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Journal of Magnetism and Magnetic Materials 316 (2007) e346e348

Magnetic properties of the geometrically frustrated spin- 12 Heisenbergmodel on the triangulated Kagome lattice $

Jozef Strec ka

Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. S afa rik University, Park Angelinum 9, 040 01 Kos ice, Slovak Republic

Available online 3 March 2007

Abstract

Geometrically frustrated spin- 12 quantum Heisenberg model on the triangulated (triangles-in-triangles) Kagome lattice is examined bythe use of variational procedure based on the Bogoliubov inequality. The investigated model system has an obvious relevance inelucidating magnetic properties of a series of polymeric coordination compounds Cu 9X2cpa 6 (X F, Cl, Br and cpa carboxypen-tonic acid). A striking magnetization curve with the 13-plateau implies an appearance of compromised non-collinear spin arrangementwithin the plateau state, whereas a presence of the 59-plateau with a collinear uud spin alignment cannot be denitely ruled out at higherelds. The thermal variation of the susceptibility times temperature data exhibits typical features of the quantum ferrimagnet.r 2007 Elsevier B.V. All rights reserved.

PACS: 05.50.+q; 75.10.Jm; 75.40.Cx; 75.50.Nr

Keywords: Heisenberg model; Triangulated Kagome lattice; Geometric frustration

1. Introduction

The quantum Heisenberg model on frustrated planarlattices represents a long-standing theoretical challenge asit exhibits the variety of exotic non-magnetic ground statesdue to zero-temperature phase transitions driven byquantum uctuations [13]. Another striking featurerelated to the geometrically frustrated planar lattices isbeing the existence of quantized magnetization plateaux inthe low-temperature magnetization curves [2,3]. It isnoteworthy that this peculiar quantum phenomenon hasalready been experimentally veried in the prototypicalexamples of several frustrated quantum spin systems, such

as the triangular lattice compounds CsFe SO 42 [4],Cs 2CuBr 4 [5] and RbFe MoO 42 [6], the Kagome latticecompound [Cu 3titmb 2CH 3COO 6].H 2O [7] and theShastrySutherland lattice compound SrCu 2BO 32 [8].

A series of polymeric coordination compoundsCu 9X2cpa 6 (X F, Cl, Br and cpa carboxypentonic

acid) [9] provides another prominent class of insulatingmagnetic materials with the frustrated lattice topology. Themagnetic lattice of this series constitute copper ionssituated at two non-equivalent positions (see Fig. 1 ).Cu 2 ions with a square pyramidal coordination ( a-sites)form equilateral triangles (trimers), which are inter-connected to one another by Cu 2 ions (monomers) withan elongated octahedron environment ( b-sites) residing thesites of Kagome lattice. Such magnetic structure can bebest summarized as triangulated (triangles-in-triangles)Kagome lattice, since the trimeric units are embedded inthe triangles of Kagome lattice. Experimental studiesreported on this family of compounds have revealed

obvious manifestations of the frustration effect. Thecompounds do not order down to 2 K [10], the magnetiza-tion shows a quantum plateau in a wide range of magneticelds and it does not saturate up to 38 T [11]. On the otherhand, the temperature dependence of reciprocal suscept-ibility indicates two temperature regimes inherent to twodifferent exchange interactions, the Weiss constantamounts Y w 230 K above 150 K, while it decreasesdown to Y w 6K below 50K [11]. Based on theconsiderations of exchange paths, a stronger antiferromag-netic interaction has been assigned to the exchange

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0304-8853/$- see front matter r 2007 Elsevier B.V. All rights reserved.doi: 10.1016/j.jmmm.2007.02.144

$ This work was supported by the scientic grants Nos. VEGA 1/2009/05, APVV 20-005204 and LPP-0107-06.

Fax: +421 55 6222124.E-mail address: jozkos@pobox.sk .

http://www.elsevier.com/locate/jmmmhttp://dx.doi.org/10.1016/j.jmmm.2007.02.144mailto:jozkos@pobox.skmailto:jozkos@pobox.skmailto:jozkos@pobox.skhttp://dx.doi.org/10.1016/j.jmmm.2007.02.144http://www.elsevier.com/locate/jmmm

8/3/2019 Jozef Strecka- Magnetic properties of the geometrically frustrated spin-1/2 Heisenberg model on the triangulated Ka

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coupling J aa

between the a-sites creating trimeric units anda weaker ferromagnetic one to the coupling J ab betweentrimeric a-sites and monomeric b-sites with the overall ratiojJ ab j=J aa % 0:025.

