.................................................................Non-collinear states inmagnetic sensorsAdrian Taga, Lars NordstroÈm, Peter James, BoÈ rje Johansson& Olle Eriksson

Condensed Matter Theory Group, Department of Physics, University of Uppsala,

Box 530, S-751 21, Uppsala, Sweden

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Certain materials have an electrical conductivity that is extremelysensitive to an applied magnetic ®eld; this phenomenon, termed`giant magnetoresistance'1±3, can be used in sensor applications.Typically, such a device comprises several ferromagnetic layers,separated by non-magnetic spacer layer(s)Ða so-called `super-lattice' geometry1±3. In the absence of a magnetic ®eld, theferromagnetic layers may be magnetized in opposite directionsby interlayer exchange coupling, while an applied external mag-netic ®eld causes the magnetization directions to become parallel.Because the resistivity depends on the magnetization direction, anapplied ®eld that changes the magnetic con®guration may bedetected simply by measuring the change in resistance. In order todetect weak ®elds, the energy difference between different mag-netization directions should be small; this is usually achieved byusing many non-magnetic atomic spacer layers. Here we show,using ®rst-principles theory, that materials combinations such asFe/V/Co multilayers can produce a non-collinear magnetic statein which the magnetization direction between Fe and Co layersdiffers by about 908. This state is energetically almost degeneratewith the collinear magnetic states, even though the number ofnon-magnetic vanadium spacer layers is quite small.

When electrons travel through a ferromagnetic super-lattice theyexperience an effective potential that depends on the spin direction.The spin direction in the magnetic layers is different from that in thenon-magnetic layers. The spin-down electrons have a higher energybarrier to pass through than the spin-up electrons and this gives

rise to a lowered transport coef®cient for the spin-down states.However, owing to the good conductance of the spin-up electrons,the total electronic transport is high. In the case of anti-parallelcoupling, spin-up and spin-down electrons experience a higheffective potential in alternating magnetic layers, which reducesthe electrical transport properties of both spin channels so that thetotal electrical resistance is increased dramatically. This difference inresistance can be used to detect small ®elds for use in, for instance,sensor applications1±3.

To detect very weak magnetic ®elds, the interaction between themagnetic layers must be weak. This is achieved by increasing thewidth of the non-magnetic spacer layer. This layer typically has athickness of 20±40 atomic layers1±3. However, an increased width ofthis non-magnetic spacer layer will decrease the number of mag-netic layers per unit volume. As it is the presence of magnetic layerswhich produces the difference in resistance between the ferro- andanti-ferromagnetic con®gurations, it is desirable to increase thenumber of magnetic layers within a certain volume.

In addition, non-epitaxial growth and increased lattice imperfec-tions (such as stacking faults) occur in traditional devices; thesediminish the giant magnetoresistance (GMR) effect. This effect iscaused by the fact that the non-magnetic and magnetic layers havedifferent lattice constants and the strain ®eld produced by the latticemismatch causes imperfections in the crystal structure. The latticeimperfections give rise to incoherent scattering, which may destroythe spin information of the electrons, which in turn reduces anddegrades the GMR effect. The thicker the intermediate non-magnetic ayers, the more lattice imperfections occur. Therefore, itis also for this reason important to make the non-magnetic layers asthin as possible, while still keeping the interlayer exchange couplingweak, to enable detection of tiny magnetic ®elds.

Here we demonstrate that, by suitable tuning of the super-latticegeometry and an appropriate choice of materials, we can stabilize anew magnetic con®guration. The successive magnetic layers do notcouple in a simple ferro- or anti-ferromagnetic way, but in anessentially perpendicular, non-collinear fashion; every second mag-netic layer has a magnetization direction pointing `out-of-plane',as opposed to the `in-plane' intermediate layers. This unusual

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Figure 1 Calculated values of the difference in magnetization directions of Fe and Co as a

function of number of V spacer layers (x). The energy landscape is shown as contours and

in colour code as indicated on the right-hand side of the ®gure. The calculations made

here employed the linear muf®n-tin orbital (LMTO) method and the Kohn±Sham equations

are solved for a general potential without any shape approximation. The calculations were

well converged with respect to all parameters involved, such as k-space sampling, basis

set truncation and total energy. Details of the method can be obtained from J. M. Wills

(unpublished work).

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magnetic con®guration is shown to have improved properties foruse in sensor applications, even when the number of non-magneticspacer layers is small. Hence the above-mentioned problem ofincoherent scattering and lattice imperfections, caused by thestrain ®eld, is greatly reduced.

