PHYSICAL REVIEW B VOLUME 60, NUMBER 6 1 AUGUST 1999-II Pinholes in antiferromagnetically coupled multilayers: Effects on hysteresis loops and relation to biquadratic exchange J. F. Bobo LPMC, UMR 5830 CNRS-UPS-INSA, INSA, De´partement de Ge´nie Physique, 31077 Toulouse Cedex 4, France H. Kikuchi Fujitsu Limited, 10-1 Morinosato-Wakamiya, Atsugi, 243-01, Japan O. Redon Laboratoire Louis Ne´el, CNRS, 25, Avenue des Martyrs, Boi te Postale 166, 38042 Grenoble Cedex 9, France E. Snoeck CEMES-CNRS, 29, rue Jeanne Marvig, 31055 Toulouse Cedex, France M. Piecuch LPM, UMR CNRS 7556, Universite´ Henri Poincare´, Boi te Postale 239, 54506 Vandoeuvre Cedex, France R. L. White Department of Materials Science and Engineering­CRISM, Stanford University, Stanford, California 94305-2205 Received 29 April 1998; revised manuscript received 3 November 1998 We present a micromagnetic study of the influence of ferromagnetic bridges between consecutive ferromag- netic layers in antiferromagnetically coupled multilayers. The model is compared with experimental results for hysteresis loops obtained from the multilayer systems Co-Cu and FeNi-Ag. The presence of pinholes in Cu-Co multilayers is confirmed by transmission electron microscopy. We demonstrate that low densities of ferromag- netic pinholes in such multilayers are sufficient to give rise to significant deviations from the expected bilinear interlayer coupling and modify the observed interlayer oscillatory exchange coupling. The effects of pinholes can be simulated in certain cases by biquadratic exchange coupling, and we propose a magnetic phase diagram which correlates the apparent bilinear and biquadratic couplings to the pinholes density, size, and interlayer exchange strength. S0163-1829 99 14529-6 I. INTRODUCTION nonmagnetic layers has been observed in a wide range of systems based on transition metals.1 Such interlayer coupling The presence of holes in a nanometer-scale thin film is is at the origin of the so-called giant magnetoresistance2,3 commonly believed to occur. The origin of pinholes may be GMR in multilayers. A study of the dependence of this due to fundamental growth processes e.g., Stransky- exchange coupling upon the thickness of the nonmagnetic Krastanov or island growth in multilayers, inducing rough- or spacer layers shows an oscillatory behavior with a first ness and holes, or twinning faults in molecular beam epitaxy maximum of antiferromagnetic coupling occurring at spacer MBE grown samples or to more extrinsic origins like the thicknesses (ts) between 6 and 15 Å for most spacer mate- presence of arrays of dislocations or the morphology of the rials. Usually, the maximum GMR is observed near this first substrate pits, steps, terraces . For a single-layer film such maximum. Camley and Barnas4 have shown theoretically holes may not play a significant role in determining the film and Die´ny et al.5 experimentally that the GMR effect is pro- properties. However, if a multilayer structure is grown with portional to cos 2 , where 2 is the angle between the mag- such defects, one can easily imagine that the holes in one of netizations of successive ferromagnetic layers in zero field the layers will get filled by the material of the subsequent this angle becomes 0 at the saturation field . Thus the zero- layer, producing an electrical or magnetic interlayer coupling field value of 2 has to be as close as possible to 180° to affecting significantly the properties of the whole structure. achieve the maximum GMR. The experimentally observed We will calculate the effect of such pinhole defects in anti- magnetization curves of AF-coupled multilayers can show ferromagnetically AF coupled multilayers, and we will several behaviors, as summarized in Fig. 1 for two Cu-Co compare it to the experimental magnetic behavior of real multilayers with equal Co thickness 12 Å and a Cu spacer samples. layer thickness equal to 7 and 9 Å. In the ideal case of bilinear uniform antiferromagnetic exchange, the magnetiza- II. DESCRIPTION OF THE PROBLEM tion curve is a straight line with no remanence illustrated in Fig. 1 a . The saturation field HS is then simply proportional Interlayer magnetic exchange coupling in multilayers or to the interlayer coupling intensity Ji . However, this linear trilayers comprised of a succession of ferromagnetic and behavior is rarely observed at the first AF maximum see 0163-1829/99/60 6 /4131 11 /$15.00 PRB 60 4131 ©1999 The American Physical Society 4132 J. F. BOBO et al. PRB 60 processing conditions of the multilayer. We will neglect the anisotropy energy in this study, assuming that the Zeeman and exchange contributions dominate. If we assume all lay- ers are identical, the coupling occurs mainly between neigh- bor layers, and i 1 i 1 i . Equation 1 is sim- plified as E N 1 J cos 2 NMStmH cos . 2 The minimization of Eq. 2 then gives linear magnetization curves displayed in Fig. 1 a with a saturation field: 4 N 1 J Hsat NM . 3 Stm However, this simplified description of interlayer exchange coupling does not describe correctly the shape of experimen- tally observed magnetization or GMR curves. For most cases the hysteresis loops have a non-negligible remanence and curvature Fig. 1 b . Such hysteresis loops can frequently be fitted by a model that includes a biquadratic interlayer ex- change. This biquadratic term has been initially proposed by Ru¨hrig et al. in the case of exchange-coupled multilayers6 and has been reported by other authors:7 FIG. 1. Magnetization curves obtained at room temperature for two Co 12 Å -Cu(t E NtmMSH cos N 1 J cos 2 B cos2 2 . 