M. Nishiyama et al- Magnetic ordering and fluctuation in Kagome lattice anti-ferromagnets, Fe and Cr jarosites

8/3/2019 M. Nishiyama et al- Magnetic ordering and fluctuation in Kagome lattice anti-ferromagnets, Fe and Cr jarosites

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Magnetic ordering and fluctuation in

Kagome lattice anti-ferromagnets,

Fe and Cr jarositesM. Nishiyama, T. Morimoto, S. Maegawa, T. Inami, and Y. Oka

Abstract: Jarosite family compounds, KFe3(OH)6(SO4)2 (abbreviated to Fe jarosite) andKCr3(OH)6(SO4)2 (Cr jarosite) are typical examples of Heisenberg anti-ferromagnets onthe Kagome lattice and have been investigated by means of magnetization and NMRexperiments. The susceptibility of Cr jarosite deviates from the CurieWeiss law due to theshort-range spin correlation below about 150 K and shows the magnetic transition at 4.2 K,while Fe jarosite has the transition at 65 K. The measured susceptibility fits well with the

calculated one on the high-temperature expansion for the Heisenberg anti-ferromagnet onthe Kagome lattice. The values of the exchange interactions of Cr jarosite and Fe jarositeare derived to be JCr = 4.9K and JFe = 23 K, respectively. The 1H-NMR spectra of Fe

jarosite suggest that the ordered spin structure is the q= 0 type 120 configuration with +1chirality. The transition is considered to be caused by a weak single-ion type anisotropy. Thespin-lattice relaxation rate, 1/T1, of Fe jarosite in the ordered phase decreases sharply withdecreasing temperature and can be well explained by the two-magnon process of spin wavewith the anisotropy.

PACS No.: 75.25+z

Rsum : Les composs de la famille des jarosites, KFe3(OH)6(SO4)2 (not jFe) etKCr3(OH)6(SO4)2 (not jCr), sont des exemples typiques de modles antiferromagntiques deHeisenberg sur rseau de Kagom et ont t tudis ici via des mesures de magntisation etRMN. La susceptibilit de jCr dvie de la courbe de CurieWeiss cause de corrlations despin courte porte sous 150 K et montre une transition magntique 4,2 K, alors que jFe aune temprature de transition 65 K. haute temprature, les mesures de susceptibilit sonten bon accord avec les valeurs calcules dans une expansion du modle antiferromagntiquede Heisenberg sur rseau de Kagom. On en dduit les valeurs de linteraction dchangepour les jarosites Fe et Cr, JCr = 4.9 K et JFe = 23 K respectivement. Le spectre 1H-NMRde jFe suggre que la structure ordonne de spin est une configuration 120 de type q=0avec chiralit +1. Nous pensons que la transition est cause par une anisotropie faible detype un ion. Le taux de relaxation du rseau de spin, 1/T1, de jFe dans la phase ordonnedcrot rapidement avec la diminution de la temprature et peut sexpliquer trs bien par unmcanisme deux magnons de londe de spin avec lanisotropie.

[Traduit par la Rdaction]

Received August 1, 2000.AcceptedAugust 8, 2001. Published on the NRC Research Press Web site on Novem-ber 6, 2001.

M. Nishiyama, T. Morimoto, and S. Maegawa.1 Graduate School of Human and Environmental Studies,Kyoto University, Kyoto, 606-8501, Japan.T. Inami. Synchrotron Radiation Research Center, Japan Atomic Energy Research Institute, Mikazuki, Hyogo679-5148, Japan.Y. Oka. Faculty of Integrated Human Studies, Kyoto University, Kyoto, 606-8501, Japan.

1 Corresponding author .

Can. J. Phys. 79: 15111516 (2001) DOI: 10.1139/cjp-79-11/12-1511 2001 NRC Canada


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Fig. 1. Temperature dependence of the susceptibilityfor KFe3(OH)6(SO4)2.

Fig. 2. Temperature dependence of the susceptibilityfor KCr3(OH)6(SO4)2.

