Classical fluctuations and anisotropy in quasi-one-dimensional antiferromagnets

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  • Classical fluctuations and anisotropy in quasi-one-dimensional antiferromagnets

    P. Santini, Z. Domanski, J. Dong,* and P. ErdosInstitute of Theoretical Physics, University of Lausanne, CH-1015 Lausanne, Switzerland

    ~Received 25 January 1996!

    The magnetization of the easy-plane quasi-one-dimensional antiferromagnets CsMnBr3 and RbMnBr3 in thepresence of an external magnetic field can be well interpreted by minimizing the classical magnetic energy.However, this type of calculation cannot reproduce the observed anisotropy of the magnetization at highmagnetic fields, which therefore is usually attributed to quantum effects. We show that the anisotropy isalready present at the classical level, provided thermal fluctuations are taken into account.@S0163-1829~96!06333-3#


    We develop a theory of the anisotropic magnetic proper-ties of quasi-one-dimensional~quasi-1D! compounds withthe ABX3 structure (A5alkaline metal cation,B5cation ofthe 3d group,X5halogen anion!. The magnetic lattice ofBions is hexagonal, forming a triangular lattice in thea-bplane, and chains parallel to thec axis. The superexchangecoupling between different chains is much weaker than theintrachain exchange, whence the quasi-1D nature of thesesystems. The intrachain coupling is antiferromagnetic inmostABX3 compounds, but it can be also ferromagnetic, asin CsNiF3 ~Ref. 1! or CsCuCl3.


    Studies on quasi-1D antiferromagnets have been stimu-lated by Haldanes conjecture4 about the singlet nature of theground state of 1D antiferromagnetic chains with integerspin.5 An inelastic-neutron-scattering experiment on a com-pound of theABX3 family, CsNiCl3,

    6 gave support toHaldanes theory.

    The great interest in these compounds stems also from thenoncollinear magnetic order occurring at low temperature,due to the weak interchain couplings.7,8 Various experimentsindicate that most of these compounds experience triangularmagnetic ordering, with adjacent spins in the basal planeforming an angle of about 120, while adjacent spins alongthe chains are antiparallel. Relativistic single-site anisotropicinteractions select the chain direction to be easy~as inRbNiCl3 or CsNiCl3) or hard ~as in CsMnBr3 orRbMnBr3!. The noncollinear nature of the magnetic state,and the competition between anisotropic single-site interac-tions and interchain exchange are responsible for the inter-esting reorientation phenomena observed when an externalmagnetic field is applied.914

    In the present paper, we deal with CsMnBr3 andRbMnBr3. For these two systems, the magnetization processin an external field is reproduced very satisfactorily9,12,14bycalculating the spin configurations which minimize the clas-sical energy of the system. There is, however, a qualitativefeature of the magnetization curves which is not reproducedby these calculations : for magnetic fields strong enough toovercome the 3D ordering, the dependence of the magneti-zation on the direction of the applied field appears to bemuch stronger than predicted. It was proposed15 that thiseffect is due to the quantum nature of the spins. This pro-

    posal is supported by the observation that in the linear spinwave approximation the quantum ground state appears to bemuch more anisotropic than the classical ground state.

    We will show in the following that it is not necessary toinvoke quantum effects to interpret the anisotropy, providedthat thermal fluctuations are included in the classical calcu-lation.


    We consider the following Hamiltonian for the spin de-grees of freedom of Mn ions:


    i , jSiSj1D(

    iSi ,z2 2h(




    2(i , j

    SSiSjr i j3 2 3~Sir i j !~Sjr i j !r i j5 D . ~2.1!The Lande factorg is equal to 2, andS55/2. Here,z is thedirection of the chains, andh5gmBH, with H the appliedmagnetic field. The first and second terms are the exchangeinteractions for spins along the chains, and between thechains, respectively. For the two compounds under consider-ation,J, J8.0. The third term is a weak electric quadrupolesingle ion anisotropy (D!J), coming from second-orderperturbation theory with the spin-orbit interaction connectingthe isotropic ground term of the ion with excited terms hav-ing L0. This anisotropy is of the easy-plane type (D.0)for CsMnBr3 and RbMnBr3. The fourth term is the Zeemanenergy and the last term is the dipolar interaction, withr i jthe vector connecting spinsi and j . We keep only termsrepresenting the dipolar interaction between nearest neigh-bors ~NN! along the chains. The dipolar interaction is oftennot considered in the study of antiferromagneticABX3 com-pounds, since it is expected to be important only for ferro-magnets. However, the anisotropy energy associated with theNN dipolar interaction has the same order of magnitude asthe single-ion anisotropy energy. Therefore it is not justifiedto neglect it in calculating the anisotropy of the magnetiza-tion.

