PHYSICAL REVIEW B VOLUME 56, NUMBER 21 1 DECEMBER 1997-I Exchange coupling between iron layers separated by silver and gold A. T. Costa, Jr., J. d'Albuquerque e Castro, and R. B. Muniz Instituto de FiŽsica, Universidade Federal Fluminense, NiteroŽi, 24210-340, Brazil Received 23 May 1997; revised manuscript received 12 August 1997 The exchange couplings between bcc Fe layers separated by fcc Ag and Au are calculated for Fe/Ag/Fe and Fe/Au/Fe 001 trilayer structures as functions of the spacer thickness for several temperatures. The calculated couplings show a short-period oscillatory behavior in Fe/Au for all temperatures investigated. For Fe/Ag a long period prevails for temperatures T 300 K, but at T 0 a strong short-period contribution is present for Ag thicknesses 30 atomic planes. These results are in very good agreement with finite-temperature experiments, but the coupling strengths as calculated by assuming perfect interfaces are much larger than those observed. It is shown that interplanar distance relaxation at the interfaces leads to a rather large effective change of the coupling amplitude in Fe/Au for Au thicknesses 20 atomic planes, but mainly causes a phase shift in the oscillatory coupling for Fe/Ag. It is found that interfacial interdiffusion substantially reduces the amplitude of the coupling in Fe/Au/Fe, but not much in Fe/Ag/Fe. S0163-1829 97 06645-9 The lattice constant of bulk bcc Fe matches the nearest- Au; at the necks, however, they are comparable. On such a neighbor distances of both fcc Ag and Au within less than basis, one expects the long-period belly contribution to be 1%. This allows the growth of Fe/Ag and Fe/Au multilayers much weaker in Au than in Ag. However, the relative im- in the 001 direction with low-stress interfaces. In the stack- portance of the two components in each system depends also ing, the fcc Ag and Au 001 planes place themselves rotated on the degree of confinement experienced by carriers in by 45° around 001 relative to the Fe 001 planes. The those FS extremum states, caused by the magnetic distances between the fcc 001 planes in both Ag and Au layers.9,12­15 The 001 bcc/fcc interfaces involve two differ- (dAu Ag ) are & times the bcc Fe 001 interplane spacing ent lattice structures which are rotated by 45° around 001 (dFe). The occurrence of atomic steps at the interfaces may relative to each other. Such a rotation imposes distinct thus cause significant misalignments. In fact, earlier attempts boundary conditions on the spacer FS states at the interfaces, to measure the interlayer exchange coupling J in Fe/Ag/Fe especially on those states around the necks, since the states at multilayers failed to observe oscillatory dependence on Ag the belly are not affected by this rotation. A theoretical thickness,1,2 presumably due to the poor quality of the analysis of these effects requires explicit calculations of the samples used. Later, however, J was measured in Fe/Ag/Fe coupling in these systems. and Fe/Au/Fe 001 structures with improved interfaces and, in both cases, it was found to oscillate, with decreasing am- For perfect interfaces, the ions on every 001 atomic plitude, between ferro- and antiferromagnetic as a function plane of the trilayer systems under consideration are ar- of the spacer thickness N.3,4 Well-defined oscillations were ranged in a square lattice. Thus, the wave vector k parallel also observed in these systems by scanning electron micros- to the layers is a good quantum number. It follows that the copy with polarization analysis.5 In those experiments, J(N) formalism developed in Refs. 14 and 16 can be used to cal- predominantly oscillates with a long period in Fe/Ag/Fe culate J, defined as the total-energy difference per surface 001 , and with a short-period in Fe/Au/Fe 001 . atom between the antiferromagnetic and ferromagnetic con- For sufficently large spacer thicknesses, the oscillation pe- figurations of the trilayer. Most of the experimental results riods of J(N) are related to the geometry of the spacer Fermi are for the bilinear exchange coupling term J1 which for surface FS .6­9 For fcc 001 noble-metal spacers such as perfectly smooth Fe/Ag and Fe/Au 001 interfaces is virtu- Ag and Au, J(N) has two oscillatory components: one with ally equal to J/2.16 a long period coming from the ``belly,'' and another with a To calculate J we have used a tight-binding model with short period associated with the ``necks'' of the spacer FS. s,p,d orbitals and hopping up to second nearest neighbors. The period of the former has been directly observed by pho- The tight-binding parameters for all fcc Au and Ag