1. Introduction

Anti-ferromagnets on the Kagome lattice are geometrically frustrated systems with the compe-tition from the anti-ferromagnetic interactions between neighboring spins. The anti-ferromagnets onthe triangular lattice have also been well-known as frustrated systems. While the triangular lattice hassix nearest neighbors and the adjacent triangles on the triangular lattice share one side, or two latticepoints, in common, the Kagome lattice has only four nearest neighbors and the adjacent triangles shareonly one lattice point in common. Thus, the spins on the Kagome lattice suffer smaller restrictionsfrom neighboring spins than the spins on the triangular lattice. The Heisenberg anti-ferromagnet on theKagome lattice exhibits infinite and continuous degeneracy of the ground state. Theoretically, the two-dimensional isotropic Heisenberg Kagome lattice anti-ferromagnet has no magnetic phase transition

at finite temperature. The thermal or the quantum fluctuation, however, resolves the degeneracy of theground state [1, 2]. This effect induces the coplanar spin arrangement and two Nel states have beendiscussed as candidates for the spin structure at zero temperature. One is a q=0 type and the other isa

3

3 type of the 120 structure. Theoretical studies suggest that the latter is favored slightly [1].When a weak Ising-like anisotropy is introduced into the Hisenberg Kagome lattice anti-ferromagnet,the system has a magnetic phase transition at finite temperature and has a peculiar spin structure [3]. Asmall perturbation, anisotropy, or distortion can resolve the degeneracy of frustrated systems and causethe phase transition.

The jarosite family compounds, KCr3(OH)6(SO4)2 andKFe3(OH)6(SO4)2, are examples of Heisen-berg Kagome lattice anti-ferromagnets [46]. We have investigated these powder samples by measure-ment of the magnetization and 1H nuclear magnetic resonance experiments to clarify the magnetic

transition and the spin fluctuation in the Heisenberg Kagome lattice anti-ferromagnet. The magneticions Cr3+ andFe3+ have spins 3/2 and 5/2, respectively. The ions form the Kagome lattice on the c-planeand interact anti-ferromagnetically with each other. The protons observed by NMR locate nearly on theKagome planes. Adjacent Kagome planes are separated by nonmagnetic ions, S, O, H, and K with longinteraction paths, so that the interplane magnetic interaction is very weak.

2. Experimental results

The magnetization was measured using a SQUID magnetometer in the temperature range between2 and 300 K. Figures 1 and 2 show the susceptibility of Fe jarosite and Cr jarosite, respectively. Thesusceptibility of Fe jarosite has the cusp at TN[Fe] = 65 K, while the susceptibility of Cr jarosite increases

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Fig. 3. Observed and calculated inverse susceptibil-ity of KCr3(OH)6(SO4)2. The closed circles are ex-perimental data of 1 under an external magneticfield of 500 Oe. The result calculated from the high-temperature series expansion up to the eighth order isshown by the dotted curve. The result obtained fromPad [4,4] approximants is shown by the continuouscurve. The broken curve shows the CurieWeiss law.

Fig. 4. Proton spin-lattice relaxation rate ofKFe3(OH)6(SO4)2. The closed circles are experimen-tal data at 75.1 MHz. The continuous curve shows thetemperature dependence of 1/T1 calculated from thetwo-magnon process using the energy gap of 25 K.

abruptly below 4.2 K.

The susceptibility of the Heisenberg Kagome lattice anti-ferromagnet has been calculated on thehigh-temperature expansion up to the eighth order and the result is extended to the lower temperature bythePad[4,4]approximants[7]. Figure3 shows thecomparisonbetween theexperimental andtheoreticalinverse susceptibility for Cr jarosite. The experimental values deviate clearly from the CurieWeiss lawbelow about 150 K and fit very well with the calculated curves above 20 K. The values of the exchangeinteraction for Cr jarosite and Fe jarosite were obtained: JCr/kB = 4.9 K and JFe/kB = 23 K. Thetheoretical susceptibility deviates remarkably from the CurieWeiss law below about 8JS(S+ 1)/kBdue to the development of short-range spin correlation [7]. For Fe jarosite, the value of 8JS(S+ 1)/kBis about 1600 K, which is far away from the experimental temperature region. For the other Kagomelattice anti-ferromagnet, SrCr8x Ga4+xO19, the value is about 860 K and is also large [7,8]. The valuefor Cr jarosite is about 150 K, which is in the experimental temperature region. Thus Cr jarosite is agood example to show the deviation from the CurieWeiss law owing to the small J and S values.

The 1H-NMR spectrum of Fe jarosite has a sharp peak in the paramagnetic phase, while the spectrumbelow 65 K becomes broader andshows a typical pattern for the powder anti-ferromagnets [9]. This tran-sition temperature coincides with the susceptibility data and the spectrum indicates anti-ferromagneticordering.

