PHYSICAL REVIEW B VOLUME 57, NUMBER 9 1 MARCH 1998-I Magnetic-sublayers effect on the exchange-coupling oscillations versus cap-layer thickness M. Zwierzycki and S. Krompiewski* Institute of Molecular Physics, P.A.N., Smoluchowskiego 17, 60-179 Poznan´, Poland Received 24 July 1997; revised manuscript received 29 October 1997 We have found that some periods of interlayer-exchange-coupling IEC oscillations as a function of cap- layer CL thickness may be suppressed if the in-plane extremal spanning vectors of the cap- and ferromagnet- material Fermi surfaces do not coincide. The suppression of the IEC oscillations versus the CL thickness holds also if the magnetic slab thickness tends to infinity. On the one hand, we have shown by means of very simple arguments that apart from the well-known selection rules concerning the spacer and cap layers another rule related to the magnetic sublayers has to be fulfilled for the interlayer coupling oscillations versus CL thickness to survive. On the other hand, the distribution of induced magnetic moments across the nonmagnetic cap and spacer sublayers has been computed and shown to reveal the underlying periodicity of the materials they are made of i.e., related to their bulk Fermi surfaces independently of whether or not the selection rules are fulfilled. This means that the IEC oscillations are of global nature and depend on all the sublayers that constitute the system. S0163-1829 98 06709-5 I. INTRODUCTION which may be viewed as a manifestation of the quantum-well states.16­19 Magnetic multilayers have been intensively studied for over a decade now.1­3 The reasons are, apart from challeng- II. METHOD ing cognitive aspects, already partially realized practical applications of superlattices as magnetoresistive sensors, an- Our earlier papers6,20,21 based on the single-band tight- gular velocity meters, recording heads, and magnetic binding model have proved that the model we use gives a memory elements. The phenomenon most of these applica- reasonable qualitative description of basic physical mecha- tions is based on is the well-known giant magnetoresistance nisms responsible for oscillatory phenomena in magnetic GMR coming from a strong electron-spin dependence of trilayers. Our Hamiltonian, described in detail in Ref. 21, resistivity in magnetic systems. To optimize devices of that consists of the nearest-neighbor hopping and spin-dependent sort, it is necessary to test the effect of all of the ingredients on-site potential terms. The systems under consideration now of the system in question including kind of materials they are trilayers capped with an overlayer, of the type are made from and thicknesses of particular sublayers either novrO/nfF/nsS/nfF, where novr , nf , and ns stand for the directly on GMR or indirectly on the interlayer exchange numbers of cap (O), ferromagnetic (F), and spacer (S) coupling IEC . Obviously, the effect of a spacer on IEC was monolayers in the perpendicular z direction. Hereinafter the established first3,4 and then that due to magnetic subscripts and superscripts ovr and s will always refer to the sublayers;5­10 finally the cap-layer CL effect has been stud- cap and spacer layers, whereas the spin-dependent param- ied quite recently.11­15 eters referring to ferromagnetic sublayers will be indexed by Before we present our original results let us briefly recall or . For simplicity, we restrict ourselves to a simple the most important facts concerning the CL's: i The IEC cubic structure and regard the lattice constant and the hop- oscillates as a function of CL thickness with a period deter- ping integral as the length and energy units, respectively. mined by extremal k spanning vectors of the CL Fermi sur- The interlayer exchange coupling has been calculated face, ii a bias of the oscillations their asymptotic value from the difference in thermodynamic potentials exactly as depends on spacer thickness,11,14,15 iii the IEC oscillations in Ref. 21. Moreover, the magnetic moments including the are strongly suppressed if stationary in-plane spanning vec- induced ones m have been expressed in terms of the eigen- tors of the CL Fermi surface do not coincide with their coun- functions u of the Hamiltonian as mi ni ni , with ni terparts of the spacer Fermi surface,14,15 and