Noncollinear order contribution to the exchange coupling in Fe/Cr„001… superlattices

M. Freyss, D. Stoeffler, and H. Dreysse´Institut de Physique et de Chimie des Mate´riaux de Strasbourg, Groupe d’Etude des Mate´riaux Metalliques, UMR 46 du CNRS,

UniversiteLouis Pasteur, 23, rue du Loess, 67037 Strasbourg Cedex, France~Received 11 July 1996!

By means of ad-band tight-binding Hamiltonian, we calculate the noncollinear distribution of magneticmoments in Fe/Cr superlattices, as a function of the relative orientationDw of the magnetic moments at thecenter of two adjacent Fe layers. All magnetic moments are computed self-consistently in both magnitude andangle. We find that for thick layers of a Cr spacer, the total energy varies parabolically as a function ofDw, inaccordance with the phenomenological proximity magnetism model proposed by Slonczewski. However, thismodel is not entirely satisfied for small Cr thicknesses because of the assumptions made in it.@S0163-1829~96!52042-4#

Cr/Fe~001! superlattices and trilayers have been widelystudied for their many interesting properties such as giantmagnetoresistance, oscillating exchange coupling, and non-collinear interlayer coupling. This latter phenomenon wasfirst observed by Ru¨hrig et al.1 on a Fe/Cr/Fe~001! wedge:90°-coupling domains were found in zones around the tran-sition between ferromagnetic and antiferromagnetic ex-change. Noncollinear couplings in Fe/Cr systems have sincebeen reported by other groups. Among them, Fullertonet al.2

and more recently, Schreyeret al.3 This latter group ob-served by polarized neutron reflectometry a 50° couplingbetween Fe layers in Cr/Fe~001! superlattices. A phenom-enological model was proposed by Ru¨hrig et al.1 to accountfor the noncollinear coupling by adding a biquadratic term tothe bilinear term in the exchange coupling energy:E(Dw)52J1mW 1•mW 2 2J2 (mW 1•mW 2)

2, where Dw is theangle between the magnetizationmW 1 andmW 2 of two succes-sive Fe layers, but the 50° coupling observed by Schreyeret al.3 is quite unexpected considering such a coupling en-ergy expression whose minimum occurs for a noncollinearspin structure withDw590°.

Another model has been suggested by Slonczewski4 toaccount for the noncollinear character of the coupling be-tween ferromagnetic layers separated by an antiferromag-netic spacer, such as Fe/Cr systems. This model, the so-calledproximity magnetism model, is based on the intrinsicantiferromagnetic structure of the spacer, on the asumptionof the existence of twisted quasiantiferromagnetic states inthe spacer, as well as on its thickness fluctuations. The ex-change coupling energy is then phenomenologically written:E(Dw)5J1(Dw)21J2 (Dw2p)2 with 0<Dw<p. Thepositive coefficientsJ1 and J2, respectively, favor ferro-magnetic and antiferromagnetic couplings. From the compe-tition between both terms result noncollinear couplings,which can be different from 90° depending on the value ofJ1 andJ2. In the case of a perfectly abrupt interface and aninteger number of spacer monolayers, one of the two coeffi-cients vanishes:J150 and J2.0 when the number ofspacer layers is even~antiferromagnetic coupling!, J250andJ1.0 when the number of spacer layers is odd~ferro-magnetic coupling!.

Here, we propose to confront Slonczewski’sproximitymagnetism modelwith band structure calculations. We cal-culate the noncollinear distribution of magnetic moments inCrn /Fe5(001) superlattices~with n52–6! as a function ofDw. Dw is defined as the angle between the magnetic mo-ments of Fe atoms at the center of two successive Fe layers,as shown on Fig. 1. We assume a perfectly abrupt interface,which means that, according to the parity ofn, eitherJ1 orJ2 vanishes in the expression of the exchange coupling en-ergy. The coupling energy is then expected to vary paraboli-cally as a function ofDw. Because of the symmetry of thesuperlattice, calculations are only performed on the three in-equivalent Fe sites and on then/2 @respectively (n11)/2]inequivalent Cr sites whenn is even~respectively, odd!. Themagnetic moments on all other sites can be determined ac-cording to the symmetry properties of the system, either byrotation or translation. In each of our calculations, we fixDw and determine self-consistently the magnitude and orien-tation of all moments.Dw is varied from 0 to 180° and thedistribution of magnetic moments, as well as the total en-ergy, is discussed as a function ofDw.