planes toemission in several noble-metal overlayers, including Ag were taken from Ref. 17, and those for ferromagnetic Fe and Au on bcc Fe 001 .10 More recently, quantum well were obtained as in Ref. 18. Results of J1(N) for Fe/Au/Fe states around the FS necks were observed in Cu films grown and Fe/Ag/Fe 001 trilayers are shown in Fig. 1 for various on fcc Co 001 .11 Thus, it is currently also possible to probe temperatures. Clearly, J1(N) is dominated in Fe/Au by the the period of the neck contribution directly by photoemis- short-period neck contribution, for all temperatures consid- sion. ered. This is evidenced in Fig. 2, where the discrete Fourier The weight of each oscillatory component depends on the transform of N2J is taken, at T 0 K, for large values of N. spacer FS curvatures and carrier velocities in the vicinity of Such a procedure is useful for obtaining the relative ampli- the spacer FS extrema states.6­9 Comparison of the Fermi tude of the various oscillatory components of J, provided surfaces of Au and Ag, shows that at the belly the electron's one knows the asymptotic behavior of J(N), and has a reli- average effective mass in Ag is 4 times larger than that in able method of calculating it in this region.19 The 1/N2 0163-1829/97/56 21 /13697 4 /$10.00 56 13 697 © 1997 The American Physical Society 13 698 BRIEF REPORTS 56 FIG. 1. Calculated exchange coupling for Fe/Au/Fe and Fe/ Ag/Fe 001 trilayers as a function of spacer thickness for tempera- tures: T 0 K a , T 200 K b , and T 400 K c . FIG. 3. Calculated exchange coupling at T 300 K for Fe/Au/Fe a and Fe/Ag/Fe b 001 trilayers as a function of spacer thick- asymptotic behavior of the coupling amplitude applies to T ness. The insets show the experimental results of Fuss et al. Ref. 4 0 K and ordered spacers only and, in most cases, it is not a , and of Celinski et al. Ref. 3 b . reached until N 20 atomic planes at least. On the other hand, for Fe/Ag/Fe 001 trilayers it is the long-period belly contribution that clearly prevails for T 300 K. At lower temperatures, however, a significant short-period contribution is visible in Fig. 1 for Ag thickness 30 atomic planes. The relative weight of both contributions is displayed in Fig. 2, where it is shown that at T 0 K, the short- and long-period components of J(N) for Fe/Ag have comparable amplitudes asymptotically. The strong tempera- FIG. 4. Calculated exchange coupling for two different interfa- cial interplanar distances: dFe-Sp dFe open circles, and dFe-Sp dSp FIG. 2. Discrete Fourier transform for 22 N 50 of N2J(N) filled circles see text . Results are obtained at T 300 K for Fe/ at T 0 K for Fe/Au/Fe solid line and Fe/Ag/Fe dashed line Au/Fe a , and Fe/Ag/Fe b trilayer systems, as a function of spacer 001 trilayers. thickness. 56 BRIEF REPORTS 13 699 FIG. 5. Calculated exchange coupling for Fe/Au/Au1 pFep / Au1 qFeq /Fe1 qAuq /Fe1 pAup /Fe 001 trilayers. Results are ob- tained at T 300 K as a function of Au spacer thickness for p q 0 perfect interfaces a ; p 0.025, q 0.05 b ; p 0.05, q 0.1 c , and p 0.1, q 0.15 d . FIG. 6. Calculated exchange coupling for Fe/Ag/Ag1 pFep / Ag1 qFeq /Fe1 qAgq /Fe1 pAgp /Fe 001 trilayers. Results are ob- tained at T 300 K as a function of Ag spacer thickness for ture dependence of the short-period component, and the fact p 0.025, q 0.05 a , and p 0.05, q 0.1. b Filled circles rep- that it manifests itself at T 0 K for large values of N, sug- resent the results for perfect interfaces (p q 0). gest that the confinement mechanism discussed in Ref. 15 is relevant for the neck contribution in Fe/Ag trilayers. The small change in the coupling amplitude of Fe/Ag/Fe 001 , periods determined from the position of the peaks in Fig. 2 for small values of N. On the other hand, it has quite a large agree perfectly with those calculated from the spacer FS ex- effect in Fe/Au/Fe, producing changes of a factor of 4 in trema in the direction perpendicular to the layers, namely: the coupling strength for relatively small Au thicknesses. pb n n Au Ag 9.2(5.3) atomic planes and pAu pAg 2.4 atomic This apparently large variation in amplitude may also result, planes. mainly, from a phase shift. Since the spacer is probed at In Fig. 3 our results are compared with experiments. The discrete intervals, plane by plane, and J basically oscillates agreement is excellent as far as the periods and phase of with a short period of about 2.4 atomic planes, a small phase oscillations are concerned, but the calculated coupling shift, in this case, can effectively produce an apparent large strengths are much larger than those observed, both in Fe/ change in