The spin-lattice relaxation rates, 1/T1, of1H in Fe jarosite are shown in Fig. 4. The rate 1/T1 inthe paramagnetic phase slightly increases as the temperature approaches TN[Fe]. The rate in the orderedphase decreases sharply as the temperature is lowered.

The NMR spectrum for Cr jarosite has a sharp peak in the paramagnetic phase, while the half widthincreases below 4.2 K. The rate 1/T1 for Cr jarosite is almost independent of the temperature in theparamagnetic phase, however, it decreases below 4.2 K as the temperature is lowered.

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3. Discussion

The NMR spectrum of Fe jarosite indicates that all the protons feel the same magnitude of theinternal dipolar field from the Fe3+ spins. This suggests that the ordered spin structure is the q = 0type 120 configuration with +1 chirality. If there existed a magnetic alignment with 1 chirality, twokinds of proton sites with different magnitudes of the internal field must exist. The neutron diffractionexperiments confirmed this magnetic structure and revealed that the spins are directed to or from thecenter of the triangle on the Kagome lattice [10].

This spin structure is considered to be caused by the single-ion type anisotropy of magnetic ions.Each magnetic ion is surrounded by an octahedron composed of six oxygens, whose principal axis cantsabout = 20 from the c-axis towards the center of a triangle and the octahedron is distorted slightly.The distortion and the canting of the octahedron must cause the single-ion anisotropy. The spin systemcan be expressed as

H = 2J

Si Sj + D

i

(Sz

i )2 E

i

{(Sxi )2 (Sy

i )2} (1)

where the local coordinate (x, y, z) for each ion is determined by the relation with each surroundingoctahedron, D > 0 and E > 0. In the case of

E > Dsin2

1 + cos2 (2)

the spin structure with the minimum energy for the system is the q=0 type with +1 chirality and thespins are directed to or from the center of the triangle on the Kagome lattice. This means that the systemcorresponds effectively to the two-dimensional Ising magnet, which has the magnetic phase transition atfinite temperature [10]. The two-dimensional ordering in the plane would induce the three-dimensionalordering, when there is at least nonzero interplane interaction.

The relaxation rate in the ordered phase can be analyzed by the two-magnon process of the spin wave

in the Heisenberg Kagome lattice anti-ferromagnet. The relaxation rate by the two-magnon process isexpressed as [11]

1

T1=

22e

2n

2i,j

Gij

m0

1 +

m

2 e/ kBT(e/ kBT 1)2 N()

2 d (3)

where e and n are the gyromagnetic ratios of the electronic and nuclear spins, Gij is the geometricalfactor of the dipolar interaction, m is the maximum frequency, 0 is the energy gap, and N() is thestate density of magnons. The dispersion relation of magnons in a system ofq=0 type spin structure hasbeen obtained by Harris et al. [7]. We adapt their method for Fe jarosite by introducing the anisotropy.The energy gap of the dispersion curve, which is effective for the relaxation, is given as

= 2S

6J+ (D + E)2cos2 1E + E cos2 D sin2 (4)Applying thelong-wave approximation forthedispersionrelation, therelaxation rate is expressedas [12]

1

T1=

22e

2n

2i,j

Gij9kB

1

(T2m T20 )3T5

Tm/TT0/T

x2

T0

T

2x2 +

Tm

T

2 ex(ex 1)2 dx (5)

where Tm = m/kB and T0 = 0/kB. We calculated the temperature dependence of 1/T1 and thecalculated values are shown in Fig. 4 by the continuous curve. The agreement between the experimental

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Fig. 5. Magnetization curve of KCr3(OH)6(SO4)2 at 2.0 K.

data and the calculated values is fairly good and we get 25 K for the energy gap. The anisotropies, Eand D, are estimated by using (2) and (4) as 0.0045 < E/J < 0.066 and D/J = 16E/J 0.068.