FIG. 1. Translational unit cell of the Cr6 /Fe5(001) superlattice.The black arrows correspond to Fe magnetic moments, the greyarrows to Cr magnetic moments.Dw is defined as the relative ori-entation between the moments of the central atoms in two succes-sive Fe layers. We consider situations corresponding to physicalconfigurations when an increasing external magnetic field is appliedrotating half the Fe magnetizations fromDw5180° to 0°. Helicalorders over the Fe layers are not considered because the total mag-netization would remain equal to zero for all values ofDw. Due tothe rotational symmetry, there are here only six inequivalent sites inthe superlattice, three Fe atoms and three Cr atoms.

PHYSICAL REVIEW B 1 NOVEMBER 1996-IIVOLUME 54, NUMBER 18

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The magnetic moments are calculated self-consistentlyusing a tight-binding model restricted to thed electrons. Theuse of such a method is particularly relevant in the study ofcomplex systems presenting a large number of inequivalentsites and where first-principle calculations would require toolarge a supercell.5,6 In the basis of atomic orbitalsu ims&~i stands for the atomic site,m for the orbital symmetry, ands56 1 for the spin!, the Hamiltonian is given by

H5 (i i 8mm8ss8

u ims&^ i 8m8s8ub i 8m8s8ims

1(ims

2D i

2su ims&^ imsu,

whereb i 8m8s8ims contains all nonmagnetic terms andD i is the

splitting defined asD i 5I iM i , I i being the exchange integralandMi the magnetic moment on the sitei .

In the case of noncollinear magnetic moments, the quan-tization axis is different from site to site. In order to expressboth parts of the Hamiltonian in the same basis, it is neces-sary to rotate the local quantization axis according to a ref-erence directionz, by applying a rotation matrix on the ex-change part ofH. The magnetic moments are obtained self-consistently with the relationD i5I i •Mi , by means of thereal-space recursive method.7 Self-consistency is imposed onboth the magnitude and the direction of the moments byiterating on the three components (Mr ,i ,M u,i ,Mw,i) of eachlocal vectorial magnetic moment. Convergence is obtainedwhen, for the (p11)th iteration:

maxi$uMr ,i~p11!2Mr ,i~p!u%,e

and

maxi$uM u,i~p!u,uMw,i~p!u%,e

with e equal to 1025. In this work, we restrict the angularvariation by settingu5p/2 constant and only lettingw vary.The hopping integrals and the exchange and Coulomb inte-grals used in our calculations are chosen to reproduce satis-factorily ab initio results obtained with the FLAPW tech-nique and have been used in previous studies8,9 wherecollinear or frozen noncollinear orders have been considered.

Figure 2 gives the distribution of magnetic moments ob-tained forn56 andD w5180, 90, and 0°. The part of thesuperlattice shown corresponds to the part between theDw-disoriented Fe moments as shown on Fig. 1. The groundstate corresponds to the collinear order, that is to say toDw5180° sincen is even. Whenn is odd, the ground statecorresponds toDw50°. The Cr displays then its usual layer-by-layer antiferromagnetic structure and Cr and Fe are anti-ferromagnetically coupled at the interfaces, as experimen-tally observed.10–12 When Dw is changed, this magneticorder cannot be satisfied anymore and a noncollinear order isinduced to minimize the frustrations. Cr almost remains an-tiferromagnetic, but the direction of the moments rotatesfrom the ~001! direction with Dw5180° to approximatelythe ~010! direction withDw50°. Whenn is odd, the centralCr moment is always exactly in the direction of the bisectorof the Fe moments directionDw. On the other hand, whenn is even, the Cr central moments are almost perpendicular

to that direction. It can also be noticed that the Fe momentswithin the same layer are not perfectly ferromagnetically or-dered anymore whenDw is different from 180°. The anglebetween the central Fe moment and the interfacial Fe mo-ment of the same layer can amount to as much as almost45° whenDw50°.