2. Model and method

In the present article, we shall aim to analyze the spin- 12Heisenberg model on the triangulated Kagome lattice inconnection with the magnetism of the series of Cu 9X2cpa 6 compounds. So far, the geometric frustrationinherent to this lattice topology has been studied in termsof the Ising model only [12]. It might be expected, however,that an application of the Heisenberg model would be moreappropriate, since the exchange coupling between Cu 2

ions is almost isotropic. Accordingly, let us dene theHeisenberg model upon the underlying triangulatedKagome lattice ( Fig. 1 )

H J aa Xi ; j S ai S a j J ab Xk ;l S ak S bl H Xi ;d2a ;bS dz i , (1)where S S x ; S y; S zstands for the spin- 12 operator and itsspatial components, the rst two summations extend overtwo different sets of the nearest-neighbor a a and a bpairs, respectively. Finally, the last summation runs overthe total number of sites (3 N ) and this term incorporatesthe effect of external eld.

Since the Hamiltonian (1) cannot be treated exactly forthe Heisenberg model, in what follows we propose avariational procedure based on the trial Hamiltonian

H 0 Xk 2 trimers l aa S ak 1S ak 2 S ak 2S ak 3 S ak 3S ak 1( X

3

i 1

ga S az ki

gb2

S bz ki h i). 2which takes exactly into account stronger antiferromag-netic interaction between the a-sites, while the weaker

ferromagnetic one is perturbatively treated within thevariational mean-eld like treatment. Unknown variationalparameters ( l aa , ga , gb) can be obtained using theBogoliubov inequality G p G 0 h H H 0i0 , where G andG 0 are free energies of the system described by theHamiltonian H and H 0 and h. . .i 0 denotes thermal

average over the ensemble dened by the trial HamiltonianH 0. After minimizing rhs of Bogoliubov inequality thevariational parameters read: l aa J aa , ga 2J ab mb H ,gb 4J ab ma H with both sub-lattice magnetizationdened as

ma 16

3sh 3xga shxga 2e3xl aa shxga

ch3xga chxga 2e3xl aa chxga , 3

mb 12 tanh xgb, 4

x 1=2k BT . The Gibbs free energy then follows from

G =N l aa =2 4J ab ma mb x1fln2ch xgb=2

ln2ch 3xga 2ch xga 4e3xl aa

chxga =3g. 5Eqs. (3)(5) provide a closed-form expression suitable forcalculation of all basic thermodynamic quantities.

3. Results and discussion

To simplify further discussion, let us dene a set of dimensionless parameters t k BT =J aa , h H =J aa , anda j J ab j=J aa , as describing temperature, external eld, andrelative strength of interaction parameters. Next, m2ma mb=3 stands for the total magnetization reducedper one site and we shall set a 0:025 with regard to thesuggestion [11] for the ratio between both couplingconstants of Cu 9X2cpa 6 compounds.

First, let us take a closer look at the low-temperaturemagnetization curve. Fig. 2 a shows the total and sublatticemagnetization as a function of the external eld. As onecan see, the low-temperature magnetization curve exhibitsa plateau in the range of moderate elds. It can easily beunderstood that the plateau at 59 of the saturation momentcorresponds to a spin conguration with the monomeric b-sites aligned to the external-eld direction, while thetrimeric a-sites rest in one of the available uud (""# ) spincongurations ( f 1 or f 3).

As recently pointed out by Dai and Whangbo [13], thegeometric frustration inherent in the triangle motif mightpossibly lead to a compromised non-collinear spinarrangement. In the absence of external eld, the four-folddegenerate ground-state eigenfunctions

f 1 ffiffiffiffiffiffiffiffi1=2p j "#"i j #""i ;f 2 ffiffiffiffiffiffiffiffi1=2p j #"#i j "##i ;f 3 ffiffiffiffiffiffiffiffi2=3p j ""#i ffiffiffiffiffiffi1=6p j "#"i j #""i ;f 4 ffiffiffiffiffiffiffiffi2=3p j ##"i ffiffiffiffiffiffi1=6p j #"#i j "##i ; 6can be mixed to obtain the four non-collinear spinarrangements shown in Fig. 3 that can be described by

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aa

a

b

bb

Fig. 1. A cross-section of the triangulated (triangles-in-triangles) Kagome lattice. Solid (broken) lines schematically reproduce the exchangeinteractions J aa (J ab ).