4 s) multilayers with, respectively, ts 9 Å a and ts 7 Å b . The term B (B 0) takes into account the biquadratic Fig. 1 b and, as developed further, corrective coupling coupling. Biquadratic coupling leads basically to perpendicu- terms are introduced to fit the experimental data. The usual lar zero-field configurations of the spins ( i i i 1 expression for the free energy of a multilayer with an inter- 2 90°) compared to the bilinear coupling which causes layer coupling J antiparallel configurations ( i j between consecutive ferromagnetic layers i i i 1 2 180°). Sev- indexed by i and j having equal thickness t eral origins have been proposed for the biquadratic term, m and bulk satu- ration magnetization M either as an intrinsic effect related to the interlayer coupling S , assuming coherent in-plane rota- tion of the magnetization within each layer is mechanism8 or due to extrinsic effects related to the mor- phology of the multilayer. In the latter case and for Fe-Cr 1 multilayers, Slonczewski demonstrated that the presence of E 2 Jij cos i j MStmH cos i Eanis. 1 atomic steps in the Cr spacer layer could cause an apparent i j i biquadratic coupling.9 His model is based on the bulk anti- The parameters of Eq. 1 are displayed in Fig. 2. Note that ferromagnetism of Cr which causes small periodicity cou- the angles are referenced with respect to the applied mag- pling oscillations versus the chromium spacer layer thickness netic field. The first term of Eq. 1 is the usual bilinear period of 2 monolayers ML 4 Å . If there is an atomic exchange, the second term is the Zeeman term, and Eanis is step in the spacer layer, then the sign of the interlayer cou- the anisotropy contribution. The magnetic anisotropy can be pling will change from positive ferromagnetic F to nega- either intrinsic magnetocrystalline or can be induced by the tive AF . A periodic array of steps will therefore induce fluctuations of the interlayer coupling. If the size of the ter- races is smaller than some critical length, Slonczewski's model predicts a behavior of the magnetization curves which can be fitted by the introduction of a biquadratic interlayer exchange coupling. Several authors10,11 have reported the fit- ting of hysteresis curves to energy expressions involving bi- quadratic terms. They report cases for which the biquadratic exchange coefficient is comparable to or even larger than the bilinear term. No theory based on fundamental interactions presently predicts biquadratic exchange magnitudes ap- proaching those used to fit the magnetization curves in the above. We believe the physical process operative, and pro- ducing in many cases a behavior mimicked by biquadratic exchange, is in fact pinhole coupling. The modification of AF coupling by discontinuities in the spacer layers related to their finite size has been invoked by Gradmann and Elmers12 FIG. 2. Schematic view of a multilayer with interlayer exchange to explain the controversy between MBE and sputtered Co/ coupling. Cu 111 samples. We simply assume in our work that spacer PRB 60 PINHOLES IN ANTIFERROMAGNETICALLY COUPLED . . . 4133 III. PINHOLE MODEL We use the standard micromagnetics approach to solve the magnetic configuration of a trilayer F1-spacer-F2 , with F1 and F2 the two ferromagnetic layers.18 This approach can be generalized to a multilayer stack if we assume that all the layers are identical and that the coupling occurs between first neighbor layers only. The energy of the system for N mag- netic layers will be the sum of N 1 interlayer coupling FIG. 3. Schematic view of a pinhole showing the twisting of the terms and N Zeeman terms. The interlayer coupling is de- magnetization symbolized by arrows through the pinhole between fined as AF everywhere in the trilayer except on the pinholes the two magnetic layers. sites where it is positive and is given by Eq. 6 . Therefore the value of the interlayer exchange coupling will depend on layers of infinite size, but having ferromagnetic discontinui- the in-plane location (x,y) across the sample surface. In- ties, can also affect the AF interlayer coupling. This has also plane domain walls will appear between F regions at the been discussed in the early works on magnetic multilayers.13 pinholes and the surrounding AF areas. The magnetic field is Basically, interlayer coupling through pinholes resembles applied in the plane of the layers along the direction x. If we the theory presented by Slonczewski concerning two films assume, from symmetry, that the magnetizations of the two having a primary AF coupling, but with a distribution of layers F1 and F2 are exactly mirroring each other with re- F-coupled local regions. However, a major difference is that spect to the applied magnetic field, we can define (x,y) as pinholes couple the ferromagnetic layers by direct exchange, the angle between the field axis and the local magnetization inducing a F coupling far stronger than the AF coupling of direction. Then 1(x,y) 2(x,y) (x,y) and the total the surrounding areas JAF is of the order of several tenths of energy of a bilayer system can be written as a function of the erg/cm2,2,14,15 while Jh is estimated two orders of magnitude local angle (x,y): higher. If the pinhole is represented by a ferromagnetic col- umn of height ts the spacer layer thickness , its magnetiza- tion, assumed to remain in plane,16 will experience a twisting E 2Atm " 2 J x,y cos2 x,y s around an axis perpendicular to the two planes see Fig. 3 . The exchange energy cost for such a twisting is then17 2HMStm cos x,y dx dy. 7 In the case of a multilayer these energy terms get cor- E 2 2 2 ex A Mx My Mz dV rected as presented in Eq. 2 . This trilayer approximation, V MS MS MS used for a multilayer, does not take into account the possi- ts ts bility that a pinhole could either propagate through the whole S A d 2dz S A d2 2dz. 5 multilayer stack or be confined into one fraction of the whole 0 dz 0 dz artificial stacking. Transmission electron microscopy TEM Performing a micromagnetics integration along the col- see Sec. V indicates that pinholes would not short all the umn, one obtains, in a first approximation d(2 )/dz is as- layers through the whole multilayer thickness, and so they sumed constant and S is the area occupied by the pinhole , have a smaller impact on the total coupling of the multilayer than the AF exchange. Thus we introduced a correction pa- 2 4SA 2A 2A rameter to the ferromagnetic interlayer coupling at the pin- E 4SA 1 cos2 t S S cos 2 . hole sites 0 1 to reduce the effective pinhole strength. s ts 2 ts ts This problem of through-thickness coupling variations has It gives an effective ferromagnetic interlayer exchange been recently presented by Kolhepp et al.19 in the case of coupling at the pinhole given by Eq. 6 : Fe-FeSi multilayers which exhibit a different interlayer cou- pling at the surface of the multilayer dominant AF coupling 2A than at the substrate side F coupling . They concluded that Jh t . 