*Corresponding author. Tel.: #48-12-617-35-39; fax: #48-12-634-00-10.

E-mail address: kulakowski@novell.ftj.agh.edu.pl(K. Ku"akowski).

Journal of Magnetism and Magnetic Materials 221 (2000) 391}396

Field-cooling induced unidirectional anisotropy in thetwo-dimensional Ising antiferromagnet Rb

2Cu

1~xCo

xF4

B. Kawecka-Magiera, K. Ku"akowski*, A.Z. Maksymowicz

Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, al.Mickiewicza 30, 30-059 Krako&w, Poland

Received 3 May 2000; accepted 10 July 2000

Abstract

Small cluster approximation and Monte Carlo Metropolis algorithm are applied to demonstrate that "eld coolinginduces a unidirectional magnetic anisotropy of small clusters of Cu in Rb

2Cu

1~xCo

xF4. Within the Ising model, this

anisotropy appears as a net magnetization at zero magnetic "eld. The e!ect is due to a coupling between the orbitalordering within clusters of Cu impurities and the antiferromagnetic ordering of Co matrix. ( 2000 Elsevier Science B.V.All rights reserved.

PACS: 05.50.#q; 75.30.Gw; 75.30.Hx

Keywords: Unidirectional magnetic anisotropy; Ising model

1. Introduction

Unidirectional magnetic anisotropy is oftena consequence of chemical inhomogeneities anda sample history. This kind of e!ects is of potentialinterest for applications of micromagnets for mag-netic recording. Here, we describe a mechanismwhich can lead to the unidirectional magnetic an-isotropy, and which up to our knowledge has notbeen investigated yet. We show that the e!ect canappear as a direct consequence of an orbital order-ing within ferromagnetic clusters of impurities andof a coupling of this ordering to the antiferromag-netic structure of the matrix. Orbital ordering

means that alternating electronic orbital states areoccupied in neighboring atoms [1].

The e!ect of the orbital ordering was found to berelevant in Rb

2Cu

1~xCo

xF

4. This system is known

to be a two-dimensional Ising S"1/2 antifer-romagnet if x'0.4 [2]. For low concentration ofCo (0.18(x(0.4) and at low temperatures a spinglass phase appears. It has been argued in Ref. [3]that near x"0.218 we get a good example of therandom-bond system; the distribution of bonds J issupposed to ful"l the condition +JP(J)"0, whichallows to apply the $J Edwards}Anderson model[4]. As it has been pointed out in Ref. [3], magneticbonds between atoms of Cu and Co do depend onthe orientation between d-orbitals of copper withrespect to the Cu}Co bond. A bond is ferro-magnetic (J"#20K in temperature units) ifthe orbital is perpendicular to the bond, and itis antiferromagnetic (J"!37K) for the parallel

0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 4 9 0 - X

orientation. For Cu}Cu pairs, one orbital of Cu isexpected to be perpendicular to another [1], andthe bond is always ferromagnetic (J"#22K).The Co}Co exchange integral J"!90.8K leadsto the antiferromagnetic structure. One of the mainpoints of Ref. [3] is that orbitals of Cu are antifer-romagnetically ordered, i.e. directions of electronicorbitals of neighboring atoms are always mutuallyperpendicular.

The aim of this paper is to demonstrate that a"eld cooling of antiferromagnetic Rb

2Cu

1~xCo

xF4

below the NeH el temperature should lead to a un-idirectional magnetic anisotropy of Cu clusters. Aswe work within the Ising model, the unidirectionalanisotropy to be discussed here manifests as a non-zero average reduced magnetization in the absenceof an external magnetic "eld. Note that the abovegiven values of the exchange integrals J rely at leastpartially on the opinion [3] that the system can bedescribed within the random-bond model. How-ever, for purposes it is enough to accept two as-sumptions: (i) the value of a Co}Cu exchangeintegral depends on the Cu-orbital orientation withrespect to the Co}Cu bond, and (ii) orbitals of Cuatoms are ordered, as described in Ref. [3].

In the next section, the e!ect is explained quali-tatively by the calculation of energies of spin con-"gurations of small clusters of Cu in theantiferromagnetic matrix of Co. Quantitativeevaluations of the e!ect by using Monte CarloMetropolis technique are also described in thissection. Results of these calculations are describedin Section 3. In the last section we compare ourmechanism with a similar e!ect produced by di!u-sion of impurities.

