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VOLUME 63, NUMBER 6 PHYSICAL REVIEW LETTERS 7 AUGUST 1989Theory of Giant Magnetoresistance Kff'ects in Magnetic Layered Structures

with Aatiferromagnetic CouplingR. E. Camley ' and J. Barnas

Institut fu rFe'stkorperforschung der Kernforschungsanlage Julich GmbH,Postfach 1913,D 5170 -Julich, West Germany(Received 30 March 1989)

We present a simple theoretical description of recently measured giant magnetoresistance effects inFe/Cr layered structures. The resistivity is calculated by solving the Boltzmann transport equation withspin-dependent scattering at the interfaces. The magnitude of the effect depends on the ratio of the lay-er thickness to the mean free path and on the asymmetry in scattering from spin-up and spin-down elec-trons. Good agreement with experiment is found for both sandwich structures and superlattices.PACS numbers: 75.50.Rr, 72. 15.Gd, 73.50.Jt, 75.70.Cn

The study of magnetic layered structures has resultedin the discovery of a fascinating variety of behaviors. 'Recently there have been some very exciting reports ofgiant magnetoresistance effects in magnetic layeredstructures with antiferromagnetic couplings. ' Baibichet al. have suggested that spin-dependent scattering atthe interfaces is responsible for the magnetoresistanceeA'ect, but there have been no theoretical calculations in-vestigating if this mechanism is consistent with the ex-perimental results. Furthermore, the general behavior(magnetic field dependence, temperature dependence,and structural influences) has not previously been ad-dressed theoretically. Here we present a simple theoreti-cal model which reproduces all the major features of thedata qualitatively and quantitatively. Furthermore, theparameters obtained from the model give some supportto the suggestion of Baibich et al.In the Fe/Cr multilayers which were studied, there is

an effective antiferromagnetic coupling between Fe filmsdue to the intervening Cr films. As a result, the magnet-ic moments of neighboring Fe films are antiparallel toeach other in zero field. With a strong enough externalfield, the antiferromagnetic coupling may be overcome,and the magnetic moments of the Fe films can be forcedto all lie in the same direction.The experimental results for the magnetoresistance onepitaxial samples can be summarized as follows: (1) Theresistance of the structure is largest when the magneticmoments in neighboring Fe films are antiparallel andsmallest when they are parallel. (2) Multilayer struc-tures with thin Fe films have a much larger magne-toresistance effect than a single sandwich structure ofFe/Cr/Fe. A magnetoresistance effect of 50% at liquidHe temperatures was reported for a multilayer structure,while the largest number reported for a sandwich is 1.5%at room temperatures. (3) Changing from room temper-ature to that of liquid He increases the magnetoresis-tance effect by about a factor of 2 to 3.The problem considered here is similar, to some ex-tent, to that of electron scattering from domain wallswhich was discussed by Cabrera and Falicov. Their re-

suits, however, cannot explain the present experimentaldata. The mechanisms explored in Ref. 8 show nochange in resistivity in a geometry where the currentflows parallel to the domain wall and parallel (or anti-parallel) to the magnetizations. Experimentally it hasbeen shown that the large magnetoresistance effect inlayered systems is nearly the same for the current Bow-ing parallel or perpendicular to the magnetizations(the small differences can be attributed to the magne-toresistivity anisotropy effect). To explain all the basicfeatures of the magnetoresistance in layered structuresone has to take into account the diffusive scattering ofelectrons due to interface roughness. Such scattering, ofcourse, does not occur in the domain-wall problem.Point (1) above suggests that a spin-dependentscattering mechanism is responsible for the effect. Points(2) and (3) suggest that the effect also depends on therelative lengths of the mean free path compared to thethickness of the various films. Our model, an extensionof the Fuchs-Sondheimer theory, uses these ideas by in-troducing spin-dependent coefficients for specular reAec-tion, transmission, and diffuse scattering at the Fe/Crboundaries.We compute the conductivity of the structure throughuse of the Boltzmann equation. Consider first a simplesandwich structure as illustrated in the inset of Fig. 1.The dashed line in the center of the Cr film is not a trueboundary; it is the position at which the change in axis ofquantization for the electron spin is calculated. In eachregion the Boltzmann equation reduces to a differentialequation which depends on the coordinate z only:

t)g+ g eE &fociz rv- mv- BvHere fo is the equilibrium distribution function and g isthe correction to the distribution function due to scatter-ing and the external electric field E in the x direction.We neglect terms in the Boltzmann equation which arisedue to magnetic fields since, for the size of the fields in-volved here, the resulting effects are much smaller thanthat discussed here. It is convenient to divide g into four