The geometry of the magnetic super-lattice discussed here con-sists of alternating iron (Fe), vanadium (V) and cobalt (Co) layersgrown in the (001) crystallographic direction, but, as will bediscussed below, we can also use other materials combinationsand growth directions. The sequence of atomic layers consideredhere is repeated to give a multilayer with the following sequence:¼Fe/V/Co/V/Fe/V/Co¼ The ®rst ferromagnetic layer is Fe, epi-taxially grown on V. The Fe layer (consisting of several atomicplanes of Fe) has a tetragonally distorted crystal structure, causedby the lattice mismatch with the V layer (that also consists ofseveral atomic planes), and exhibits a magnetization parallel to thelayer (in-plane). The in-plane magnetization of the Fe layer is aneffect of the relativistic spin-orbit coupling (resulting in magneto-crystalline anisotropy, MAE) in combination with the structuralstrain. (The strain was calculated to give a c/a ratio of 0.84 for Feon V, assuming that all the atomic layers of the multi-layer adoptthe in-plane lattice parameter of V and that the volume of Fe andCo is conserved and is the same as in the bulk.) Our calculationsfrom ®rst principles of the energy difference between in-plane andout-of-plane magnetization for strained Fe on V is 0.6 meV per Featom.

For the second ferromagnetic layer a Co layer (with severalatomic planes) epitaxially grown on the V layer is used. The Colayer also has a tetragonally distorted crystal structure due to thelattice mismatch with the V spacer layer (calculated to give a c/aratio of 0.80). For Co, the magneto-elastic coupling produces amagnetization parallel to the growth direction (out-of-plane) andperpendicular to the magnetization direction of the Fe layer. Thecalculations from ®rst principles of the energy difference betweenin-plane and out-of-plane magnetizations for strained Co is1.4 meV per Co atom; that is, more than twice as large as thevalue for Fe.

If there were no other physical effects, the magnetization wouldalternate between the Fe and the Co layers, pointing in-plane for theFe layer and out-of-plane for the Co layer; this is a con®guration werefer to as a perpendicular magnetization, in contrast to the simpleferro- or anti-ferromagnetic con®guration. However, the ordinarybilinear interlayer exchange interaction between the two ferro-magnetic layers (Fe and Co) tends to turn the magnetization andarrange them in parallel, in a ferromagnetic or anti-ferromagneticarrangement4. If the thickness of the spacer layer is tuned, theinterlayer exchange coupling can be made to be of the same size asthe magneto-crystalline contribution. The two competing inter-actions can thus be made to produce complex non-collinear

magnetization pro®les, with a tiny energy difference betweendifferent magnetic orientations.

To quantify our analysis we have calculated the strength of theinterlayer exchange coupling of the proposed Fe/V/Co multilayer(with 3 Fe and 3 Co atoms). This was done in the same way as for thecalculations of the MAE, that is, from ®rst principles. The interlayerexchange coupling was found to oscillate in a RKKY-like fashion as afunction of V thickness; the calculated exchange coupling was ®ttedto a functional form Eex � I�x�cos�fFe 2 fCo� where I(x) is thestrength of the interlayer exchange coupling, x is the number of Vlayers, and fFe(Co) is the angle between the magnetization axis for theFe(Co) layers and the normal of the Fe(Co) layer. Calculations ofthis kind almost always give the correct oscillatory behaviour withcorrect period, but the magnitude of I(x) is over-estimated by afactor of 5±10 (ref. 4). Hence, we have scaled our calculatedinterlayer exchange coupling by a factor of 5. (We will discuss theconsequences of different scalings below; we argue that they donot in¯uence the basic physics involved here.) We can nowcombine the different energy terms to obtain a simple expressionfor the total energy of the multilayer as a function of the angles ofmagnetization of the Fe and Co layer. The MAE energies areparameterized in a traditional way, as EMAE 2 Fe � Acos2�fFe� for Feand EMAE 2 Co � Bcos2�fCo� for Co. The total energy can then bewritten as Etotal � Eex � EMAE 2 Fe � EMAE 2 Co. By minimizing theenergy as a function of angles fFe and fCo we calculate themagnetization directions of the Fe and Co layers as a function ofthe thickness of the V spacer layer.