2. Calculations

The approximation of small clusters means thatwe calculate thermodynamic averages, taking intoaccount the magnetic degrees of freedom of Cuclusters only. The spins of the antiferromagneticmatrix of Co are kept constant. This approxima-tion can be justi"ed if temperature is su$cientlylow. Also, the Zeeman energy should be su$cientlyweak, to preserve the Co}Co antiferromagneticground state.

Assuming perfect orbital ordering in the sensegiven in the Introduction, two orbital con"gura-tions are possible for each cluster of Cu atoms.Each of these con"gurations can be obtained fromanother one by a rotation of each orbital by p/2.Our method is to calculate thermodynamical aver-ages of magnetization against "eld for each cluster.During this calculation, spin coordinates of a clus-ter are treated as thermalized, i.e. all spin con"gura-tions are taken into account in appropriatepartition functions. On the contrary, orbital coor-dinates are treated as quenched, as well as positionsof atoms in a cluster.

To "nd the anisotropy, the probabilities of twoorbital states of all possible clusters are needed.These probabilities are calculated with an assump-tion that the orbital degrees of freedom of theclusters are in a thermodynamic equilibrium withthe surrounding antiferromagnetic structure of thematrix during annealing at a given temperature¹

!//and in the presence of a given "eld h

!//. Then,

the probabilities of two possible orbital states ofeach cluster are obtained by summing up over thespin coordinates of the cluster atoms.

To calculate the probabilities of the orbital andmagnetic states of the clusters, we need the energiesof magnetic states of the clusters. The cluster shapes(up to four atoms) are given in Fig. 1. Their energiesare

E(1)"!h(s1#s

2)!J

0s1s2

!J1(2s

1!s

2)g!J

2(s1!2s

2)g, (1)

E(2)"!h(s1#s

2#s

3)!J

0(s1s2#s

2s3)

!J1(2s

1#s

3)g!J

2(s1!2s

2#s

3)g, (2)

E(3)"!h(s1#s

2#s

3)

!J0(s1s2#s

2s3)

!J1(s2!s

1!2s

3)g

!J2(s2!2s

1!s

3)g, (3)

E(4)"!h(s1#s

2#s

3#s

4)

!J0(s1s2#s

2s3#s

3s4)

!J1(2s

1#2s

3!s

4)g

!J2(s1!2s

2!2s

4)g, (4)

392 B. Kawecka-Magiera et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 391}396

Fig. 1. Clusters of impurities in a quadratic lattice, from 2 to4 atoms in a cluster.

E(5)"!h(s1#s

2#s

3#s

4)

!J0(s1s2#s

2s3#s

2s4)

!J1(2s

1#2s

3#s

4)g

!J2(s1!s

2#s

3#2s

4)g, (5)

E(6)"!h(s1#s

2#s

3#s

4)

!J0(s1s2#s

2s3#s

3s4)

!J1(s1!s

2#s

3!2s

4)g

!J2(2s

1!s

2#s

3!s

4)g, (6)

E(7)"!h(s1#s

2#s

3#s

4)

!J0(s1s2#s

2s4#s

3s4)

!J1(s1!s

2!s

3#s

4)g

!J2(s1!s

2!s

3#s

4)g, (7)

E(8)"!h(s1#s

2#s

3#s

4)

!J0(s1s2#s

2s3#s

3s4#s

4s1)

!J1(2s

1#s

3!2s

4)g

!J2(s1!2s

2#s

3!s

4)g, (8)

where the argument j of E(j) is the cluster number,as in Fig. 1, s

iis an ith spin, J

0, J

1and J

2are the

exchange integrals for Cu}Cu, Cu}Co for an or-bital parallel and perpendicular to the bond, re-spectively. Eqs. (1)}(8) are valid for only one of twopossible orientations of the orbitals. Appropriateequations for another orientation can be obtainedwhen J

1and J

2are mutually exchanged in these

equations. The variable g equal to $1 is introduc-ed for two di!erent orientations of antiferromag-netic sublattices around a given cluster. Forexample, one atom of Co is surrounded by four Coatoms with spins either up (g"1) or down(g"!1). Note, however, that a monoatomic clus-ter has the same energy, whatever is the orientationof its orbital. This is why monoatomic clusters donot give any contribution to the unidirectional an-isotropy.