iv the direct E ui, (E) 2, where the summation runs over occupied and inverse photoemissions16,17 on various combinations of states. overlayers deposited on different films show a periodic dis- tribution of the so-called quantum-well states QWS's with III. ASYMPTOTIC LIMITS periods determined by extremal spanning vectors of the overlayer Fermi surface. We shall refer to the latter only In this section we present some analytic formulas that will indirectly, by exploiting the fact that the QWS's lead to some be useful for the interpretation of rigorous numerical results spin polarization of nonmagnetic cap layers. of Sec. IV. As has been shown in Ref. 22, the IEC can be The aim of the present paper is to emphasize the rel- Fourier transformed with respect to ns and n . That proce- evance of magnetic sublayers to IEC oscillations as a func- dure can be quite straightforwardly generalized to include tion of CL thickness. In addition, we shall comment on in- the CL thickness as well. The resulting asymptotic within duced magnetic moments in the nonmagnetic sublayers, the stationary phase approximation expression consists of 0163-1829/98/57 9 /5036 4 /$15.00 57 5036 © 1998 The American Physical Society 57 BRIEF REPORTS 5037 the terms of the form A s pqrn(k ,EF)exp 2i pkzns (qkz rk ovr z)nf nkz novr summed over all the in-plane wave vec- tors for which the exponential is stationary. The A coeffi- cients are defined analogously to those in Ref. 22. Their exact numerical values are not important for qualitative con- siderations; we note only that all the amplitudes of oscilla- tions vanish asymptotically with the given sublayer thickness going to infinity.22 There exist, however, some additional restrictions imposed by the asymptotic behavior of the IEC. In particular, a direct generalization of the results of Ref. 22 to the present case, with the cap layer, gives A0qrn 0 no coupling for ns ). Another limit to be taken is nf , when, in view of the above-mentioned asymptotic behavior, all the terms tend to zero except for A FIG. 1. Exchange coupling vs cap-layer thickness for ns 5, p000 and A p00n . Since the oscillations versus spacer thickness survive in this limit nf 3, EF 2.1, Vs V 0, and V 2.0. Stationary in-plane in contrast to the ones versus the CL thickness that decay spanning vectors of the spacer and both the ferromagnetic FS's are k see below , we conclude that A ( , ). For curves a and b the stationary in-plane vector of p000 0 and A p00n 0. the cap FS remains the same, opposite to curves c and d, for which Finally, taking into account the above-mentioned restric- k tions and keeping for simplicity only the lowest-order har- (0, ),( ,0) which results in suppressing the oscillations. monics, we arrive at the formula It is evident from formula 1 that the bias of oscillations with CL thickness depends not only on the spacer- and magnetic-layer thicknesses but on the on-site Vovr potential J A sn sn 1000e2ikz s A1100e2i kz s k nf as well. The latter observation results from the fact that the A 1 coefficients in the second and third terms of Eq. 1 depend A sn n sn n ovrn on the value of the reflection coefficient at the cap- 1010e2i kz s kz f A1101e2i kz s kz f kz ovr ferromagnet interface, which in turn depends on the cap ma- 2 3 terial electronic structure. A sn n ovrn The stationary spanning vectors, for a sublayer character- 1011e2i kz s kz f kz ovr ***, 1 ized by the potential V, can be determined in a very simple 4 way by minimizing with respect to k the following Fermi where the 's are the sets of in-plane wave vectors for which surface equation for the sc lattice: the relevant exponentials are stationary. For the case of the CL thickness dependence this allows us to formulate the kz k ,EF arccos V EF /2 coskx cosky . 