The variation of the magnitude and the relative angle ofthe moments as a function ofDw are shown in Fig. 3. Due tothe symmetry of the superlattices, only the inequivalent sitesare represented. Two cases are shown:n55 andn52. Wesee that in both cases, and in all other cases considered aswell (n53,4,6), themagnitude and the angle of the mo-ments behaves in the same way as a function ofDw. Theydisplay the following features:~1! The magnitude of the cen-tral and intermediary Fe moments are slightly enhanced(;2.4mB) compared to the bulk value~2.20mB!. ~2! Themagnitude of the interfacial Fe moments (;1.5mB! is re-duced compared to bulk, due to hybridization with the Crdstates.~3! The Cr moments are also enhanced when the col-linear order is not much perturbed (Dw,90° whenn is odd,Dw.90° whenn is even!, especially at the interfaces.~4!More generally, forDw,90° when n is even and forDw.90° whenn is odd, the magnitude of the moments isalmost constant and their relative angle varies linearly as afunction of Dw. ~5! On the other hand, forDw.90° when

FIG. 2. Distribution of the magnetic moments obtained in theCr6 /Fe5(001) superlattice withDw5180°, 90°, and 0°@~a!, ~b!, and~c!, respectively#. The radius of the circles is proportional to themagnitude of the moments. The scale gives the magnitude of themoments in Bohr magnetons. The horizontal dashed lines corre-spond to the bulk value of the Fe and Cr moments~respectively, 2.2and 0.6mB!.

R12 678 54M. FREYSS, D. STOEFFLER, AND H. DREYSSE´

n is even and forDw,90° whenn is odd, the magnitudedecreases and the moments rotate more slowly as a functionof Dw, except for the Cr central moment which always ro-tates linearly withDw. For these values ofDw , the collinearorder is very perturbed.

Figure 4 shows the total energy curve obtained forn56andn55. The curves show that the ground state correspondsto the collinear order, that is forDw5180° for evenn and

Dw50° for oddn, as already mentioned. The energy curveshave been fitted to a parabolaE5A2Dw2 on the intervalDw5@0°,90°# for oddn, andE5A2(Dw2p)2 on the inter-val Dw 5@90°,180°# for evenn. The curve obtained forn56 displays an almost perfectly parabolic behavior, in ac-cordance with Slonczewski’s model. On the other hand, forn55 the curve is parabolic for small distortions of the col-linear order withDw,90°, but deviates from the parabolawhenDw.90°. The same feature is obtained forn54,3,2.

Table I gives the coefficients of the fits to the total energycurves. Besides the parabolic fit on half theDw interval pre-viously mentioned, the curves have also been fitted on thetotal Dw interval @0°,180°# by the following functions:E5A3Dw21B3Dw3 ~3rd order! and E5A4Dw21B4Dw3

1C4Dw4 ~4th order!, when n is odd. For evenn, Dw isreplaced by (Dw2p). The more the Cr thickness decreases,the more theB3 coefficient in the third order function and theC4 coefficient in the fourth one increase, that is the more thebehavior of the total energy differs from the parabola. ThecoefficientA2 in the second order function corresponds toJ1 whenn is odd, and toJ2 whenn is even in Slonczews-ki’s model. These coefficients are found to be of the order of30 meV per crystallographic cell. Moreover, ann-parity ef-fect can be noticed on the behavior of the total energy. Whenn is odd, the deviation from the parabolic behavior of thetotal energy curves is faster then whenn is even: Whenndecreases, theB coefficient increases faster for odd values ofn. A reason for this effect could be that whenn is odd, thecentral Cr moment is exactly in the direction of theDw bi-sector: Its orientation is thus fixed by symmetry propertiesand adds a constraint in the orientation of the other magneticmoments, which does not exist whenn is even.

FIG. 3. Variation of the magnitude and the angle of the mag-netic moments as a function ofDw, in the Cr2 /Fe5(001) andCr5 /Fe5(001) superlattices. The anglew are given relatively to thecentral Fe atom. The empty symbols correspond to Fe sites, and thefilled symbols to Cr sites. The circles correspond to the central sites,the squares to the intermediate sites, and the diamonds to the inter-facial sites.

