Physical mechanism of oscillatory interlayer exchange coupling

Transcription

1 Journal of Magnetism and Magnetic Materials 116 (1992) L13-Ll7 North-Holland Letter to the Editor Physical mechanism of oscillatory interlayer exchange coupling P. Bruno Institut d Electronique Fondamentale, CNRS UA 22, B2t. 220, ljnic,ersiik Paris-Sud, F Orsay Cedex, France Received 8 July 1992 A theory of the exchange coupling between ferromagnetic layers separated by a non-magnetic metal spacer is proposed by analogy with the Friedel-Anderson theory of localized moments in dilute magnetic alloys, based on the concept of virtual bound state. This model provides a physically transparent interpretation of both the period, phase and intensity of the oscillatory interlayer coupling The recent experimental discovery of oscillatory exchange coupling between ferromagnetic layers separated by a non-magnetic metal spacer [l] has stimulated an important theoretical activity [2-41. In particular, a model based on a Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism allowed to predict in an essentially correct manner the oscillation periods for noble metal spacers [4]. However, the RKKY theory fails in predicting correctly the phase and intensity of the interlayer coupling, essentially because it uses the rather crude approximation of a contact interaction between the spins and the conduction electrons. Actually, this problem bears much resemblance with that of the interaction between magnetic impurities in a non-magnetic host metal, which has been first discussed by Caroli [5] on the basis of Friedel-Anderson picture of localized moments [6,71. This model, based on the concept of virtual bound state [61, has been very successful in explaining the properties magnetic and electric properties of dilute alloys. Within this model, the localized energy levels of the magnetic alloys are broadened into virtual bound states by the hybridization (the so called s-d mixing) with the Correspondence to: Dr. P. Bruno, Institut d Electronique Fondamentale, CNRS UA 22, Bit. 220, Universitt Paris-Sud, F Orsay Cedex, France. conduction electrons of the host metal; the stability of the local moments is described by a Stonerlike condition 171. The s-d mixing induces an oscillatory spin polarization around the magnetic impurities which is in turn responsible for the exchange interaction between impurities [5]. When trying to adapt these ideas to the problem of interlayer coupling, we are faced with the difficulty that the energy levels of the magnetic layers are not localized levels; rather, they form a two-dimensional band structure, so that the meaning of a virtual bound state for this situation seems unclear at first sight. The clue to the problem is to take advantage of the in-plane translational invariance of the problem, i.e. of the fact that the total Hamiltonian is diagonal in k,,: now, for a given k,, the energy levels of the magnetic layers are localized levels and it is clear that the virtual bound states are to be defined locally in the kl, plane. Then, the calculation of the coupling is completely analogous to that of Caroli for impurities [5]. The remarkable result is that only a few vectors k; of the kll plane contribute to the coupling: those for which the measure q, of the Fermi surface of the spacer metal is stationary; the oscillation periods are given by the corresponding q,* and are in agreement with those predicted by the RKKY theory [41. The phase and the intensity of the coupling depend respec /92/$ Elsevier Science Publishers B.V. All rights reserved

2 .1r,u 1.14 I' Ihr, 1 0 / Physicul mechunism of oscillutory interluyiv exchange coupling tively on the occupation and on the spin polarization of thc virtual bound state at ki. The analogy with the problem of dilute riagnetic alloys, upon which the present theory is based, proves to be cxtrcmcly fruitful and provices, for the first time, a realistic and consistent delxription of the overall characteristics of the inte :layer coupling: oscillation periods, phase, intensit.0 and thermal dependmce; it is almost completely analytic, so that the physical mechanisms involved in the phenomenon of oscillatory interlayer coupling are displayed transparently. Investigations of the intedayer coupling on the basis of s-d mixing have been previously attempted by Wang et al. and by Lacroix and Gavigan [2], who calculated the fourth order perturbation in the s-d mixing,. However, these authors neglected the width of the magnetic layers energy band; this is a rathe1, crude approximation which is thus inappropriate to describe correctly the coupling intensity and the oscillation phase. The model consists of two ferromagnetic monolayers F1 and F2 embedded in a non-magnetic metal. The distance between F1 and F2 is z = (N + l)d, where d is the spacing between atomic planes, and N the number of atomic planes of the spacer. The magnetic atoms are assumed to be located at the atomic positions of the host metal. The Anderson-like Hamiltonian [7] contains three terms describing respectively (i) the conduction electrons of the host material yc = c Ek,l,k,C~,,k.,,~rCkl,.~ ;,u 3 (1) k.k ~ (ii) the ferromagnetic layels = c Eklld~kll,~di,kll.~ + c utzt,rll, T ',,RIl, 1 k,l3ct RII (i = 1, 2), (2) (iii) the hybridization between the ferromagnetic layers and the host material v;= c (eikzic 'kil,k di?kll kll,k:.ir,p kll,k ~ ) + h.c. (i= 1, 2). (3) For the sake of simplicity, we restrict ourselves to non-degenerate bands, assume the host material to be symmetric with respect to the layer planes, and take the two magnetic layers as identical. These restrictions, however, are not essential and can be removed for a more realistic approach. In contrast with the magnetic impurity problem, the magnetic layers are not described by localized energy levels, but by two-dimensional energy bands. In eqs. (1)-(3) the summations over kll are performed on the two-dimensional first Brillouin zone of the layers (2DBZ), whereas those over k, run from - ~ / d to T/d; for the host material, this ammounts to using a periodic zone scheme and replacing the first Brillouin zone (FBZ) by a (completely equivalent) prismatic unit cell in the reciprocal space, which is better suited to the problem than the FBZ [41. In eq. (3), V,,k is proportional to 1/ ik, where N, -j +cc Ys 'the total number of layers of the host - material; it is convenient to define pkl,,kz = \in2 Vkll.k,, which is thus independent of Nz. Using the Hartree-Fock approximation, the on-site Coulomb interaction is taken into account by shifting the band energy levels of the magnetic layers according to Et,k,l,u = Ekll + un;.-u. (4) Following Anderson [7] and Caroli IS], the problem is then easily handled by using the Green's functions formalism. If the magnetic layers are infinitely apart, they do not interact, so that the problem reduces to that of a single magnetic layer. Since the total Hamiltonian is diagonal in k,,, the problem can be treated locally in the k,, plane. Due to the mixing interaction V;, the localized state E,,kl,,c, is broadened into a uirtual bound state, with a density of states equal ot the imaginary part (divided by -T) of the Green's function G,'!k l,,u ( ) iot- E;,kll,u - rkll(e) -t iakll(e)