6 this inhomogeneous behavior leads to a mimic of biquadratic s coupling. The reduction of the pinhole interlayer coupling Then, if we use commonly accepted constants for the can also be explained by the small size of the column of the saturation magnetization MS , the exchange stiffness A ferromagnet reducing its effective ferromagnetic stiffness. ( 10 6 erg/cm), and a standard value for the spacer layer This aspect of the pinhole problem was recently developed thickness ts( 10 Å), one obtains Jh 20 erg/cm2. There- by Fulghum and Camley.20 They used a more sophisticated fore, the coupling strength between two ferromagnetic layers method to determine the magnetic ground state of a pinhole- through a pinhole is ferromagnetic and between one and two coupled structure, calculating the local value and orientation orders of magnitude larger than the AF coupling away from of the spin versus the temperature, the applied field, and the the pinhole. We can develop a model based on a distribution density of pinholes. They demonstrated that the strength of of pinholes in between AF-coupled magnetic multilayers and the pinhole coupling is strongly temperature dependent as a determine the influence of various parameters such as pin- direct result of their reduced size. Another reason why we hole size, pinhole density, and the intensity of their coupling introduce a reduction to the exchange through the pinholes to calculate the magnetization curves of magnetic multilay- comes from the approximation used to derive Jh in Eq. 6 ers. where we assumed a linear twisting of the spins along the 4134 J. F. BOBO et al. PRB 60 ferromagnetic column. This assumption is valid for small twisting and perfectly cylindrical columns, but the actual case may lead to lower coupling strength. IV. SOLUTIONS TO THE PINHOLE MODEL A. Analytical approach From Eq. 7 , it is possible to write an equivalent Euler's equation: 2Atm 2 x,y 2J x,y sin x,y cos x,y HMStm sin x,y . 8 FIG. 4. View of our numerical model. Pinholes black columns Equation 8 could not be solved analytically for the as- are randomly scattered over the meshed area. They connect both considered pinhole configuration, but it provides insight of layers by direct ferromagnetic exchange, while the magnetization in the as expected solution, and so leads to a characteristic each cell of the mesh is coupled to its first four neighbors by direct length exchange and to the other layer cell by indirect antiferromagnetic coupling. At l2 m J , 9 10 6 erg/cm, and ferromagnetic layers of 10­20 Å, li 40­ 400 Å, whereas lh 5 ­ 10 Å for the usual cases at the which defines the solution space of the equation. Since there maximum AF interlayer exchange occurring commonly for are two distinct domains, l will have two different values: ts 10 Å. The behavior of the system clearly depends on the lh in the pinhole area where J is positive and li in the AF spacing between pinholes and the strength of the pinhole area where J is negative. li is associated with the domain coupling. As we will subsequently see in Sec. VII, the ratio wall width in the AF area surrounding a pinhole and is the of dh , the distance between pinholes, to li and the ratio of distance of relaxation from the ferromagnetic to the antifer- the pinhole diameter, r ( a), to lh provide useful param- romagnetic alignment from the pinhole to the unpinned re- eters for the construction of a phase space in which the so- gion. The concept of the characteristic length lh or the do- lutions to the pinhole model can be classified. Parentheti- main wall width within a pinhole is somewhat more subtle, cally, we note that Eq. 8 has been solved analytically for a but can be understood as follows. Imagine a pinhole such somewhat simpler quasi-one-dimensional case where the that the spins in the magnetic films at the perimeter of the bridging structures are long thin sheets.21 The magnetization pinhole are fixed in the antiferromagnetic alignment an un- curves and spin configurations for this case are similar to the physical situation, but useful for a gedanken experiment . same properties calculated for the pinhole configuration ob- The ferromagnetic coupling through the pinhole induces a tained by the numerical solutions given below. parallel configuration of the spins, therefore rotated by 90° from the orientation in the AF area. If lh r, the pinhole B. Numerical approach radius, the spins within the pinhole area will indeed be torqued around so that the spins at or near the center of the To solve the general case, we created a square meshed pinhole will achieve the ferromagnetic configuration. If l area containing up to n h cells 2500 cells of lateral size a. A r, the spins in the pinhole will be only partially torqued given number of pinholes nh , each occupying one cell, are around and will not achieve full ferromagnetic alignment. each randomly arranged over the meshed area. Periodic The parameter r/l boundary conditions are used at the edges of the meshed area h is a measure of the rotation of the spins within the pinhole area. A large , achieved when r is see Fig. 4 . The mean distance between two pinholes is large or l d h is small, corresponds to a strong coupling through h a ncells /nh. Size fluctuations of the pinholes were the pinhole. Inserting this relationship into Eq. 9 and in- not taken into account by the model, but several runs with cluding the pinhole stiffness reduction factor , we finally constant pinholes densities xh nh /ncell , li /lh ratios and obtain varying li /a could be performed to probe the influence of the pinholes sizes. In each cell, indexed by (i, j) the local ex- t pression of the energy is given by l2 mts h 2 with 0 1. 