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VOLUME 63, NUMBER 6 PHYSICAL REVIEW LETTERS 7 AUGUST 19890.08-

0.06-Et

N=8N=6N=5N = 3.5

z=a,Gg f =T fga f +R fgg+ f,gP+ f T fgQ + f +R fgBf

(7)Q. 0.04-QI

0.02-N=P

=15

0.2 I I I0.4 0.6l

I0.8 I10FIG. 1. The maximum normalized change in resistance, as afunction of the diAusive scattering constant Df and N. pfi isthe resistivity when the magnetizations in the two Fe films areantiparallel and pff is the resistivity for the case where themagnetizations are parallel. The inset shows the geometry.

separate portions in each region of space. We calculateseparately the contributions to g for spin-up or spin-down electrons moving to the right (positive v, ) or left(negative v, ). Thus g for electrons with positive v, in re-gion A is given byg~+ I (vx, z )+g~ + I (v. ,z ) .

Similar equations hold for the down spins at z =a, anda similar set of equations holds for the z =+a interface.Finally, an electron in film A with spin up whichenters film D will have a certain probability to be an upor down spin in film D because the magnetic moments inthe two films are in different directions. We take thisinto account by introducing a change in quantizationaxis at z =0. Thus

ga f =T f fgcf+ T f igci,g8- i = l lgc- l+ l fgc- fgc+f Tffga+f+TflgB+l ~gc+ i j lg~+ i+ j fg~+ f

(io)(i2)

where Tif is the probability for an electron of spin up(with respect to the magnetization in layer A) at z =0to continue as a spin down (with respect to the magneti-zation direction in layer D) at z =+0 and the other sym-bols are defined similarly. The transmission coefficientsare given by

Similarly g for electrons with negative v, in region A isg=gg I(v.,z)+gg I(vz). TI I =TII =cos (8/2),TI I =TII =sin (8/2), (i4)The solutions for g&+- f are given by

eEr r)fo Wzgq+ I(v, ,z) = ]+A+ I expm vx ziv, (4)where 0 is the angle between the magnetization vectorsin the two Fe films.Having found the various g's, we can find the current

density in the direction of the field by using

g~+ f =Rpg~ f at z =b,gDf =RpgD+ f at z =+b .

(s)

Similar equations hold for the down spins. At the Fe/Crinterface, the distribution function g for an electron witha given spin leaving the interface contains terms from apossible reflection event (with a probability RI for upspins and a probability R I for down spins) and a possibletransmission event (with a probability TI for up spinsand T I for down spins). We obtain for up spins at

The solutions for g~+. I are the same except with t re-placed by ] everywhere. The solutions in the other re-gions will have similar forms.The coefficients A ~ f (i) and similar coefficients for thesolutions in the remaining regions 8, C, and D are as yetunknown parameters which are to be determinedthrough the boundary conditions. At the outer surfacesof the sandwich, the distribution function g for an elec-tron leaving the surface is equal to the distribution func-tion g for an electron of the same spin striking the sur-face multipled by the probability of a specular scatteringevent, Rp. Thus we have

J(z) =q vg(vz)d'v. (is)The current in the entire structure is then found by in-tegrating J(z) over the coordinate z. It is then simple tofind the effective resistivity of the entire structure.The resistivity for an infinite superlattice can be calcu-lated similarly. One simply replaces the true outerboundaries of the Fe film with perfectly reAectiveboundaries at the midpoints of the Fe films.There are still a large number of unknown parameters.We make some simplifying assumptions in order to con-centrate on the most important factors: (I) We assumea model system of two equivalent simple metals, i.e., theyhave the same Fermi energies, same mean-free-pathvalues, etc. In this case it is appropriate to neglect angu-lar dependence' of scattering at the Fe/Cr interfaces.Also, it is natural to then assume that there is onlytransmission or difl'usive scattering'' at the Fe/Cr inter-faces, i.e., RI =RI =0. (2) We assume that the scatter-ing at the outer boundaries is purely diffusive.Despite the simplicity of these assumptions, our modelnonetheless gives a good account of the magnetic depen-