    The NN interaction can be divided into an isotropic andan anisotropic Ising part:


    540163-1829/96/54~9!/6327~6!/$10.00 6327 1996 The American Physical Society

  • Hdip5d(i

    S 13SiSi112Si ,zSi11,zD , ~2.2!with d5(3g2mB

    2/R3). The distanceR between nearest neigh-bors is near 3.26 in both compounds. The isotropic part ofHdip gives only a slight renormalization ofJ in Eq. ~2.1!.Note that the small ferromagnetic Ising anisotropy is notexpected to change the easy-plane character.16 As far as thelong-range part of the dipolar interaction is concerned, itwould not be possible to include it in our treatment of theHamiltonian. Anyway, it is much weaker than the NN part,and we believe that it does not add any new qualitative fea-ture, given the antiferromagnetic character of these com-pounds.

    A more complete Hamiltonian should include also afourth-order~electric hexadecapole! single-ion axial anisot-ropy term, of the formD8( iSi ,z

    4 and higher-order exchangeinteractions along the chains. The fourth-order anisotropy isnot necessarily negligible with respect to the second-orderone, but is usually not considered in these compounds sinceit is not expected to add any new qualitative feature. Wehave found in our calculation that its effect on the magneti-zation is additive to that of the second order anisotropy, sothat we will not consider it in the following.

    Exchange interactions of more than second order areknown to exist from the analysis of the inelastic-neutron-scattering spectra of CsMnxMg12xBr3.

    17 Also, it was shownin Ref. 18 that biquadratic exchange plays a role in the high-temperature zero-field magnetic susceptibility of CsMnBr3.We will analyze its effect on the low-temperature finite-fieldmagnetic response in Sec. V.

    Whenh50, a single antiferromagnetic chain of quantumspins with easy-plane single-site anisotropy and small ferro-magnetic Ising couplings is not expected to possess long-range order, not even atT50.16 The weak interchain cou-pling in Eq. ~2.1! produces 3D long-range triangularantiferromagnetic order~characterized by six different sub-lattices!, with the transition temperature given approximatelyby19


    3kBS~S11!~JJ8!1/2. ~2.3!

    The behavior withh0 andT50 was studied classically inRefs. 9 and 12, with the dipolar interaction neglected. Whenthe anisotropyd5D/3J8.1, as it is the case for the twocompounds under consideration, it was found that forhalong thez axis, as the field increases the spins cant conti-nously to the field direction, with their transversal compo-nents conserving the 120 structure. Full saturation occursfor h[u hu.8SJ118SJ812SD. When the field is appliedin the basal plane, the situation is more complex because thesix sublattices are not equivalent with respect to the fielddirection. The sublattice magnetizations undergo a reorienta-tion, with the spins remaining in the basal plane, and ath5hc5(48S

    2JJ8)1/2 a collapse of two pairs of sublatticesoccurs. The value ofhc is about 0.45 meV in RbMnBr3, and0.75 meV in CsMnBr3. Forh.hc the reorientation proceedsas in the case h along z, with saturation forh.8SJ118SJ8. However, the magnetization withh in theplane is greater by a factor of aboutD/4J than the magneti-

    zation withh alongz. This theoretical result is the oppositeof what is experimentally observed. Also, the measured an-isotropy (;5210%) ~Refs. 9, 12, and 14! is about one orderof magnitude larger than in this classical theory.

    We have done the classicalT50 calculation with the di-polar interaction included, and we found that this interactionintroduces only small quantitative differences@for instance,the anisotropy parameter becomesd5(D1d)/3J8#. Themagnetization per Mn ion forh.hc is given withh parallelto thez axis by


    8J12D22d1O~J8/J2!, ~2.4!

    whereas forh perpendicular to thez axis one has


    8J1O~J8/J2!. ~2.5!

    For CsMnBr3 and RbMnBr3 d.D, soMhiz.Mh'z , as isexperimentally observed. However, the calculated anisotropyis still much smaller than the measured one.

    From this, it might be concluded, as in Refs. 12 and 15,that the classical theory is not adequate to describe the an-isotropy observed, and that quantum fluctuations have to beinvoked. Such a conclusion is not warranted, because noexperiments have been made belowT051.5 K. SincekBT0is larger than the energy associated with the anisotropicterms in Eq.~2.1!, calculations done for the anisotropy set-ting T50 are not conclusive. We will show that the anisot-ropy calculated from the classical model atT;1.5 K is in-deed much greater than atT50.


    In the following, we consider only magnetic fieldsh.hc , for which the 3D ordering associated with theJ8term in Eq.~2.1! is overwhelmed by the external magneticfield. Indeed, forh.hc , Eqs.~2.4! and~2.5! are the same asthose obtained by a one-dimensional calculation using asingle magnetic chain, apart from residual 3D contributionsO(J8/J2). We restrict ourselves to a 1D system, by settingJ850 in Eq. ~2.1!. We will estimate the leading correctiondue to these omitted terms in Sec. VI. For a single chain, theclassical partition function at temperatureT51/kBb is

    Z5E )i

    dV i4p )l exp@2bH~ l ,l11!#. ~3.1!