10 Eij NAtm Note that the pinhole strength increases for thin spacer a2 a2 cos ij i j layers, as it should. Since J within the pinhole is much larger i ,j neighbors than J in the antiferromagnetically coupled region, lh is NMstmH cos ij N 1 Jij cos 2 ij , 11 smaller than li . The ratios between, respectively, li and the pinhole spacing and between lh and the pinhole diameter where the first and last terms are the interlayer and interlayer play an important role in characterizing the magnetic behav- exchange terms, respectively. Jij is the local value of the ior of the coupled films. We can estimate the values of lh and interlayer exchange coupling in one cell whose location is li : with standard interlayer exchange coupling values of specified by the i and j indices the indices i and j repre- 0.1­0.8 erg/cm2, a ferromagnetic exchange stiffness A sent the neighboring cells in the same layer : PRB 60 PINHOLES IN ANTIFERROMAGNETICALLY COUPLED . . . 4135 FIG. 5. Zero-field spin configuration of a bilayer with three FIG. 6. HRTEM of a Co-Cu multilayer showing the epitaxial pinholes only the top layer is displayed, the field direction is hori- growth of Cu and Co, but a very low contrast between the different zontal . layers. i Jij J0 0 outside of pinholes as a result of the AF haves symmetrically with respect to the horizontal magnetic coupling, field axis its spins are therefore pointing downwards . The ii Jij 2 A /ts 0 at the pinhole sites. orientation of the local moment is almost parallel to the ap- In order to save computation time, the cosine expressions plied field in the pinholes while it tends to align perpendicu- in Eq. 11 are replaced by the projections along x and y axes lar in the AF area. However, the parameters chosen for this of the magnetization unit mean vector mij i j x cos ij and my calculation are such that the pseudo-domain-wall in the film sin ij : determined by li is greater than the spacing between the pinholes, and so the spins never become perpendicular the H Eij NAtm i j i j i j i j relaxed entirely to the antiferromagnetic configuration any- a2 a2 mx mx my my i ,j neighbors where over the mesh . The equilibrium configuration was obtained this way for each value of the applied magnetic NM i j i j2 i j2 stmHmx N 1 Ji j mx my . 12 field up to the saturation field i.e., until all the spins are aligned along the applied field . The total magnetization is It is convenient to write this expression in terms of the defined as the sum over the cells of M reduced field h H/H S cos . We used our sat where Hsat is given by Eq. 3 and model to fit the experimental data using J the characteristic lengths l 0 , nh , and as h and li : parameters, and we show23 that low pinhole area densities are required to fit the experimental magnetization curves. Eij i j i j i j i j NAt mx mx my my m i j neighbor cells V. ORIGIN OF PINHOLES: EXPERIMENTAL EVIDENCE BY TRANSMISSION ELECTRON MICROSCOPY 4 N 1 a2 ij a2 ij2 ij2 N N 1 m . l2 hmx 2 mx y i N lij The in-plane morphology of the films was investigated by 13 transmission electron microscopy. The samples presented here were deposited by rf sputtering.24 We present as most Note that in the last term of Eq. 13 we have l2ij where lij relevant data concerning very thin spacer layer thickness may have either the value li if the cell i,j is outside a around the first AF maximum of Cu-Co. In order to perform pinhole or the value lh if cell i,j contains a pinhole. TEM, the samples were first prepared as standard cross- The total energy was minimized by the torque method18 sectional specimens and the final thinning was done by ion which consists of aligning the local magnetization vector milling with a 77-K-cooled sample holder to minimize along an effective field which is the gradient of the local preparation damage. The micrographs were collected with a energy with respect to the magnetization direction: m ¯ ij Phillips CM 30 microscope operating at 300 kV. As shown ij "m(Eij) recursively at all the sites of the mesh in Fig. 6, high-resolution HR confirms the epitaxial growth until the discrete magnetization vectors converge to the of copper and cobalt. Since cobalt and copper have very lowest-energy configuration. Our code22 makes a minimum close atomic numbers respectively, Z 27 and Z 29 , they of 2000 iterations over the mesh allowing a convergence have a low contrast in HRTEM. If TEM is used in out-of- better than 10 6. For an example of a spin configuration, focus conditions to enhance the Fresnel contrast,25 the stack- Fig. 5 shows the remanent magnetic configuration obtained ing of Cu and Co becomes visible as shown in Fig. 7. We with a Co 12 Å -Cu 8 Å -Co 12 Å trilayer with a pinhole can also distinguish discontinuities in the copper layers, density of 0.0068, a mesh size of 10 Å, and an AF exchange which appear darker. These discontinuities occur rather ran- of 0.5 erg/cm2. The three pinholes presented in this figure domly on the cross section and are assumed to be cobalt are labeled by circles, and only the top layer configuration is pinholes. Their size is close to the thickness of the copper reported. The Co underlayer, not shown in this figure, be- layers ts 8 Å in this sample , and the mean distance be- 4136 J. F. BOBO et al. PRB 60 FIG. 7. Defocused TEM