As it is the difference in magnetization directions that is ofinterest, we show in Fig. 1 the calculated difference in angle,Df � fFe 2 fCo, as a function of V thickness. There are twoparameters that control the energy of the system, Df and the Vthickness, and these parameters give rise to an energy landscape, asFig. 1 shows. The minimum energy is indicated by a thick black line.The angle between the magnetization of the Fe and Co layers doesnot simply switch between 08 and 1808, as is the case for theconventional multilayers. Instead a more involved behaviouroccurs, where, starting from a few atomic V layers, the couplingoscillates in a damped fashion, approaching 908 for thick V layers.This ®nding is a direct consequence of the competition between theMAE and the interlayer exchange coupling. Our result has con-sequences for the detection of small external ®elds, because theproposed multilayer, for low V thicknesses, is already more sensitivethan a conventional device. For instance, at a V thickness of 5±6atomic layers, a very small energy is needed to change the magnet-ization direction from about 1308 to 908. This can be seen in Fig. 1 asonly two contours in the energy landscape must be passed. In asimilar way, V layers of about 4 and 7 atomic widths give rise toangles of about 308 and 708, respectively, which can be changed to08 by a minute external ®eld (ferromagnetic coupling). As the

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Figure 2 Calculated values of Df, as a function of V thickness in atomic widths. The

dashed curve is the same as the black curve in Fig. 1, from a calculation where the

strength of the interlayer exchange coupling has been reduced by a factor of 5. The dotted

curve represents a calculation where no scaling has been made for the interlayer

exchange coupling, and the solid curve represents a calculation where the magneto-

crystalline anisotropy contribution has been neglected.

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resistance of multilayers depends on the magnetic coupling, a smallexternal ®eld can be registered.

Finally, we note that although the basic idea behind this work isthat new non-collinear magnetic con®gurations of multi-layersystems can be stabilized, the actual numbers calculated here forappropriate thicknesses are connected to some uncertainties. Wehave assumed, for instance, that the contribution given by the MAEin Fig. 1 is insensitive to the in¯uence of the spacer layer. Inaddition, we have neglected the magnetic dipole contribution5

(which is indeed small for this system). The largest cause of errorpresented here is the calculated value of the interlayer exchangecoupling, since previous work shows that the calculated strength istypically overestimated, compared to experimental measurements,by a factor of 5±10. In order to mimic an experimental situation asclosely as possible we therefore reduced the calculated strength ofthe interlayer exchange coupling by a factor of 5. In order to analysehow different values of the interlayer exchange coupling in¯uencesthe proposed multilayer we show in Fig. 2 the difference in angle,Df, as a function of V thickness, for calculations with differentstrengths of the interlayer exchange coupling. The most conspic-uous feature of the behaviour of Df as a function of spacer thicknessis not modi®ed by the strength of the exchange coupling; merely, thedamping strength is in¯uenced. Also, without the in¯uence of theMAE the magnetization simply oscillates between ferro- and anti-ferromagnetic, whereas a non-negligible MAE produces a smootherbehaviour in the coupling between different layers. Hence, thesesources of error do not change the basic idea put forward here, thatnon-collinear, essentially perpendicular magnets may be grown, andthat this is caused by a competition between interlayer exchangecoupling and the MAE. This competition produces a magnetizationpro®le that varies more smoothly with spacer thickness, and alsoreduces the energy between different magnetic con®gurations. Wenote here the difference from a conventional exchange-coupledsensor, that in theory could be tuned to a small coupling energy bygetting close to a node in the oscillating exchange coupling curve.Owing to the large sensitivity to the exact thickness, such a set-up isuseless in reality and has not been realized experimentally. In ourproposed device, we suggest tuning the spacer-layer thickness to alocal maximum in coupling strength; this is done in practice inconventional sensors.

The suggested multilayer sensor has, in certain aspects, a similarperformance to conventional sensors, but with increased sensitivity,and from a technical process viewpoint we can draw on theexperience of these systems. The precise number of atomic layersneeded to optimize the proposed device must be determinedexperimentally. An extensive experimental study is needed to ®ne-tune the optimal choice of materials combinations and thicknesses.Other candidates of interest, in the sense that they have MAEstabilizing perpendicular magnetism in a strained state, are Niand Co separated by an fcc spacer layer such as Cu, Pd or Pt. Inaddition, Pt has a large spin-orbit coupling the enhances the MAEvalues. M

Received 1 October 1999; accepted 17 May 2000.

1. Baibich, M. N. et al. Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Phys. Rev.

Lett. 61, 2472±2475 (1988).

2. Parkin, S. S. P., More, N. & Roche, K. P. Oscillations in exchange coupling and magnetoresistance in

metallic superlattice structures: Co/Ru, Co/Cr, and Fe/Cr. Phys. Rev. Lett. 64, 2304±2307 (1990).