    The spin at sitei is described by two polar angles (u i ,f i),and dV i5sin(ui)duidfi . H( l ,l11)5H(u l ,f l ,u l11 ,f l11)is the contribution toH coming from thel th bond. Since inour numerical calculation we discretize the variables(u,f), any explicit calculation of the integrals appearing inthis section is made in terms of discrete sums.

    We calculateZ by the integral operator method. Thismethod can be applied to any one-dimensional classical sys-tem with short-range interactions. It is usually formulatedusing periodic boundary conditions,20,21and in this case it isthe continous version of the transfer-matrix method for theIsing model. We reformulate the method for open boundaryconditions. These are equivalent to periodic conditions for a


  • very large system, but allow to associate the largest eigen-vector of the integral operator to a reduced density. Otherclassical transfer-matrix studies ofABX3 compounds may befound in Refs. 22 and 23.

    We start at one end of the chain~site 0! and integrateexp(2bH) over successive spins, except the first and the last.The result of the firstN21 integrations is

    CN~uN ,fN ,u0 ,f0!5E )l51

    N21dV l4p )k50




    This expression will be expanded using the eigenvectors ofthe following symmetric homogeneous Fredholm equation ofthe second kind:

    lC~u,f!5E dV84p exp@2bH~u8,f8,u,f!#C~u8,f8!.~3.3!

    Let us calll l and hl(u,f) the eigenvalues and the corre-sponding normalized eigenvectors. Note, that in the theory ofintegral equations, usually the reciprocals ofl l are called theeigenvalues. Then the kernel is expanded as


    l lhl~u,f!hl~u8,f8! ~3.4!

    and Eq.~3.2! becomes

    CN~uN ,fN ,u0 ,f0!5(l

    l lNhl~uN ,fN!hl~u0 ,f0!. ~3.5!

    The partition function of a chain ofN11 spins is

    ZN5E dVN4p dV04p CN~uN ,fN ,u0 ,f0! ~3.6!5(

    ll lNS E dV4p hl~u,f! D



    and in the limit

    Z5 limN`

    ZN5S E dV4p hm~u,f! D2

    lmN , ~3.8!

    with lm the largest eigenvalue.We remark that the eigenvectorhm(u,f) is proportional

    to the reduced densityr1(u,f) for the free site of a semi-infinite chain, i.e., the density integrated over all degrees offreedom apart from those corresponding to the free site:

    r1~uN ,fN!}E dV04p CN~uN ,fN ,u0 ,f0!5(

    ll lNhl~uN ,fN!E dV04p hl~u0 ,f0!


    and in the limit

    r1~u,f!5 limN`

    r1~uN ,fN!}lmNhm~u,f!E dV84p hm~u8,f8!.


    The reduced density for any site of an infinite chain can bederived by regarding the chain as composed of two semi-infinite chains. The reduced density is then proportional tohm2 (u,f).While in the caseD50, h50 Eq. ~3.3! may be solved

    exactly in terms of angle spheroidal wave functions,21 noanalytic solution exists in the general case, so that a numeri-cal calculation has to be made.

    Whenhiz, the integral operator in Eq.~3.3! is invariantunder rotations aboutz, implying that its solutions have toform a basis for irreducible representations of the two-dimensional rotation group.

    Since hm(u,f) is a reduced density, it is in principlephysically observable, and it has to belong to the identityrepresentation and not to depend onf. By using this prop-erty, one can integrate overf8 in Eq. ~3.3! for any given pair(u,u8) and reduce the integral equation to a one-dimensionalone:

    lC~u!5E du8 sin~u8!4p K~u,u8!C~u8!, ~3.11!with

    K~u,u8!5E df8exp@2bH~u8,u,f82f!#. ~3.12!Because of the cylindrical symmetry,K does not depend onf.

    Equation~3.11! is discretized by using a Gaussian quadra-ture for the integral:



    WtK~us ,u t!sin~u t!

    4pC~u t!, ~3.13!

    whereus areM Gaussian abscissae, andWs the correspond-ing Gaussian weights. ForT.1 K, takingM5100 is largeenough to give a very good convergence. This matrix eigen-value equation is easily symmetrized by multiplying it by@Wssin(us)#

    1/2, and may be diagonalized by a standard rou-tine.

    When the magnetic field is not along thez axis, the cy-lindrical symmetry is lost, and the full two-dimensional in-tegral equation has to be solved. The integral is discretizedby using a two-dimensional Gaussian quadrature formula.The lower the temperature, the larger the number of Gauss-ian points necessary to have accurate results, of the order of104 for T;1.5 K. Since only the largest eigenvalue isneeded, we used the Lanczos method24 to diagonalize thelarge matrix eigenproblem obtained through the discretiza-tion.