Now we discuss Cr jarosite. As is seen in Fig. 2, the susceptibility for Cr jarosite increases sharplybelow 4.2 K. Below this temperature the difference between the susceptibility measured after the zero-field cooling (ZFC) and that measured after the field cooling (FC) was observed. The same behaviorhas been reported by Keren et al. [13]. The magnetization curve was measured at 2.0 K to clarify thisanomaly and is shown in Fig. 5. Here, we find a small hysteresis loop, which suggests the existenceof weak ferromagnetic moments. The difference in the susceptibility between ZFC and FC comesfrom the hysteresis loop below 4.2 K. The long-range ordering has been observed below 4.2 K in ourneutron diffraction experiment for Cr jarosite. Lie et al. have also reported the weak long-range anti-

ferromagnetic ordering observed by the neutron experiments [14]. We conclude that the transition ofCr jarosite at 4.2 K is not a spin-glass-like transition but a magnetic-ordering one.The weak ferromagnetic moment in Cr jarosite is considered to be caused by the canting of the 120

arrangement perpendicular to the c-plane due to the anisotropy. On the other hand, the ferromagneticmoment was not observed for Fe jarosite. These results can be explained by the antiparallel stackingof the net moments on the c-plane for Fe jarosite and the parallel stacking of the net moments for Cr

jarosite. This is consistent with the result from the neutron experiments that the magnetic unit cell inCr jarosite is equal to the chemical unit cell, while that in Fe jarosite is double the chemical unit cellalong the c-axis.

4. Summary

WehaveinvestigatedtheKagomelatticeanti-ferromagnetsKFe 3(OH)6(SO4)2 andKCr3(OH)6(SO4)2by means of magnetization, NMR, and neutron experiments. The susceptibility data of these sam-ples agree well with the susceptibility calculated by the high-temperature expansion for the two-dimensional Heisenberg Kagome lattice anti-ferromagnet. Long-range magnetic ordering occurs at65 K for KFe3(OH)6(SO4)2. The spin structure in the ordered phase is the 120 structure with q = 0,+1 chirality and the direction being to or from the center of the triangle. The order is considered tobe caused by the single-ion-type anisotropies, D and E. This system corresponds effectively to thetwo-dimensional Ising system, which has two-dimensional magnetic ordering at finite temperature. Thefluctuation in the ordered phase is found to be caused by the spin wave. The nuclear spin-lattice relax-ation is governed by the two-magnon process. For KCr3(OH)6(SO4)2, the magnetic transition occurs at

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4.2 K. The transition is not a spin glass one but due to the magnetic ordering. The weak ferromagneticmoments are observed. These arise from the canted 120 spin structure, and all the net moments in theplanes are parallel to the c-axis.

References

1. J.N. Reimers and A.J. Berlinsky. Phys. Rev. B: Condens. Matter Mater. Phys. 48, 9539 (1993).2. A. Chubukov. Phys. Rev. Lett. 69, 832 (1992).3. A. Kuroda and S. Miyashita. J. Phys. Soc. Jpn. 64, 4509 (1995).4. M. Takano, T. Shinjo, and T. Takada. J. Phys. Soc. Jpn. 30, 1049 (1971).5. M.G. Townsend, G. Longworth, and E. Roudaut. Phys. Rev. B: Condens. Matter Mater. Phys. 33, 4919

(1986).6. S. Maegawa, M. Nishiyama, N. Tanaka, A. Oyamada, and M. Takano. J. Phys. Soc. Jpn. 65, 2776

(1996).7. A.B. Harris, C. Kallin, and A.J. Berlinsky. Phys. Rev. B: Condens. Matter Mater. Phys. 45, 2899 (1992).8. G. Aeppli, C. Broholm, A. Ramirez, G.P. Espinosa, and A.S. Cooper. J. Magn. Magn. Mater. 90/91, 255

(1990).9. Y. Yamada and A. Sakata. J. Phys. Soc. Jpn. 55, 1751 (1986).

10. T. Inami, M. Nishiyama, S. Maegawa, and Y. Oka. Phys. Rev. B: Condens. Matter Mater. Phys. 61,12 181 (2000).

11. T. Moriya. Prog. Theor. Phys. 16, 23 (1956).12. S. Maegawa. Phys. Rev. B: Condens. Matter Mater. Phys. 51, 15979 (1995).13. A. Keren, K. Kojima, L.P. Le, G.M. Luke, W.D. Wu, Y.J. Uemura, M. Takano, H. Dabkowska, and

M.J.P. Gingras. Phys. Rev. B: Condens. Matter Mater. Phys. 53, 6451 (1996).14. S.-H. Lie, C. Broholm, M.F. Collins, L. Heller, A.P. Ramirez, Ch. Kloc, E. Bucher, R.W. Erwin, and N.

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M. Nishiyama et al- Magnetic ordering and fluctuation in Kagome lattice anti-ferromagnets, Fe and Cr jarosites