amplitude. Au/Fe and Fe/Ag/Fe 001 trilayers. The discrepancies may Mossbauer spectroscopy has shown that some interdiffu- be due to interface roughness, which can drastically affect sion occurs, during the deposition of Fe over Ag and Au.22 the coupling amplitude.20 However, for Fe/Ag and Fe/Au The amount and extension of interfacial diffusion depend on 001 , interplane distance relaxation near the interface can the substrate's temperature during deposition. For growth at also play an important role, because of the relatively large T 300 K, it is expected to be limited and resticted to very difference between dFe and dAu Ag . In fact, there is some few interfacial atomic planes, because Fe and Ag Au are evidence of tetragonal distortion of the Fe atoms at the inter- known to be immiscible in the bulk. Actually, it differs from faces in Fe/Ag superlattices.21 To investigate such an effect one interface to the other, as none seems to occur during the we have varied the interplanar distance dFe-Sp between the Fe deposition of Ag on to Fe.21,22 and spacer Sp 001 planes at the interface. We have con- The occurrence of interdiffusion produces a disordered sidered two extreme cases, namely, dFe-Sp dFe and alloy at the interfaces. Translational symmetry parallel to the dFe-Sp dSp . In each case, the tight-binding parameters have layers is then broken, and it is usually necessary to take been scaled according to the distance-dependence prescrip- configurational averages of the quantities of interest. As far tion of Andersen et al.,23 and the Fe-Sp hoppings were taken as the interlayer coupling is concerned, Bruno et al.24 have as the average between the Fe and spacer hoppings. The recently shown that, to an excellent approximation, one can results are shown in Fig. 4, where one sees that interfacial still use Eq. 1 of Ref. 18 to calculate J across a disordered interplane relaxation basically causes a phase shift, and a spacer, provided the Green functions involved are replaced 13 700 BRIEF REPORTS 56 by their corresponding configurational averages. Kudrnovsky In summary, we have calculated the interlayer exchange et al.20 have discussed the effect of interface roughness in coupling in Fe/Ag/Fe and Fe/Au/Fe 001 trilayers for sev- Co/Cu 001 systems. They found that the amplitude of the eral temperatures. We have found that the coupling oscillates short-period component of the interlayer coupling is reduced, with a long period in Fe/Ag, and with a short period in Fe/Au by nearly an order of magnitude, when only 10% of interfa- systems. Our results agree with experiment, as far as the cial diffusion occur in Co/Cu/Co 001 trilayers. Here we periods of oscillations are concerned, but the coupling investigate the effect of interfacial interdiffusion in a similar strengths as calculated by assuming perfect interfaces are way, by treating the interfacial atomic planes as disordered much larger than those observed. We have shown that the alloys compatible with a given concentration profile. We re- effect of interplanar distance relaxation at the interfaces can strict ourselves to small interfacial admixtures, and assume be very important for relatively small spacer thicknesses. It that it takes place at two planes on each side of the leads to a rather large effective change of the oscillations Au Ag /Fe interface only. The disorder is treated within a local average t-matrix approximation, which, in the dilute amplitude in Fe/Au/Fe, but basically causes a phase shift in limit, is equivalent to the coherent-potential approximation the oscillatory coupling for relatively small Ag thicknesses used in Ref. 20. Our results for Fe/Au/Fe are presented in in Fe/Ag/Fe. We have also shown that a weak interfacial Fig. 5, for various interfacial alloy compositions. The reduc- interdiffusion substantially reduces the coupling amplitude in tion in the coupling amplitude, though large, is not as dra- Fe/Au/Fe, but has little effect in Fe/Ag/Fe. We conclude that matic as that obtained in Co/Cu by Kudrnovsky, amounting a combination of interfacial interplane relaxation and inter- to a factor of about two for 5­10 % Au/Fe interfacial inter- diffusion, together with possible occurrence of terraces at the diffusion. In Fig. 5 we see that for sufficiently large interface interfaces in real samples, probably accounts for the remain- diffusion, a long-period oscillatory behavior begins to show ing discrepancies between calculated and observed values of up with the suppression of the short-period component, as the coupling strength in these systems. expected. 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