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VOLUME 63, NUMBER 6 PHYSICAL REVIEW LETTERS 7 AUGUST 1989

1.2-ICO

0.8-0oa-0

I t I I I l i I I I l l I I l I I I I

ory

I

X

Lcw TI 2 suit~ II

50

10

75t I

120A Fe/10A CrI I I

I I

~OA Fe/9A I. r

90

-'l.8 -1.2 -0.6 0 0.6 'l. 2 1.8B [10 "T]

I t I I k IFIG. 2. The percentage change in resistance as a function ofapplied field. The deviation of the experimental data from zeroat high field is a measure of the size of the magnetoresistanceanisotropy effect, neglected in the theory. The parameters forthe figure are N =6 and Dt =0.48. 3.2

1.8

120A Fe/10A Cr/120~ FeLow Tresult'

dence of the resistivity. We are now able to characterizethe resistivity of magnetic layered systems with just twoparameters he diffusive scattering parameter 1TI=Dl, and N=DI/DI. Dl should be a measure of theroughness of the interface. For a perfect interface thereis translational invariance parallel to the layers and thewave vector parallel to the layers must be conserved. Inthis case the resistivity would not be changed due toscattering of electrons at the interfaces, and there wouldbe no magnetoresistance effect. As interface roughnessincreases, D t should also increase. N determines theasymmetry in up-spin and down-spin scattering. If thesuggestion of Baibich et al. is correct then N=6, avalue obtained from measurements of Fe samples withCr impurities. 'We now turn to the results of the theoretical calcula-tions. Figure 1 shows how the magnetoresistance effectdepends on the diffusive scattering parameter and on N.These calculations are for a (120-A Fe)/(10-A.Cr)/(120-A Fe) structure with a mean free path X = 180A. As Dl increases or as N increases the magnetoresis-tance effect also increases. Since the experimental re-sults show approximately a 1.5% effect for this case, wecan put a lower limit on the value of N at 1.6.Figure 2 shows resistivity versus applied field for a(120-A Fe)/(10-A Cr)/(120-A Fe) sandwich structure.The theoretical curve is produced with parametersX=180 4, appropriate to room temperature, %=6, andDt =0.48. The angle 0 between the magnetizations inthe two Fe films is calculated by minimizing the sum ofthe exchange, anisotropy, and Zeeman energies for thisstructure. The parameters for the calculation were asfollows: interface exchange constant A ]2= 8.0&10J/m, K=3.8X10 J/m, and the surface is a (110)plane. The experimental results are those from Ref. 6.

0gt I I I I I I0.1 'l. 0 1.9 2.8 3.7 0.6 5.5X [10 Aj

FIG. 3. Maximum percent change in resistance as a func-tion of mean free path X for different structures and differentparameters Dt and N. (---) Dt =0.5, N =12; ( )Dt =0.48, N=6; (-.- ) Dt =0.5, N=0.2. The upper two setsof curves are infinite superlattices with units cells as shown.The lower set of curves is for the sandwich structure. Also in-cluded are experimental results (e) from Ref. 6 and those es-timated from Ref. 7. The low-temperature results are placedat the side of the figure as we do not have sufficient data to ac-curately estimate the low-temperature mean free path.