3 P. Bruno / Physical mechanism of oscillatory interlayer exchange coupling L1.5 The width Akil(e) and energy shift rkll(~) of dependence of the coupling on the distance z is the virtual bound state and the phase shift entirely contained in the function T~,~,,,~(E) are given by Fkll(E, ) and For k,t << AkJcF), the phase shifts at the Fermi level are related to the occupation number n,,k and magnetic polarization ml,k,l.of the virtual bound state of vector k,, by the simple relations [61 By using complex contour integration methods, one obtains (for large 2) Fk,,(e, 2) = -A:,( ) 4 P, (E)eiq~, ( 12) P, where q,, = kr - k; and kr (respectively k;) is such that ek,kp = E (respectively E9.k; = E ), with the z compchent of the velocity u, > 0 (respectively u, > 0); the dimensionless quantity T]r,kII,r(EF) = t P(ni,kll +amt,kll). (8) They must be determined self-consistently from the Hartree-Fock equations (4)and the center of the band E,,, must be adjusted to satisfy the Friedel sum rule for the total screening charge [61. As the distance z is reduced, F1 and F2 interact with each other; following Caroli [5], one obtains the interlayer exchange energy per unit area El,2 = I1,2 cos el,,, (9) where el,* is the angle between the magnetizations of F1 and F2, and the interlayer coupling constant is given by M Z ) 1 xfkii(ey z)g~,k,l,cr (c)] 7 (10) where we have retained only the leading term, in the limit of large spacer thickness 2. The reader is referred to the paper by Caroli [5] for details on the derivation of the coupling expression. The of the order of 1, depends only on the band structure of the host material. It is also convenient to define an effective velocity ~1, by For z larger than + (llkll - mk,l) T/2)] (15) where the velocity is taken at ef, the factor exp(iqr. z) oscillates rapidly at energies close to the Fermi level so that the other factors can be kept constant, equal to their value at cf. The result of the integration over E in eq. (10) is then proportional to exp(iq,, z), with qf. taken at E = E ~ When. integrating over kll, the factor exp(iqc. z) oscillates rapidly, yielding a vanishing contribution, except near vectors k; where 4: is stationary. The integration over kll is thus performed using the stationary phase approximation, by expanding 4: around its stationary value 4; up to second order in (kl,- k;), i.e.