micrograph of a Cu-Co multilayer with contrast enhancement between Co and Cu due to Fresnel contrast. Several discontinuities arrowed in the copper layers pinholes are visible. tween two consecutive pinholes is approximately 100 Å. Note that some of these pinholes propagate through several bilayer periods of the stacking. The same kind of behavior is shown in Fig. 8 for another Cu-Co multilayer, confirming the existence of pinholes in Co-Cu multilayers. VI. MAGNETIZATION CURVES: COMPARISON WITH EXPERIMENT A. Co/Cu multilayers These multilayers were deposited by dc magnetron sput- FIG. 9. MH curves for Co-Cu multilayers with ten bilayers at tering on oxidized Si 100 wafers.26 The base pressure and various spacer layer thickness near the first AF peak fitted to the Ar pressure during deposition were 5 10 8 Torr and 3.0 pinhole model solid lines are fits . mTorr, respectively. The deposition rates were about 1.0 Å/s for Cu and 0.3 Å/2 for Co. The Co(50 Å)/ Cu(ts Å)/ Co(10 Å) of the Co buffer layer is subtracted from these curves. For N /Cu(50 Å) films, with tS in the range from 8.3 to 11.1 Å, were 111 textured. The values of t N 10, the maximum MR ratio 26% is smaller than for S are close to the first peak of the well-known GMR vs Cu thickness curve. N 20 38% . For both values of N, the GMR ratios peak The MR ratio was measured using a four-point probe near 10 Å. Table I shows J0 and the pinhole density xh method. The magnetic properties were measured with a vi- obtained from our calculation for N 10 and N 20. J0 is brating sample magnetometer. Figure 9 shows the M the AF interlayer exchange for tS in the absence of pinholes. f(H) curves measured on samples with N 10 and the as- Its value for tS 10 Å is very close to those reported in Refs. calculated ones using the pinhole model. The magnetization 14 and 15 and in agreement with these studies, J0 increases with decreasing tS and is maximum for tS 8.3 Å.14,15 How- ever, in contrast with the NiFe-Ag system, we did not try to extract an oscillation period with the Bruno-Chappert model27 because the samples were not highly textured and the oscillatory interlayer coupling is known to be strongly dependent on the crystallographic orientation of the layers. We used the same value of the pinhole reducing coefficient for all samples. The best fits were obtained for 0.225. For N 10, as expected, xh increases monotonically as the spacer thickness is reduced. For N 20 the variation with tS is not as clear; our calculation is probably less reliable for predict- ing xh for large N because of a more complex evolution of the pinhole propagation and of the structural disorder through the multilayer. The pinhole model can explain the shape of the magnetic hysteresis MH curves of Co/Cu multilayers, including the remanence observed. The density of pinholes rises rapidly below tS 10 Å. A density of pinholes of a few percent leads FIG. 8. Fresnel contrast TEM micrograph on another Cu-Co to significant changes in the hysteresis curves. The pinhole multilayer showing again pinholes. model explains naturally a variety of hysteresis loop shapes PRB 60 PINHOLES IN ANTIFERROMAGNETICALLY COUPLED . . . 4137 TABLE I. Antiferromagnetic interlayer exchange J0 , pinhole area density, and GMR ratio as a function of Cu thickness for the Cu-Co multilayers. N 10 bilayers N 20 bilayers ts J0 xh GMR J0 xh GMR Å erg/cm2 % % erg/cm2 % % 8.3 0.83 13.8 12.8 a 8.9 0.80 13.0 17.0 9.4 0.72 11.5 12.9 9.6 0.52 7.3 24.7 10.0 0.44 6.6 19.8 0.42 5.1 38.7 10.5 0.24 3.8 21.8 0.34 5.1 31.5 11.1 0.16 2.4 25.9 0.25 5.2 3.1 aSample not available. without invoking biquadratic exchange. The value of J0 , the layers. We have fitted the magnetization curves to the pin- interlayer AF exchange, is still increasing for the thinnest hole model for all Ag spacer layer thickness. The results spacer layers, 8.3 Å, even though the maximum GMR ratios gave an increase of the AF interlayer coupling J up to 0.09 occur near tS 10 Å. This shows that pinholes affect the erg/cm2 for tS 7.5 Å, but with an increasing pinhole density GMR curves, that the maximum in the GMR ratios does not from xh 0.25% for tS 9.1 Å to xh 1.25% for tS 7.5 Å, coincide with a maximum in the interlayer AF exchange, and while the pinhole size was kept close to 8 Å therefore close that the true dependence of the interlayer exchange is to the Ag layer thickness . The coefficient was found to masked by the pinhole effects. Changes in the character of give the best fits for a value close to 0.5. These conclusions GMR curves for thin spacers, from rounded at tS 10 Å to were recently corroborated by Bouat and Rodmacq28 with cusped at smaller spacer thickness, support this picture. samples of better structural quality which exhibited large saturation fields for tS 7.5 Å. B. NiFe/Ag We were unable to observe the second AF coupling oscil- lation for t Permalloy/silver multilayers were grown by triode sput- S 20 Å, and so we analyzed the oscillation based on the shape of the first AF peak, fitting the dependence of J tering on glass substrates cooled at 100 K with an amorphous vs t SiO S to the predicted oscillatory expression for a fcc Ag 111 2 buffer layer. The Ar pressure was 0.5 mTorr and the spacer layer by Bruno and Chappert:27 deposition rates were 1 Å/s, which led to 111 -oriented multilayers. A series of samples with N 18 bilayers, con- m* t stant NiFe thickness (t d2 S M 27 Å), and several Ag thickness J tS 3I0 2 sin 2 tS exp , 14 around the first AF oscillation t me ts tc S from 7.5 to 12.7 Å were prepared. Room-temperature magnetoresistance measure- where I ments showed a maximum GMR ratio of 12.2% for t 0 13 erg/cm2 according to Bruno and Chappert S is the theoretical interlayer coupling prefactor, d 2.35 Å in 9.1 Å. For tS 9.1 Å, MH curves have a low remanence the lattice spacing for Ag 111 , m*/m and a nearly linear shape characteristic of bilinear coupling. 