We see here that the theory is in reasonable agreementwith experimental results. One reason for the differencesmay be that the theoretical calculations did not includemagnetoresistance anisotropy. In comparison to theeffect discussed here in Fe, the contributions of the mag-netoresistance anisotropy is small; however, in other ma-terials it may be considerably larger. ' 'In Fig. 3 we explore the behavior of the magnetoresis-tance effect as a function of mean free path and as

dependent on the number of layers. We also show exper-imental results on the sandwich structure of Ref. 6 andthe multilayer structure of Ref. 7. As the mean freepath increases (a decrease in temperature) the magne-toresistance effect grows larger. For the sandwich struc-ture a change in mean free path from 180 to 6000 A(change from room temperature to impurity-dominatedscattering at low temperatures) gives an enhancement ofthe magnetoresistance effect by a factor of 2.5 in goodagreement with experiments. The theoretica1 resultsshow that the infinite superlattices have a significantly

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VOLUME 63, NUMBER 6 PHYSICAL REVIEW LETTERS 7 AvrvsT 1989larger magnetoresistance effect (a factor of 10-30 timeslarger depending on the mean free path and N) than thesingle sandwich, again in agreement with the experimen-tal results. This simply reflects the increased role ofspin-dependent interface scattering in the multilayeredsystems with a large mean free path.We stress that a determination of N cannot be madefrom a single sandwich structure alone. For example,the N 6, Dt =0.48 curve for the sandwich structureshown in Fig. 3 is very similar to one calculated usingparameters N =12, Dt =0.44. However, in the superlat-tice structure the N=12 case would give (p11p11)/p11=137% at k =6000 A, a value much too high to agreewith the data. The fact that we obtain a reasonable fit toboth the sandwich and the superlattice results with a sin-gle set of parameters (with N=6) gives considerablesupport to the case N=6. A conclusive determination ofN would require detailed experimental results, particu-larly on many-layered structures.In conclusion, we have demonstrated that our modelcorrectly describes all the major features of the experi-mental data. In particular, we have seen that increasingthe number of layers and increasing the ratio of meanfree path to thickness both significantly increase the sizeof the magnetoresistance eff'ect.The authors would like to thank P. Griinberg andAchim Fuss for helpful discussions. The work of R.E.C.was partially supported by U.S. Army Research OfficeGrant No. DAALO3-88-K-0061." Permanent address: Department of Physics, University ofColorado, Colorado Springs, CO 80933-7150.

"Permanent address: Institute of Physics, Technical Uni-versity, 60-965 Poznan, Piotrowo 3, Poland.'C. F. Majkrzak, J. M. Cable, J. Kwo, M. Hong, D. B.McWhan, Y. Yafet, and C. Vettier, Phys. Rev. Lett. 56, 2700(1986).2M. B. Salamon, S. Sinha, J. J. Rhyne, J. E. Cunningham,R. W. Erwin, and C. P. Flynn, Phys. Rev. Lett. 56, 259 (1986).3B. Heinrich, S. T. Purcell, J.R. Dutcher, K. B.Urquhart, J.F. Cochran, and A. S. Arrott, Phys. Rev. B 38, 12879 (1988).4L. L. Hinchey and D. L. Mills, Phys. Rev. B 33, 3329(1986).5R. E. Camley and D. R. Tilley, Phys. Rev. B 37, 3413(1988); S. Tsunashima, T. Ichikawa, M. Nawate, and S. Uchi-yama, in Proceedings of the International Conference onMagnetism, Paris, 1988 [J. Phys. (Paris), Colloq. (to be pub-lished) ].G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn,Phys. Rev. B 39, 4828 (1989);G. Binasch, Diplomarbeit, Insti-tute fur Festkoperforschung der Kernforschungsanlage Julich,1988 (unpublished).7M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,F. PetroA', P. Eitenne, G. Creuzet, A. Friederich, and J. Chaze-las, Phys. Rev. Lett. 61, 2472 (1988).sG. G. Cabrera and L. M. Falicov, Phys. Status Solidi (b)61, 539 (1974); 62, 217 (1974).9K. Fuchs, Proc. Cambridge Philos. Soc. 34, 100 (1938);also see the review by E. H. Sondheimer, Adv. Phys. 1, 1(1952).' V. Bezak, M. Kerdo, and A. Pevala, Thin Solid Films 23,305 (1974).' 'J. W. C. de Vries, Solid State Commun. 65, 201 (1988).'2A. Fert and I. A. Campbell, J. Phys. F 6, 849 (1976).' E. Velu, C. Dupas, D. Renard, J. P. Renard, and J. Seiden,

Phys. Rev. B 37, 668 (1988).' E. Dan Dahlberg, Kevin Riggs, and G. A. Prinz, J. Appl.Phys. 63, 4270 (1988).

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