The results obtained within the small clusterapproximation are veri"ed by a comparison withthe ones of a numerical calculation. StandardMonte Carlo [6] scheme is applied to a 64]64lattice of spins, each spin equal to $1. Appropri-ate clusters were formed by means of a modi"cationof some exchange integrals J, otherwise taken as forCo}Co. The values of all exchange integrals are thesame as given above.

We also need the concentration-dependent prob-abilities of the Cu clusters. These probabilities arefound after Ref. [5], where the technique of per-imeter polynomials was applied to the quadraticlattice. Appropriate formulae are given in Table 1.For simplicity, we limit the size of clusters con-sidered here, assuming that a cluster contains notmore than four atoms. This limitation is justi"ed ifthe concentration of Cu atoms is small enough. Asit can be deduced from Table 1, for small concen-trations of copper the most relevant contributioncomes from clusters No. 1. For larger content of

B. Kawecka-Magiera et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 391}396 393

Table 1Probabilities of the clusters 1}8, as dependent on the concentra-tion y"1!x of Cu

Cluster no. Probability

1 2y2(1!y)62 2y3(1!y)83 4y3(1!y)74 2y4(1!y)105 4y4(1!y)86 4y4(1!y)87 y4(1!y)88 8y4(1!y)9

Fig. 2. Reduced magnetization curves of the clusters 1, 2, 4 and6, at ¹"4.2 K, calculated within the small cluster approxima-tion. Maximal, i.e. saturation values of the magnetization is thenumber of atoms in a cluster: 2 for cluster 1, 3 for cluster 2, 4 forclusters 4 and 6. The results represent the averages over twopossible orbital con"gurations of each cluster. Antiferromag-netic structure of the matrix is "xed. As we see, the annealing"eld used here (1¹) is too weak to activate non-zero magnetiz-ation of the cluster 2. However, this cluster gives a contributionfor higher annealing "elds.

Fig. 3. Thermal dependence of reduced magnetization per atomfor the cluster 1 (dotted curve). Its orbital con"guration is "xed.The result is compared with the one obtained with the MonteCarlo method (full curve). As we see, here the small clusterapproximation is reliable below the helium temperature.

Cu, large clusters of copper atoms cannot be ne-glected.

3. Results

The calculations performed within the smallcluster scheme show that there are only four clus-ters which can produce contributions to the uni-directional magnetic anisotropy. The numbers ofthese clusters in Fig. 1 are 1, 2, 4 and 6. At zeromagnetic "eld, the contributions of remaining clus-ters 3, 5, 7 and 8 to magnetization vanish whenaveraged over the antiferromagnetic sublatticesg"$1 and * with appropriate weights * overthe orbital con"gurations. The magnetic momentsof the listed clusters are shown in Fig. 2, as depen-dent on the applied "eld energy h (in temperatureunits). The data for this plot are: annealing "eld isequal to 1¹, which is equivalent to thermal energyat the temperature of 1K. The annealing temper-ature is equal to 20 K. The measurement temper-ature is 4.2K. As we see in Fig. 2, this annealing"eld is too weak to activate the magnetic momentof cluster No. 2.

In Fig. 3, we show the magnetic moment ofcluster No. 1, i.e. a pair of Cu atoms, as dependenton temperature of measurement. The applied "eldis zero. The results are compared with those of theMonte Carlo calculations. Note that these datarepresent one cluster at a given orbital state, andnot an average over clusters within di!erent sublat-tices nor over orbital states. As we see, the smallcluster approximation gives correct results below

394 B. Kawecka-Magiera et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 391}396

Fig. 4. The magnetization at zero "eld, obtained by means of thesmall cluster approximation, against the concentrationy"1!x of Cu (full curve). Saturation value of the magnetiz-ation is 1. The results are reliable in the range 0)y)0.2, wherewe can treat y as a small parameter, and neglect contributionfrom clusters larger than 4 atoms. For a comparison, the curve(n"2) is shown, which represents the contribution from thepairs of atoms only, i.e. cluster 1 (dotted curve).

helium temperature. Obviously, this value of tem-perature is a direct consequence of the order ofmagnitude of the exchange integrals of the system.

In Fig. 4 we evaluate the magnetic moment ofa sample of an alloy at zero applied "eld, as depen-dent on the concentration of copper y"1!x. Theplot shows the result of averaging over clusters withweights from Table 1. For this calculation, we ac-cept the annealing "eld energy equal to 1K inthermal units. As before, the annealing temperatureis 20K, and the measurement temperature is 4.2K.