0 0.17 is the reduced effective mass for Ag 111 , is the period of the interlayer But for tS 7.5, 8.3, and 9.1 Å, as shown in Fig. 10, there is coupling oscillations is the phase shift of the interlayer a significant remanence followed by a low susceptibility coupling oscillations with respect to the Rudenman-Kittel- magnetization curve at higher field up to the saturation field. Kasuya-Yosida RKKY oscillations obtained from a spheri- This behavior is very similar to the one of the Cu-Co multi- cal Fermi surface, and the last exponential factor is a damp- ing factor related to structural imperfections or roughness. tc is a damping length characteristic of the system above which the interlayer coupling vanishes. The results of our fits to the Bruno-Chappert model are presented in Fig. 11 and summa- rized in Table II. We fitted the J values obtained from the pinhole model either with I0 held constant and equal to 13 erg/cm2 or with I0 free. In both cases, the agreement of the pinhole model data with the theoretical predictions from the Bruno-Chappert model is better than the simple saturation- field-based determination. The lower value of tc found for all fits and the different phase shifts are not major parameters according to Ref. 27. Since it was impossible to measure any significant MR at the expected position of the second AF coupling oscillation (tS 20­ 30 Å), we had to extract the FIG. 10. Magnetization curves for NiFe-Ag multilayers around periodicity of the interlayer coupling from a fitting to Eq. the first AF peak fitted to the pinhole model. 14 of a smooth portion of curve with four parameters, but 4138 J. F. BOBO et al. PRB 60 TABLE II. Intensity of the interlayer AF coupling, oscillation period, phase shift, and damping factor for NiFe-Ag multilayers obtained from the Bruno-Chappert theory Ref. 27 , experimentally obtained from the saturation field determination and from the pin- hole model see text . I0 erg/cm2 Å deg tC Å Theorya 13.00 13.96 90 13.02 Deduced from Hsat 5.66 12.62 14.9 4.94 Pinhole model I0 held 13.00 15.29 55.5 4.32 Pinhole model I0 free 10.30 15.58 62.2 4.86 aReference 27. from the minimization of Eq. 5 with respect to : 2B 4B h Jeff eff eff J 1 J m m3 . 15 eff Jeff The prefactor (Jeff /J) of the expression of h vs m takes FIG. 11. Interlayer coupling strength J vs the silver thickness for into account the possibility to have a magnetization curve NiFe-Ag multilayers fitted to the Bruno-Chappert model solid line, with a lower apparent AF coupling. The example set of cal- Bruno-Chappert theory; open circles, data obtained from the satu- culated magnetization curves, zero-field spin configurations, ration field measurement; solid circles, data obtained from the pin- and phase diagrams presented here was obtained using a ra- hole model . tio li /a 5. The distance between pinholes, dh , was changed the deduced value of the oscillation period is in good agree- by increasing the dimension of the mesh total number of ment with Ref. 27, supporting the validity of the pinhole cells in the mesh , but not changing the number of pinholes model to extract accurate values of interlayer coupling pa- or the cell size. By this technique, was varied from 0.2 all rameters. cells have pinholes to 2.0. The pinhole size was kept con- stant at r a, but the effective strength of the pinhole cou- VII. MAGNETIC PHASE DIAGRAM pling was varied by changing lh so that varied from 0.04 to FOR THE PINHOLE MODEL 2. The remanence mR and the saturation field hS were calcu- lated over this range of and and are plotted in gray levels A systematic computation of the magnetization curves in Fig. 12 with and as the coordinates defining the phase and the spin configurations versus the ratios d/li and space. In Fig. 12 it is useful to divide the phase space into r/lh was done and each magnetization curve was analyzed four different domains F, AF, AF F, and AF*. The domain to extract the saturation field, remnant magnetization, and referred to as F corresponds to strong ferromagnetic coupling equivalent biquadratic and bilinear couplings. The param- by the pinholes and has mR mS 1 Fig. 13 a . The AF eters we chose for the identification of the various domains domain, by contrast, corresponds to the situation for which of the phase diagram were either hS and mR , which are the the pinhole coupling is weak, the spin configuration is basi- measured saturation field and remnant magnetization, or Jeff cally antiferromagnetic, mR 0, and m is linear in h Fig. and Beff , which are the apparent bilinear and biquadratic 13 b . The AF F domain is intermediate between F and coupling factors. So we could plot these macroscopic values AF. Significant regions of the film are basically AF, but there versus the phase space parameters and . The agreement of are small ferromagnetic regions near each pinhole. The spin the pinhole model to the biquadratic fits was also estimated configuration shown in Fig. 14 for 2 is in the AF F as a standard deviation 2, each computed m(h) curve being area. It clearly shows the in-plane local magnetization fitted to Eq. 15 this equation relates h to m and is obtained aligned along the field axis at the pinhole surrounded by a FIG. 12. Phase diagram of the pinhole mod- el: remnant magnetization a and saturation field b of calculated magnetization curves with varying reduced pinholes sizes and distance between pinholes . PRB 60 PINHOLES IN ANTIFERROMAGNETICALLY COUPLED . . . 