3 present selection rule, which in its general form for ns and nf large and fixed and novr large and varying reads Hence the in-plane extremal spanning vectors are k (0,0) for 6 EF V 2, ( ,0) and (0, ) for 2 EF kovr s V 2, ( , ) for 2 E z 0, p kzns q kz r kz nf 0, 2 F V 6, and with nonvanishing p and either q or r ( is the two- kz arccos V Ef /2 , 4 dimensional gradient in kx-ky space . This means that out of all the stationary vectors of the cap material Fermi surface with 2, 0, and 2 for the corresponding k , respectively. FS only those that simultaneously satisfy the above- Thus the period of oscillations versus the sublayer with the mentioned conditions for the in-plane gradients give rise to potential V) thickness is just /kz or /( kz) . the oscillations with CL thickness. Equation 2 is the main result of the present paper. This condition becomes even IV. NUMERICAL RESULTS simpler in the particular case of the single-band simple cubic We shall now present our exact numerical results see model considered hereinafter, when the second part of Eq. Ref. 21 for details of the method and show how they can be 2 separates and all the individual in-plane gradients must interpreted in terms of the analytical formulas from the pre- vanish cf. Ref. 22 . ceding section. Figure 1 confirms the well-known fact that The origin of the present selection rule becomes clear if the IEC oscillations versus CL thickness have a period de- we qualitatively interpret Eq. 1 in terms of the quantum termined by the kind of material the cap is made of and get interference model.23 The first term corresponds to the states suppressed if there is a mismatch in the corresponding in- reflected once at each of the spacer-ferromagnet interfaces, plane spanning vectors of the CL and the spacer. The sup- the second and third terms to the states penetrating one of the s magnetic layers and reflected back at the cap-ferromagnet pression takes place in cases c and d, where k k interface, and the last two terms describe states reaching the ( , ), opposite to k over (0, ),( ,0). The de- outer boundary of the cap layer ``vacuum'' . It is quite clear pendence of the bias values on Vovr is also clearly visible. therefore that the n The magnetic-sublayers effect is presented in Fig. 2, which f -dependent phase factor also must be taken into account while performing the stationary-phase ap- shows that the suppression may be due to the misfit in the proximation. k 's corresponding to the overlayer and magnetic sublayers, 5038 BRIEF REPORTS 57 FIG. 4. Induced magnetic moments with FIG. 2. IEC vs CL thickness for n B 1) for ns novr s 5, n f 10, EF 2.1, Vs 40 and n Vovr 0.3, and V 2.0. Stationary in-plane spanning vectors f 10 for parallel full line and antiparallel dashed line configurations. The other parameters as in Fig. 1 c . Thick vertical for the spacer and the cap layer are k (0, ),( ,0). For lines mark the interfaces. curves a and b the minority spin FS stationary points coincide with those of the spacer and the overlayer. For curves c and d both the majority and minority spin Fermi surfaces have the k bigger. This is shown in Fig. 3 and, to our knowledge, has ( , ) spanning vector and consequently the IEC oscilla- not been discussed so far, although such a trend could be tions are suppressed. predicted on the basis of analytical formulas of Ref. 14. In fact, this finding means that in order to avoid undesirable respectively curves c and d), whereas for curves a and b effects of cap layers which may be of different thickness in the periodicity is quite pronounced owing to the matching of an experiment on the IEC oscillations one should work with the above-mentioned spanning vectors. It can be also readily thick magnetic sublayers. It is also noteworthy that the oscil- seen from Fig. 2 that the phases of oscillations as well as the lation bias value depends on the magnetic layer thickness, as bias values depend on the potentials of the ferromagnetic could be predicted from Eq. 1 . layers exchange splitting . It is noteworthy that Figs. 1 and Finally, in connection with the quantum-well state 2 show that the selection rule works quite well, even when concept,16­19 we have studied the distribution of induced the relevant layer thicknesses are rather small: ns 5 and magnetic moments in the CL and in the spacer . A typical nf 10, respectively. This confirms our previous result is presented in Fig. 4. The induced magnetic moments observation21 that relatively small systems in the z direction are measured in dimensionless units ( B 1) and are of the may reveal the asymptotic behavior. A detailed inspection of order of 0.1% with respect to the magnetic layer magnetiza- curves c and d suggests that the selection rule is slightly tion. As expected, the period of the induced-moment distri- more rigorously