4. Discussion

As we see in Fig. 4, the reduced magnetizationinduced by annealing is not larger than 3%. Thecontribution of pairs of atoms also shown in this"gure, is not much smaller. Such small values of themagnetization do not promise e$cient applica-tions. This limitation could be overcomed if a su-perlattice of pairs of Cu atoms is formed.A necessary condition is that each pair must beembedded in antiferromagnetic matrix of Co, and

that the cobalt atoms are not isolated from thematrix. It is easy to check that such a superlatticeneeds at least four atoms of Co per each pair of Cuatoms. For the same annealing and measurement"eld and temperature parameters as above, thiscondition gives the reduced magnetization equal toabout 0.24 for the quadratic lattice.

However, our numerical results show that thesmall cluster approximation also gives quantitativeresults only at helium temperature. This shows thatthe probabilities of particular orbital states, cal-culated for the case of annealing temperature of20K, give an overestimation of the in#uence of "eldannealing on the orbital states. On the other hand,this in#uence is certainly stronger if larger magnetic"eld is applied during annealing. The probabilitiesof the orbital states could be, in principle, evaluatedby means of the Monte Carlo method. However, wehave found that this kind of calculation is parti-cularly time-consuming.

A similar mechanism of "eld-annealing inducedunidirectional anisotropy has been already pro-posed long time ago in Ref. [7]. It is a di!usion offerromagnetic impurities in an antiferromagneticlattice in the presence of magnetic "eld. The drivingforce of the di!usion is the di!erence of energy ofthe spin parallel to the "eld, when it is embedded ina sublattice with spins up or down. The contribu-tion from this mechanism to the reduced magneti-zation is proportional to the concentration (and notto its square) of the ferromagnetic impurities. Thatis why, in principle, this mechanism could be theleading one for small concentration of ferromag-netic impurities. However, a question remains if theatomic di!usion can be active below the NeH el tem-perature, because the energy barriers for the di!u-sion are of Coulombic origin, and forces of thiskind are usually larger than the magnetic exchange.

We note that the mechanism discussed here isactive for three-dimensional structures as well. Suf-"cient conditions for the unidirectional anisotropyare: orbital order of ferromagnetic clusters in anti-ferromagnetic matrix and the "eld annealing. Thevalues of the "eld of annealing, the annealing tem-perature and the measurement temperature mustbe carefully chosen, appropriately to exchange inte-grals which characterize a given system. Experi-mental data on the unidirectional magnetic

B. Kawecka-Magiera et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 391}396 395

anisotropy can provide a valuable information onthese exchange integrals and the temperature belowwhich the orbital order can be observed. Such datacan be helpful in interpreting quite complex sys-tems, because the orbital order itself is often accom-panied to charge and structural order [8].

Concluding, we have proved that samples of a di-luted antiferromagnet with the orbital order, an-nealed in "eld below the NeH el temperature, areexpected to show the unidirectional magnetic an-isotropy. The e!ect should be visible in particularin antiferromagnetic Rb

2Cu

1~xCo

xF4, if the con-

centration of copper is about 25%. In disorderedsamples the reduced magnetization at zero "eld isof the order of a few percent. This value may beenhanced by contributions of clusters of Cu largerthan four atoms. However, the computational com-

plexity of the problem increases abruptly with theconcentration of Cu.

References

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[2] C. Dekker, A.F.M. Arts, H.W. de Wijn, Phys. Rev. B 38(1988) 11512.

[3] C. Dekker, A.F.M. Arts, H.W. de Wijn, Phys. Rev. B 38(1988) 8985.

[4] S.F. Edwards, P.W. Anderson, J. Phys. F 5 (1975) 965.[5] M.F. Sykes, M. Glen, J. Phys. A 9 (1976) 87.[6] D.W. Heermann, Computer Simulation Methods in Theor-

etical Physics, Springer, Berlin, 1990.[7] L. Pal, T. Tarnoczi, Impurity e!ects in antiferromagnets,

Acta Tech. 55 (1}2) (1975) 9.[8] Fan Zhong, Z. D. Wang, Phys. Rev. B 61 (2000) 3192.

396 B. Kawecka-Magiera et al. / Journal of Magnetism and Magnetic Materials 221 (2000) 391}396