4139 FIG. 14. Zero-field spin configuration obtained for the inhomo- geneous AF F domain of the phase diagram. a/li . However, since the pinholes affect the magnetization in the layers over a distance li , the transition AF F F occurs for rh li 1 in Fig. 12 . Finally, if both the AF interlayer coupling and the pinhole coupling are weak i.e., for small values of and , we have what we call the AF* domain. In this domain, referred to as a homogeneous domain, the pinholes are significantly coupled to one another and compete with the AF exchange for spin orientation throughout the film area. In the AF* region the remanence is small or zero and the magnetization curves are rounded with a significantly lowered saturation field. However, since this region is constrained in the lower- left corner of the phase diagram, small variations of or induce important changes of mR or hS . Note in Fig. 13 that the computed magnetization curve is best described by the biquadratic fit in the AF* domain d whereas the fit of the AF F magnetization curve is bad c . This is due to the linear slope of the inhomogeneous m(h) curve which cannot be accurately simulated by the biqua- FIG. 13. Calculated magnetization curves with the pinhole dratic coupling model. The image plots vs and of Beff , model corresponding to the various situations of the phase diagram. Jeff , and 2 are shown in Fig. 15. In Fig. 15 a , the ratio The dashed lines are obtained with the pinhole model, while the Jeff /J is lower than 1 for most of the calculated , points, solid lines are their best fits to the bilinear-biquadratic model these except in the AF domain. For the AF F and AF* domains, fits are worse in the AF F domain . this decrease of Jeff is accompanied by an increase of Beff up to 30% of the initial bilinear coupling as shown in Fig. 15 b . pseudowall of width 5 cells, in agreement with the ratio However, the relevance of the biquadratic coupling for the l pinhole model is not perfect in the AF F domain as shown i /a 5. The magnetization curve in this domain shows a finite remanence followed by a magnetization linear in h by 2 Fig. 15 c . So the best agreement between the pin- Fig. 13 c . We refer to this domain as an inhomogeneous hole model and biquadratic coupling is for and 1, where domain because, within it, identifiable separate ferromag- 2 is the lowest and Beff /Jeff remains non-negligible netic and antiferromagnetic domains coexist and the magne- (Beff /Jeff 0.2­ 0.5). These values of and correspond to tization curves have two distinct parts too. The boundary of realistic values of the dimensioned parameters for a standard the AF F and F domains is rather sharp. Intuitively, one multilayer if 1, for a pinhole size of a 10 Å, the dis- would expect it to correspond to the percolation limit of the tance between pinholes is dh 50 Å and the corresponding pinholes which is obtained for r d a or lh /li , AF interlayer coupling is J 0.4 erg/cm2 . This domain is 4140 J. F. BOBO et al. PRB 60 VIII. DISCUSSION Exchange-coupled multilayers with small spacer layer thickness are likely to contain pinholes, and in the previous sections, we have demonstrated that experimental magneti- zation curves can be simulated with our micromagnetic model based on pinhole coupling. Our model presents a lim- ited numerical description of the influence of pinholes. For instance, we have ignored size or shape fluctuations of the pinholes, the only randomized parameter was their distance over the mesh; the effect of thermal fluctuations, studied in Ref. 20, was not treated here. Features like the domain struc- ture of AF-coupled multilayers due to anisotropy fluctuations or magnetostatic interactions were also ignored; we assumed that the layers were infinite and the magnetization was in plane. According to Persat et al.,29 during the field decrease from the saturated state of the AF-coupled bilayer, the mo- ments of both layers may open either way clockwise or anticlockwise randomly over the area of the layers, gener- ating a complex symmetry-based domain structure with the magnetization of the domain walls aligned along the applied magnetic field. These authors point out that when artificial antiferromagnetic AAF layers29 are used for sensors appli- cations, the device performance is best when the magnetiza- tion vectors in both ferromagnetic layers are perfectly anti- parallel at zero field. The presence of pinholes can be a precursor for the rotational symmetry-based domain struc- ture, generating local pinning centers for the field-aligned domain walls and leading to a decrease of the magnetic sen- sitivity of the devices. Other interlayer coupling perturbations could be pro- posed. The first one, the so-called orange-peel coupling, a dipolar interaction induced by the roughness of the layers which decreases exponentially versus the spacer layer thick- ness was proposed by Ne´el in 1963.30 However, since it is a global ferromagnetic coupling, it would just add to the anti- ferromagnetic bilinear coupling and reduce its strength with- out leading to biquadratic coupling. Furthermore, if we try to estimate the intensity of the orange-peel coupling with ferro- magnetic layers having roughnesses and thicknesses similar to the ones shown in Figs. 5 and 6, we will end up with a coupling of 0.01­0.1 erg/cm2, which is too low to actually FIG. 15. Apparent bilinear coupling Jeff a , biquadratic cou- compete with the observed AF coupling. Therefore, for small pling Beff b , and error 2 between the pinhole model and the bilinear-biquadratic coupling model over the phase diagram. The spacer thickness, orange-peel coupling is assumed to play a dashed area in b corresponds to the best agreement between both minor role in the modifications of interlayer exchange. The models. It is located at the border between the F, AF F, and AF* same kind of argument too low biquadratic contribution domains of the phase diagram. can be opposed to the more recent model developed by Demokritov et al.31 Other features like thickness fluctuations or loose spins at the interfaces may affect the interlayer cou- close to the F domain, and small variations of the pinhole pling since the intensity of exchange coupling directly de- size and density via and may strongly affect the behavior pends on the spacer thickness . But