    The l th component of the magnetization per spinmay be calculated either by numerical evaluation ofMl52] f /]Hl , with f the free energy per spin, or directly fromthe reduced densityhm

    2 (u,f) as


  • Ml5gmB*dVSl~u,f!hm

    2 ~u,f!

    *dVhm2 ~u,f!

    . ~3.14!

    We have checked that these two procedures give the sameresult. Also, we have calculated by this method the zero-fieldsusceptibility forD50, d50 and verified that its value isthe same as obtained from the analytic expression.25

    Note, that the free energy and the magnetization calcu-lated by this method can be made to be as close as desired totheir exact theoretical values by simply increasing the num-ber of Gaussian points in the discretization of the integral.The error of the magnetization in the figures we present issmaller than the width of the lines.


    We show in Figs. 1 and 2 the magnetization as a functionof h for h parallel and perpendicular toz, respectively. Themagnetic field is just abovehc . The parameters are thosederived by inelastic neutron scattering.26,27These parametersresult from fitting a spin-wave model to the measured fre-quencies. As discussed in Ref. 12 they are not necessarilyexactly the same as those which would be used in a classicalmodel to fit the magnetization experiments. They are, how-ever, adequate to allow one to see the qualitative effect oftemperature on the anisotropy of magnetization.

    It appears that while atT50 single-ion and dipolar an-isotropy have opposite effects on the anisotropy of magneti-zation, the effects tend to become the same asT increases. Infact, if D50, d0, one has alwaysM i.M' . If D0,d50 one hasM i,M' for very low T, but M i.M' forT*T150.1 K. T1 appears to be roughly proportional toD.Moreover, it appears thatM' is very close to the magneti-zation of the isotropic model ifT is not too high.

    The main mechanism affecting the magnetization at finiteT is fluctuations of the canting angle around its value atT50. These fluctuations tend to decrease the magnetization.Whenh 'z, the anisotropic terms do not influence fluctua-tions of the canting angle~since the spins remain in thexyplane!, which therefore remain the same as for the isotropicmodel. These anisotropic terms influence only fluctuations ofthe angle specifying the direction of the spins with respect tothez axis. However, these fluctuations affect the magnetiza-tion only in second order, so that if they are not too large,i.e., if T is not too high, they give only a small contribution.For this reason, the magnetization forh'z is very close tothe magnetization forD50, d50 for smallT.

    Whenh iz, fluctuations of the canting angle are energeti-cally more costly than forh 'z because they increase boththe single-ion and the dipolar anisotropy energy. For thisreason they tend to be suppressed by a finiteD or d, with theresult thatM i.M' . Note, that the effects ofD andd are notadditive, since the results for (D0,d50), (D50,d0),and (D0,d0) are similar. Our calculation predicts thatM i2M' should increase withT up to;5 K, and then de-crease slowly.

    The calculated anisotropy compares well with the experi-mental one for both RbMnBr3 @Fig. 2~a!# and CsMnBr3@Fig. 2~b!#, although the classical model with the parametersderived from scattering experiments cannot reproduce theabsolute value of the magnetization. The difference is likelyto be a quantum effect, although it is not at all obvious whythis effect should be greater in CsMnBr3 than in RbMnBr3,whose parameters do not differ very much.

    As discussed at the beginning of this section, it would bepossible to fit the value ofJ to obtain the correct absolutevalue of the magnetization, as was done atT50 in Ref. 14.

    FIG. 1. Calculated magnetizationM in units of the saturationmagnetizationMS as a function of the temperatureT with the mag-netic fieldH perpendicular~lower curve! and parallel~upper curve!to z. The dashed line corresponds to the isotropic model (D50,d50). ~a! H54 T, J50.82 meV, d50, D50.009 meV. ~b!H54 T, J50.82 meV,d50.018 meV,D50. For the meaning ofthe parameters, see~2.1! and ~2.2!.

    FIG. 2. Same as Fig. 1, but~a! H54T, J50.82 meV,D50.009 meV @parameters for RbMnBr3 ~Ref. 26!#, andd50.018 meV. MeasuredM i ~open diamond!, andM' ~filled dia-mond! for RbMnBr3 ~Ref. 14!. ~b! H56.5 T, J50.88 meV,D50.014 meV @parameters for CsMnBr3 ~Ref. 27!#, andd50.018 meV. MeasuredM i ~open diamond!, andM' ~filled dia-mond! for CsMnBr3 ~Ref. 12!.



    As we mentioned above, the existence of biquadratic in-teractions between spins along the chains has been demon-strated by inelastic-neutron-scattering experiments onCsMnxMg12xBr3 with 0.14

  • *Permanent address: Department of Physics, University of Nan-jing, 210 008 China.

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