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means of c i a i f 1 bi f 2 ci f 3 d i f 4 (i 14, themixing coefcients are given in Table III of Ref. [13]). Asa result of the non-collinear ordering, the sub-latticemagnetization ma does not contribute to the totalmagnetization which consequently exhibits the plateau at13 of the saturation moment ( Fig. 2 b). This magnetizationscenario is in agreement with the experimental datareported on the Cu 9X2cpa 6 series [11]. Naturally, it is inquestion whether the 13-plateau persist up to the saturationeld (strictly speaking, the non-collinear order should exist just in the zero external eld), or the 59-plateau develops athigher elds.

Next, we shall focus on the thermal dependence of susceptibility times temperature ( wt) product as shown inFig. 4 . Zero-eld curve displays an obvious manifestationof the quantum ferrimagnet; wt data exhibit upon cooling around minimum followed by a steep low-temperaturedivergence. Since any non-zero eld opens an energy gap,

hence, even small external eld changes the divergence to alocal maximum followed by an abrupt decrease of wt to beobserved nearby zero temperature. The position of thismaximum shifts towards higher temperatures with increas-ing the eld strength until it vanishes above h % 0:05. Notethat the aforementioned scenario is in a qualitative accordwith the behavior of Cu 9X2cpa 6 family [10].

In summary, we have provided the magnetic data of thespin- 12 Heisenberg model on the triangulated Kagome lattice that brings insight into the frustrated magnetismof the series of Cu 9X2cpa 6 compounds. The magnetiza-tion curve of this system turned out to exhibit a quantizedplateau at 13 (

59) of the saturation magnetization with the

non-collinear (collinear) spin arrangement of the trimeric(triangle) units. The 13-plateau observed experimentallyimplies a real existence of the non-collinear spin arrange-ment [11], however, a possibility that the 59-plateau emergesat higher elds cannot be denitely ruled out. To the best

of our knowledge, such fractional value has not beenreported yet neither theoretically, nor experimentally.

References

[1] G. Misguich, C. Lhuillier, in: H.T. Diep (Ed.), Frustrated SpinSystems, World Scientic, Singapore, 2004.

[2] J. Richter, J. Schulenburg, A. Honecker, Lecture Notes in Physics,vol. 645, Springer, Berlin, 2004.

[3] A. Honecker, J. Schulenburg, J. Richter, J. Phys.: Condens. Matter16 (2004) S749.

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T. Ono, et al., J. Phys.: Condens. Matter 16 (2004) S773.[6] L.E. Svistov, et al., Phys. Rev. B 67 (2003) 094434.[7] Y. Narumi, et al., Europhys. Lett. 65 (2004) 705;

Y. Narumi, et al., J. Magn. Magn. Mater. 272276 (2004) 878.[8] H. Kageyama, et al., Phys. Rev. Lett. 82 (1999) 3168;

H. Kageyama, et al., J. Phys. Soc. Japan 69 (2000) 1016.[9] R.E. Norman, et al., J. Chem. Soc.: Dalton Trans. (1987) 2905; R.E.

Norman, et al., Acta Crystallogr. C 46 (1990) 6.[10] S. Maruti, et al., J. Appl. Phys. 75 (1994) 5949;

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M. Mekata, et al., Can. J. Phys. 79 (2001) 1409.[12] J. Zheng, G. Sun, Phys. Rev. B 71 (2005) 052408.[13] D. Dai, et al., J. Chem. Phys. 121 (2004) 672.

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Fig. 2. Total and sub-lattice magnetization as a function of the external magnetic eld for a 0:025 and t 0:05. The intermediate plateau correspondsto: (a) collinear, (b) non-collinear, spin arrangements of the trimeric units.

Fig. 3. Four possible non-collinear spin arrangements of the triangle unitwith the zero total magnetization.

Fig. 4. Thermal variations of the susceptibility times temperature data forseveral values of the external eld.

J. Strec ka / Journal of Magnetism and Magnetic Materials 316 (2007) e346e348e348