � Jeff) is satisfied.The second excited states are the spin-quintet |21〉 stateslocated at 205 K. The large energy difference between thefirst and second ESs generates the 12 quantum magnetizationplateau in Fig. 5(a).We roughly estimated the errors of the J1, J2, and Jeffvalues and listed them in Table I. A discrepancy between theexperimental and calculated χ (T ) appears around 80 K whenJ1 deviates from 240 K. The experimental and calculated χ (T )are not in agreement with each other when J1 = 220 or 260 K.A discrepancy between the experimental and calculated χ (T )FIG. 6. Eigenenergies of the excited states measured from theground state (|02 > state) in the isolated spin- 12 tetramer expressedby Eq. (1). The J1 and J2 values are listed in Table I.174421-4MAGNETISM OF THE ANTIFERROMAGNETIC SPIN- . . . PHYSICAL REVIEW B 94, 174421 (2016)appears around 11 K when J2 deviates from −142 K. Thepeak heights of the experimental and calculated χ (T ) are notin agreement with each other when J2 = −132 or −152 K. Themagnetic field at which the 12 magnetization plateau appearsincreases with increasing Jeff . We roughly estimated the errorof Jeff to be ±4 K.The results calculated for spins on the Cu1 and Cu2 sitesare shown in Figs. 4(b) and 5(a). We used the isolated and in-teracting spin tetramers in the calculations of χ (T ) and M(H ),respectively. Cu2 spins show much larger magnetization thanCu1 spins at low T . The maximum χ (T ) around 11 K andthe rapid decrease up to T = 40 K mainly originate fromthe Cu2 spins. As T is increased further, the susceptibilityof Cu1 spins increases up to T = 130 K, whereas that of Cu2spins decreases. Therefore, the total susceptibility shows weakT dependence between 50 and 100 K. The most dominantinteraction is the J1 interaction. The spin state of Cu1 spins issimilar to the singlet state in AF dimers [32–34]. Therefore,the magnetization of Cu1 spins is small at low T . The twoCu2 spins in a tetramer are weakly and antiferromagneticallycoupled to each other through a Cu1-Cu1 dimer in the sametetramer. Thus, the magnetization of Cu2 spins is large.The susceptibility and magnetization of CuInVO5 resemblethose of Cu3(P2O6OH)2, which has spin- 12 trimerized chainsexpressed as the sequence -Cu(1)-Cu(2)-Cu(2)- [35,36]. TheAF exchange interaction is largest between two neighboringCu(2) spins (111 K). The magnetization of Cu(2) spins is smallat low T . In each chain, two Cu(1) spins are weakly coupledto each other through an intermediate Cu(2)-Cu(2) AF dimer.The magnetization of Cu(1) spins is large.In CuInVO5, the low-energy triplet excitation is expectedto have a finite gap above TN as in Cu2CdB2O6 [21]. Whenthe temperature is decreased, the gap closes at TN and thetriplet excitation splits into a longitudinal mode and twofolddegenerate transverse modes at T < TN. Slightly below TN, theordered moment is small and the longitudinal mode is expectedto be in the low-energy region (on the order of 1 meV). Thus,the ordered phase in CuInVO5 corresponds to the pressure-induced ordered phase in TlCuCl3 [1,2,9,10,12] and KCuCl3[6,11]. CuInVO5 may be useful for studying the longitudinalmode under the atmospheric pressure.The magnetic structure is necessary to calculate susceptibil-ity and magnetic excitations. In future, we will determine themagnetic structure of CuInVO5 by neutron powder-diffractionexperiments. 115In atoms (natural abundance 95.7%) stronglyabsorb neutrons [the thermal absorption cross section is 202(2)barn for 0.0253 eV]. A thin sample with a large area is nec-essary for neutron-diffraction experiments to decrease effectsof neutron absorptions. Powder is filled between two coaxialcylinders with different diameters (a double-wall container) toobtain a thin sample. It is expected to be possible to obtaindiffraction patterns to determine the magnetic structure.We will confirm the signs of J1 and J2 from the mag-netic structure. The value of |J2/J1| is the same as thatof M1/M2 [20,22]. Here, M1 and M2 are magnitudes ofordered magnetic moments on Cu1 and Cu2 sites, respectively.After determination of the |J2/J1| value, we will evaluateagain J1 and J2 values from the experimental susceptibilityat high temperatures. We will consider which intertetramerinteractions are effective to stabilize the magnetic structure.We will calculate the magnetic susceptibility of more realisticmodels using quantum Monte Carlo techniques. We willconfirm that the spin model for CuInVO5 is an interactingspin- 12 tetramer model. It is difficult to observe magneticexcitations by INS experiments because of the strong neutronabsorption by 115In atoms. We intend to form single crystals ofCuInVO5 and perform Raman scattering experiments on them.We expect to observe one-magnon Raman scattering indicatinglongitudinal-mode magnetic excitations as in TlCuCl3 [8] andKCuCl3 [11].IV. CONCLUSIONWe measured the temperature dependence of the mag-netic susceptibility and specific heat and the magnetic-fielddependence of the magnetization of CuInVO5. An antiferro-magnetically ordered state appears below TN = 2.7 K. Weobserved a 12 quantum magnetization plateau above 30 T at1.3 K. An isolated antiferromagnetic spin- 12 tetramer modelwith J1 = 240 and J2 = −142 K can closely reproducethe magnetic susceptibility above 30 K. We were able toexplain the magnetization curves using the interacting spintetramer model with the effective intertetramer interactionJeff = 30 K. We consider that the probable spin model forCuInVO5 is an interacting spin- 12 tetramer model. 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