FIG. 4. Variation of the total energy as a function ofDw, in theCr6 /Fe5(001) and Cr2 /Fe5(001) superlattices~squares and solidline!. The dashed line corresponds to the fit of the curves by asecond-order function,E5ADw2, on the@0°, 90°# interval for oddn, andE5A(Dw2p)2, on the@90°, 180°# interval for evenn.

TABLE I. Coefficients of the energy fit functions. The energycurves have been fitted on the@0°,90°# ~respectively,@90°,180°#!interval for odd~respectively, even! values ofn, by the followingexpression: Forn odd, E5A2Dw2 ~2nd order!; and on the@0°,180°# interval by: E5A3Dw21B3Dw3 ~3rd order! and E5A4Dw21B4Dw31C4Dw4 ~4th order!. For n even, Dw is re-placed by~Dw2 p!. E is expressed in meV andDw in p units.

2nd order 3rd order 4th order

A35 25.1 A45 27.5Fe5 /Cr6 A25 26.0 B4525.3

B351.3 C45 4.1

A35 31.6 A4541.1Fe5 /Cr5 A2531.3 B45229.6

B3523.9 C45 16.6

A35 26.1 A4528.7Fe5 /Cr4 A2527.6 B4524.5

B352.6 C454.6

A35 54.7 A4590.1Fe5 /Cr3 A2549.5 B452119.2

B35223.4 C45 61.7

A35 22.0 A4524.8Fe5 /Cr2 A2525.7 B4520.3

B357.4 C4549.5

54 R12 679NONCOLLINEAR ORDER CONTRIBUTION TO THE . . .

Our results show that Slonczewski’s phenomenologicalexpression of the exchange coupling energy,E(Dw)5J1(Dw)21J2(Dw2p)2, is satisfied for spacer thick-nesses larger than 5 monolayers. Good accordance was alsofound experimentally by Schreyeret al.3 with Slonczewski’smodel. Indeed, they were able to fit their polarized neutronreflectivity ~PNR! data for a@Fe52/Cr17#9 superlattice withSlonczewski’s model with very good agreement. The fit withthe bilinear-biquadratic coupling energy model was not asgood. This shows that in Fe/Cr systems, theproximity mag-netism modelbetter account for the exchange coupling thanthe usual bilinear-biquadratic coupling model.

On the other hand, for small spacer thicknesses, the ac-cordance between our results and the model occurs only in alimited range ofDw values, corresponding to small disorien-tation of the collinear order. This discrepancy can be ex-plained by deviations to the asumptions made in the model.First, the moments are considered constant in magnitude,whatever the value ofDw is. Second, the magnetic order inthe ferromagnetic layer remains collinear. And finally,Slonczewski supposes only small deviations of the collinearorder, that is quasicollinear states. All these three asumptionsare not satisfied by our results for small thicknesses of Cr.We have seen on Fig. 3 that the magnitudes of the moments

decrease for important distortions of the collinear order. Wehave also already mentioned that the collinear order withinthe ferromagnetic layer was much perturbed and that theangle between the central Fe moment and the interfacial Femoment of the same layer could amount to as much as al-most 45°. Finally, for very small Cr thicknesses, deviationfrom the collinear order is greater whenDw differs from theground state value.

In this work, all noncollinear local magnetic moments ofthe Fe/Cr superlattices have been computed self-consistentlyin magnitude and angle. By this electronic structure calcula-tion we have shown that exchange coupling in Fe/Cr super-lattices can be satisfactorily described by Slonczewski’sproximity magnetism model. Nevertheless, this model is notentirely satisfied when the thickness of the Cr spacer is toosmall, that is inferior to 6 monolayers. We explain that thisdiscrepancy results from the asumptions in the model that arenot satisfied for small spacer thicknesses. As Slonczewski’smodel is valid for any ferromagnetic layers separated by anantiferromagnetic spacer, it would be interesting to comparethe case of Fe/Cr superlattices with, for instance,~FeCo!/Mnsuperlattices. Noncollinear couplings have also been experi-mentally observed in those systems.13

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R12 680 54M. FREYSS, D. STOEFFLER, AND H. DREYSSE´