enforced in Fig. 2 due to nf 10) than in bution within the CL is exactly that anticipated for the bulk Fig. 1 due to ns 5), but the effect is tiny indeed and hardly CL material FS. The effect of the other sublayers is minor, visible. Incidentally, all the periods of oscillations obtained except that the magnitude of the induced moments is also by the numerical computations and visualized in Figs. 1­4 magnetic-slab dependent. This might seem, at a first glance, can be pretty well reproduced in terms of the asymptotic to be in conflict with the IEC behavior, which shows no equations 3 and 4 ; e.g., for EF 2.1 and V 0.6, oscillations for the parameters of Fig. 4 cf. Fig. 1 c . Yet 0.3, 0.0, 0.3, and 0.6, we get 3.6, 4.9, 9.9, 6.9 , and 4.3 the spin polarization in nonmagnetic layers is related to just ML, respectively. one system with the fixed sublayer thicknesses and the given Another rather obvious but noteworthy effect consists in alignment of magnetic sublayers, whereas the IEC results the disappearance of the IEC oscillations versus CL thick- from the total energy thermodynamic potential balance be- ness when the magnetic sublayer thickness gets bigger and tween the two possible ferromagnetic layer alignments and has to do with the series of samples with changing CL thick- nesses. This observation implies that the induced magnetic moments in the nonmagnetic cap layer as well as the QWS give in general the whole set of periods, of which only those survive, as far as the IEC is concerned, that fulfill the selec- tion rules referring to the entire system. In other words, the IEC oscillations are the global characteristic of the whole system, whereas the induced spin polarization in the cap layer is strictly of local nature. The selection rules completed herein by the extra condi- tion related to the extremal spanning vectors of the magnetic sublayers are quite general and apply to real systems too. In particular they allow one to explain why in the case of the Cu/Co/Cu/Co multilayer the short period of oscillations with FIG. 3. Effect of magnetic sublayer thickness on the IEC oscil- Cu cap-layer thickness is absent11 in spite of theoretical lations as a function of cap-layer thickness the parameters are the predictions14 and the photoemission results concerning same as in Fig. 1 except for Vovr 0.6). QWS.16,17,19 In fact, the explanation is simple and quite 57 BRIEF REPORTS 5039 analogous to that of Ref. 22 about IEC oscillations as a func- plane spanning vectors. If this selection rule is not fulfilled, tion of ferromagnetic layer thickness. Of the two in-plane the period anticipated from the bulk cap-layer material will extremal spanning vectors of the Cu Fermi surface only the not occur in the exchange coupling, although it will still be ``belly'' one at k 0 ) coincides with the extrema of the present in the induced moment distribution across the cap majority and minority sheets of the Co Fermi surface, giving layer. Another finding of this paper is that the IEC oscilla- rise to the long period of oscillations. The ``neck'' spanning tions versus CL thickness vanish if the magnetic sublayer vector has no counterpart in the Co FS and this is why there thickness tends to infinity. are no short-period oscillations. V. CONCLUSION ACKNOWLEDGMENTS In conclusion, we have shown that in order for the inter- This work has been carried out under the KBN Grants layer exchange coupling oscillations versus cap-layer thick- Nos. 2P03B-165-10 M.Z. and 2-P03B-099-11 S.K. . We ness to exist, it is necessary that both the cap-layer and thank the Poznan´ Supercomputing and Networking Center magnetic-layer Fermi surfaces share the same extremal in- for the computing time. *Corresponding author. FAX: 48-61 8684524. Electronic address: Phys. Rev. Lett. 75, 4306 1995 . stefan@ifmpan.poznan.pl 12 S.N. Okuno and K. Inomata, J. Phys. Soc. Jpn. 64, 3631 1995 . 1 P. Gru¨nberg, R. Schreiber, Y. Pang, M.B. Brodsky, and H. Sow- 13 J. Barnas´, Phys. Rev. B 54, 12 332 1996 . ers, Phys. Rev. Lett. 57, 2442 1986 . 14 P. Bruno, J. Magn. Magn. Mater. 164, 27 1996 . 2 S.S.P. Parkin, in Ultrathin Magnetic Structures, edited by B. Hei- 15 J. Kudrnovsky´, V. Drchal, P. Bruno, I. Turek, and P. Weinberger, nrich and J.A.C. 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