3. Binasch, G., GruÈnberg, P., Saurenbach, F. & Zinn, W. Enhanced magnetoresistance in layered magnetic

structures with antiferromagnetic interlayer exchange. Phys. Rev. B 39, 4828±4830 (1989).

4. Bruno, P. Theory of interlayer magnetic coupling. Phys. Rev. B 52, 411±439 (1995).

5. Aharoni, A. Introduction to the Theory of Ferromagnetism (Oxford Science, Oxford, 1996).

Acknowledgements

Support from J. M. Wills is acknowledged. This project has been ®nanced by the SwedishNatural Science and Technical Research Councils (NFR and TFR). Support for theEuropean programme, Training and Mobility of Researchers (TMR) is acknowledged.

Correspondence and requests for materials should be addressed to B.J.(e-mail: borje.johansson@fysik.uu.se).

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.................................................................Imaging the vortex-lattice meltingprocess in the presence of disorderAlex Soibel*, Eli Zeldov*, Michael Rappaport², Yuri Myasoedov*,Tsuyoshi Tamegai³¶, Shuuichi Ooi ³, Marcin Konczykowski§& Vadim B. Geshkenbeink

* Department of Condensed Matter Physics, The Weizmann Institute of Science,

Rehovot 76100, Israel² Physics Services, The Weizmann Institute of Science, Rehovot 76100, Israel³ Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku,

Tokyo 113-8656, Japan

§ CNRS, UMR 7642, Laboratoire des Solides Irradies, Ecole Polytechnique,

91128 Palaiseau, FrancekTheoretische Physik, ETH-Honggerberg, CH-8093 Zurich, Switzerland,

& L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia

¶ CREST, Japan Science and Technology Corporation (JST), Japan

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General arguments1 suggest that ®rst-order phase transitionsbecome less sharp in the presence of weak disorder, whileextensive disorder can transform them into second-order transi-tions; but the atomic level details of this process are not clear. Thevortex lattice in superconductors provides a unique system inwhich to study the ®rst-order transition2±6 on an inter-particlescale, as well as over a wide range of particle densities. Here we usea differential magneto-optical technique to obtain direct experi-mental visualization of the melting process in a disordered super-conductor. The images reveal complex behaviour in nucleation,pattern formation, and solid±liquid interface coarsening andpinning. Although the local melting is found to be ®rst-order, aglobal rounding of the transition is observed; this results from adisorder-induced broad distribution of local melting tempera-tures, at scales down to the mesoscopic level. We also resolve localhysteretic supercooling of microscopic liquid domains, a non-equilibrium process that occurs only at selected sites where thedisorder-modi®ed melting temperature has a local maximum. Byrevealing the nucleation process, we are able to experimentallyevaluate the solid±liquid surface tension, which we ®nd to beextremely small.

We ®rst discuss the expected vortex-lattice melting process in theabsence of disorder. Under equilibrium magnetization conditions inplatelet-shaped samples in perpendicular applied ®eld Hakz, theinternal ®elds B(x,y) and H(x,y) across the sample have a dome-shaped pro®le with a maximum at the centre7. This is because in theabsence of bulk pinning, the equilibrium shielding currents ¯owonly along the sample edges. As Ha or temperature T is increased,the ®eld H in the central part of the sample reaches the melting ®eldHm(T) ®rst, and thus a small round `puddle' of vortex liquid shouldbe formed in the centre, surrounded by vortex solid. Because of the®rst-order nature of the transition, the vortex-lattice melting isassociated with a discontinuous step in the equilibriummagnetization6, 4pDM = D(B - H). Because in our geometry H iscontinuous across the solid±liquid interface, the ®eld B in the liquidis enhanced by DB relative to the solid. In Bi2Sr2CaCu2O8 (BSCCO)crystals DB is typically6 0.1±0.4 G. Conventional magneto-optical(MO) imaging techniques8±10 (Fig. 1 legend) cannot resolve suchsmall ®eld differences. We have therefore devised the followingdifferential method.

An MO image is acquired by averaging typically ten charge-coupled device (CCD) images at some Ha and T. Then Ha isincreased by dHa p Ha, or T is increased by dT p T, a secondaveraged image is obtained, and subtracted from the ®rst. Thisprocess is averaged typically 100 times, yielding a differential ®eldresolution of about 30 mG, approximately two orders of magnitudebetter than the standard MO method. By recording the differential

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