these are only variations of an indirect coupling and they cannot become as strong as of the multilayer, causing either a dramatic reduction of the those caused by pinholes. interlayer exchange coupling or even the disappearance of AF coupling in favor of a ferromagnetic apparent coupling. We think this is the reason why the GMR multilayers do not often display highly reproducible interlayer exchange cou- IX. CONCLUSION pling at low spacer layer thickness. Indeed, the defect density Our model reveals that a low pinhole area density may pinholes, layer thickness fluctuations, etc. is intimately re- lead to non-negligible remanence or even to ferromagnetic lated to the preparation conditions of the multilayer high apparent coupling. The pinhole model explains the shape of vacuum evaporation versus sputtering and the choice of most of the MH curves at low spacer thickness, without in- buffer layers. troducing a biquadratic exchange term, and yields a higher PRB 60 PINHOLES IN ANTIFERROMAGNETICALLY COUPLED . . . 4141 apparent value of the intrinsic bilinear interlayer coupling behavior of real multilayers and it is likely to appear when than the one deduced from the saturation field. This may the spacing layers are of a few atomic planes. reconcile the discrepancy between theoretical models and ex- perimental data. As shown in the case of NiFe-Ag multilay- ACKNOWLEDGMENTS ers, the oscillation periodicity of the interlayer coupling de- duced from the pinhole model also leads to values in better The authors acknowledge Dr. B. Rodmacq and Dr. J. agreement with theoretical models. Finally, we have shown Pierre for technical support and fruitful discussions. Support in this paper that a distribution of ferromagnetic point for this work from CNRS and NATO under Grant No. CRG defects-the pinholes-is a good approach to describe the 971645 is also acknowledged. 1 S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 1991 . 17 W. F. Brown, Magnetostatic Principle in Ferromagnetism 2 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen van Dau, F. North-Holland, Amsterdam, 1962 . Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, 18 A. S. Arrott and B. Heinrich, J. Magn. Magn. Mater. 93, 571 Phys. Rev. Lett. 61, 2472 1988 . 1991 . 3 S. S. P. Parkin, N. More, and K. Roche, Phys. Rev. Lett. 64, 2304 19 J. Kohlhepp, M. Valkier, A. van der Graaf, and F. J. A. den 1990 . Broeder, Phys. Rev. B 55, R696 1997 . 4 R. E. Camley and J. Barnas, Phys. Rev. Lett. 63, 664 1989 . 20 D. B. Fulghum and R. E. Camley, Phys. Rev. B 52, 13 436 5 B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. 1995 . Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 1991 . 21 J. F. Bobo, H. Fischer, and M. Piecuch, in Magnetic Ultrathin 6 M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. Mosler, J. A. Wolf, S. Films: Multilayers and Surfaces Interfaces and Characteriza- Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 125, 635 tion, edited by B. T. Jonker, S. A. Chambers, R. F. C. Farrow, C. 1991 . Chappert, R. Clarke, W. J. M. de Jonge, T. Egami, P. Gru¨nberg, 7 B. Rodmacq, K. Dumesnil, Ph. Mangin, and M. Hennion, Phys. K. M. Krishnan, E. E. Marinero, C. Kau, and S. Tsunashima, Rev. B 48, 3556 1993 . MRS Symposia Proceedings No. 313 Materials Research Soci- 8 P. Bruno, J. Magn. Magn. Mater. 121, 248 1993 . ety, Pittsburgh, 1993 , p. 467. 9 J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 1991 . 22 The source code and compiled application for workstation or Ma- 10 H. Fujiwara and M. R. Parker, J. Magn. Magn. Mater. 135, L23 cintosh is available on request from the authors. 1994 ; H. Fujiwara, W. D. Doyle, A. Matsuzono, and M. R. 23 J. F. Bobo, M. Piecuch, and E. Snoeck, J. Magn. Magn. Mater. Parker, ibid. 140-144, 519 1995 . 126, 440 1993 ; J. F. Bobo, E. Snoeck, M. Piecuch, and M-J. 11 J. J. Krebs, G. A. Prinz, M. E. Filipkowski, and C. J. Gutierrez, J. Casanove, in Polycrystalline Thin Films: Structure, Texture, Appl. Phys. 79, 4525 1996 . Properties and Applications, edited by K. Barmak, M. A. 12 U. Gradmann and H. J. Elmers, J. Magn. Magn. Mater. 137, 44 Parker, J. A. Floro, R. Sinclair, and D. A. Smith, MRS Symposia 1994 . Proceedings No. 343 Materials Research Society, Pittsburgh, 13 O. Massenet, F. Biragnet, H. Juretschke, R. Montmory, and A. 1994 , p. 423. Yelon, IEEE Trans. Magn. MAG-2, 533 1966 . 24 J. F. Bobo, B. Baylac, L. Hennet, O. Lenoble, M. Piecuch, B. 14 J. F. Bobo, L. Hennet, M. Piecuch, and J. Hubsch, J. Phys.: Con- Raquet, and J-C. Ousset, J. Magn. Magn. Mater. 121, 291 dens. Matter 6, 2689 1994 . 1993 . 15 B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 1993 and 25 John C. Spence, Experimental High Resolution Electron Micros- references therein. copy Clarendon, Oxford, 1981 . 16 This assumption is based on the fact that for very thin films, the 26 H. Kikuchi, J. F. Bobo, and R. L. White, IEEE Trans. Magn. aspect ratio of the pinhole is not too unfavorable limited gain in MAG-33, 3583 1997 . demagnetizing energy if the magnetic moments are perpendicu- 27 P. Bruno and C. Chappert, Phys. Rev. Lett. 67, 1602 1991 ; 67, lar to the films in the pinhole , but the loss in exchange energy at 2592 E 1991 ; Phys. Rev. B 46, 261 1992 . the two interfaces with the magnetic layers is very large. If one 28 S. Bouat and B. Rodmacq private communication . assumes a total loss of the demagnetizing energy the volume of 29 N. Persat, H. A. M. van den Berg, K. Cherifi-Kodjaoui, and A. the pinhole multiplied by the square of the saturated magnetiza- Dinia, J. Appl. Phys. 81, 4748 1997 . tion , this energy will only compensate the energy needed to 30 L. Ne´el, C. R. Hebd. Seances Acad. Sci. 255, 1676 1962 . restore the in plane magnetic moment in magnetic film for a 31 S. Demokritov, E. Tsympbal P. Gru¨nberg W. Zinn, and Y. K. thickness of the order of 10 nm for the individual films. Schuller, Phys. Rev. B 49, 720 1993 .