4 / Physical mechunism of oscilla~ory interlayer exchangc~ coupling where the cross terms have been canceled by a propcr choice of the,y and y axes. Here and bclow. the index (Y is used instead of (p, u) for all quantities taken at,ti;. After integration ovcr kl,. the final expression of thc coupling in the asymptotic rtgime 191 (i.e. for z at least larger than D<,) is 11 The oscillation periods A, = 27~/I q: I given by the present s-d mixing model are exactly the same as from the RKKY approach [4,8]; this is not unexpected, since the same holds for the impurity problem too. In eq. (171, the intensity of the coupling is given by Icy = sin (.rrm,), (18) where rn, is the spin polarization of the virtual bound state at k; and depends only on the Fernii surface of the host material at k;. The phase is given by i+h, =!J:, I + 27~n,, (20) where n, is the occupation of the virtual bound state at k; and I,!J~ equals 0: -ir/2 or IT when qq is a maximum, a saddle point or a minimum, respectively. The temperature dependence of the coupling due to the rounding of the Fermi-Dirac function f(c) is the same as obtained from the RKKY theory 141 with the attenuation length L, = hcg/27rkbt. (22) The results of the above theory for the inter- layer coupling are completely analogous to those obtained for the exchange interaction between magnetic impurities in dilute alloys [5,10]. The respective influence of the spacer material and of the ferromagnetic layers on the various characteristics of the coupling (oscillation periods, intensity, phase, thermal variation) is well displayed: (9 The oscillation periods and the thermal variation of the coupling depend only on the Fermi surface characteristics of the spacer material; they are independent of the detailed mechanism of the interaction between the magnetic layers and the conduction electrons of the spacer. This is basically why they are correctly described by the RKKY theory [41. (ii) The intensity and the phase of the coupling depend both on the Fermi surface characteristics of the spacer metal (through 1: and I/J~, respectively) and on the interaction between the latter and the magnetic layers (through m, and n,,, respectively). (iii) A remarkable difference between the present result and that for impurities is that only a few k,, vectors contribute to the coupling: those corresponding to stationary values ql? of the Fermi surface measure along the z direction. Thus the intensity and phase of the coupling arc respectively determined by the polarization and occupation of the virtual bound state locally in the kl, plane, not by global values as for impurities. Among the recent experimental results in this field is the observation that (111) Fe/Cu/Fe and Co/Cu/Co multilayers exhibit oscillatory interlayer coupling with the same period (A = 6 monolayers), but with phases differing by about -IT [ll]. The above theory provides a natural explanation for this behavior, as due to the valence difference between Fe and Co; the argument, however is only qualitative, so far, for a quantitative discussion would require a degenerate-band theory. This theory also provides a useful frame for interpreting the results of ab initio calculations, in terms of the spin- and k,,-projected density of states of the ferromagnetic layers, taken at k;. This opens promising outlooks towards a better understanding of the interlayer coupling phenomenon.

5 P. Bruno / Physical mechanism of oscillatory interlayer exchange coupling L17 References [1] B. Heinrich, Z. Celinski, J.F. Cochran, W.B. Muir, J. Rudd, Q.M. Zhong, A.S. Arrott, K. Myrtle and J. Kirschner, Phys. Rev. Lett. 64 (1990) 673. S.S.P. Parkin, N. More and K.P. Roche, Phys. Rev. Lett. 64 (1990) W.R. Bennett, W. Schwarzacher and W.F. Egelhoff Jr., Phys. Rev. Lett. 65 (1990) S.S.P. Parkin, R. Bhadra and K.P. Roche, Phys. Rev. Lett. 66 (1991) J. Unguris, R.J. Celotta and D.T. Pierce, Phys. Rev. Lett. 67 (1991) 140. S.T. Purcell, W. Folkerts, M.T. Johnson, N.W.E. McGee, K. Jager, J. aan de Stegge, W.B. Zeper, W. Hoving and P. Griinberg, Phys. Rev. Lett. 67 (1991) 903. S.S.P. Parkin, Phys. Rev. Lett. 67 (1991) [2] Y. Wang, P.M. Levy and J.L. Fry, Phys. Rev. Lett. 65 (1990) C. Lacroix and J.P. Gavigan, J. Magn. Magn. Mater. 93 (1991) 413. [3] D. Stoeffler and F. Gautier, Progr. Theor. Phys. Suppl. 101 (1990) 139. F. Herman, J. Sticht and M. van Schilfgaarde, J. Appl. Phys. 69 (1991) D.M. Edwards, J. Mathon, R.B. Muniz and M.S. Phan, Phys. Rev. Lett. 67 (1991) 493. [4] P. Bruno and C. Chappert, Phys. Rev. Lett. 67 (1991) 1602, 2592(E); Phys. Rev. B (in press). [5] B. Caroli, J. Phys. Chem. Solids 28 (1967) [6] J. Friedel, Nuovo Cimento Suppl. 7 (1958) 287. [7] P.W. Anderson, Phys. Rev. 124 (1961) 41. [8] Note that Wang et al. and Lacroix and Gavigan [2] found that the coupling contains a RKKY-like term and a superexchange-like term, whereas we obtain here only a RKKY-like contribution. This is because the superexchange term decreases faster than the RKKY one, and thus does not appear here where we have retained only the leading term in the limit of large space thickness. [9] Note that 4: is defined algebraically. If 1 q," I happens to be larger than ~ / d it, must be folded into the interval [- T/d; ~ / d by ] adding or subtracting Z.rr/d, for periods smaller than 2d are physically meaningless. This is discussed in detail in ref. [4]. [lo] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer-Verlag, Berlin, 1985) chap. 6. [ll] F. Petroff, A. Barthilemy, D.H. Mosca, D.K. Lottis, A. Fert, P.A. Schroeder, W.P. Pratt Jr., R. Laloee and S. Lequien, Phys. Rev. B 44 (1991) 5355.