Oscillatory behavior of interface exchange coupling caused

Transcription

1 :ELSEVIER Computational Materials Science 1 (1998) COMPUTATIONAL MATERIALS SCIENCE Oscillatory behavior of interface exchange coupling caused by finite caps of variable thickness J. Kudrnovskfa-b**, V. Drchal a.b, P. Brunoc, I. Turekd, P. Weinbergerb a Institute of Physics, Academy ofsciences ofthe Czech Republic, Nu Slovance 2, CZ-18 4 Praha 8, Czech Republic Institute for Technical Electrochemistv, Technical tinibersig of Vienna, Getreidemarkt 9, A-16 Vienna, Austriir Institut d'electronique Fondamentule, CNRS tim 22, tmt. 22, tiniverszti Paris-Sud, F-9145 Orsa?; France Institute of Physics of Materials, Academy of Science.c. ofthe Czech Republic, iiikovu 22, CZ Brno, Czech Republic The effect of non-magnetic cap-layers on the periods and the amplitudes of the oscillations of interlayer exchange coupling (IEC) is studied theoretically using an ab initio spin-polarized surface Green function technique within a tight-binding linear muffin-tin orbital method and the Lloyd formulation of the IEC Applications are made to the free-electron like model as well as to Co/Cu/Co(O 1) trilayers with a cap interfacing vacuum through the dipole barrier. The results are analyzed in terms of a discrete two-dimensional Fourier transformation which confirms a pronounced oscillatory behavior of the IEC with respect to the thickness of the cap- and the spacer-layers. The result's are in agreement with available experimental data as well as with predictions of the electron confinement model of the IEC. Copyright 1998 Elsevier Science B.V. 1. Introduction The oscillatory interlayer exchange coupling (IEC) between magnetic layers separated by a non-magnetic spacer hlas recently attracted considerable attention. The physical origin of such oscillations is attributed to quantum interferences due to spin-dependent confinement of electrons in the spacer [ 141. An important conclusion, namely tihat the periods of the oscillations with respect to the spacer thickness are determined by the spacer Fermi surface, bas been confirmed by numerous experiments. Recently increasing interest was devoted to the study of the influence of the cap [5-71 on the IEC, and first model dheories have appeared based on simple free-electron-like models [6-91. These theories predict a novel oscillatory behavior of the IEC with respect to the thickness of the cap, which is attributed to electron confinement in the non-magnetic cap due to the vacuum barrier 191. The peiriods of these oscillations are related to the Fermi-surface of the cap material. Experimental observations of the oscillations of the IEC with respect to the cap thickness in a -~ * Corresponding author. Address: Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ- 184 Praha 8, Czech Republic. Tel.: (15 295; fax: $ ?27; e-~n~iil: kudmov@fzu.cz /98/Sl9. Copyright I998 Elsevier Science B.V. All rights reserved PI1 S ( 9 7 ) 9 7-9

2 J, Kudronovsk?; et al. /Computational Materials Science I (1998) variety of materials [S-71 support this view of the IEC. In this paper we present a detailed ab initio study of huch oscillations. 2. Formalism The system considered consists of a stack of layers, namely from the left to the right: (i) a semi-infinite (nonmagnetic) substrate, (ii) a left ferromagnetic slab of thickness M (in monolayers, MLs), (iii) a nonmagnetic spacer of thickness N, (iv) a right ferromagnetic slab of thickness M, (v) a nonmagnetic cap of thickness P, and (vi) a semi-infinite vacuum. In general, the various parts of the system may consist of different metals, including disordered cubstitutional alloys. The thickness of the left ferromagnetic slab may be extended to infinity Ah initio formulation The system described above is formally partitioned into two subsystems, denoted left (L) and right (R). The left subsystem contains the non-magnetic substrate and the left ferromagnetic slab. The right subsystem con\ists of the spacer, the right ferromagnetic slab, the cap, and the vacuum. We employ the Lloyd formulation of the IEC combined with a \pin-polarized surface Green function technique as based on the tight-binding linear muffin-tin orbital (TB-LMTO) method [lo]. Considering M and M as implicit, system specific parameters, the exchange coupling energy &y (N, P) for a given set (N. P) of spacer and cap layers can be written as the difference betaeen the configurationally averaged grandcanonical potentials of the antiferromagnetic and the ferromagnetic alignment of the magnetic slabs, (1) Here j(z) is the Fermi-Dirac distribution function and trl denotes the trace over angular momentum indices L = (tirn). In (I), the energy integration is performed along a contour in the upper half of the complex energy plane, the kli-summation runs over the surface Brillouin zone (SBZ), and Nil is the number of sites in the two-dimensional lattice. The quantity M(N. P; kll, L) is defined as where N[ = ( I - A(,B;)-~(I - A ~B~)(I - A ;B~)-I(~ - A;B;). (2) A{ = St(kll)G~(M; kil, :)S(kl). B{ = C&(M. N, P; kll, z). (3) In Eq. 1(3), the quantities St (kll ) and S(k11) are the screened structure constants which couple neighboring (principal) layers [I 11, G z(m; kll, z ) is the configurationally averaged surface Green function of the left magnetic subsystem, while!:$(m, N, P: kll. z ) is its counterpart for the right magnetic subsystem which also contains the spacer, the cap, and the vacuum. Finally, o denotes the spin index (a =f. &). We refer the reader to [ 111 for an efficient evaluaiion of the configurationally averaged surface Green function in the framework of the present formalism. The use of a Green function formulation of the IEC is essential for describing randomness in the spacer, magnetic slabs, and the cap within the coherent potential approximation (CPA). The calculations are significantly simplified [ 12 I by using the vertex-cancellation theorem [ 131. It should be noted that due to the block tridiagonal form of the inverse of the Green function matrices, inherent to the TB-LMTO method, the evaluation of the surface Green functions, and hence the evaluation of the IEC itself, scales linearly with M, M, N and P.

3 19 J. Kudronovsk?j et d. /Computational Materials Science 1 (1998) Electron conjnement in a non-magnetic cap For a qualitative interpretation of the ab initio calculations it is convenient to summarize first the simple freeelectron like model of the cap proposed in [9]. The expiression for &,(A', P) in the limit of large spacer and cap thicknesses, and for the simplest case when the cap anid the spacer are formed by the same material. is given by Here, A, and B, are the (complex) amplitudes of the oscillations expressed in terms of reflection and transmission matrices and qa is a stationary spanning wave vector of the bulk Fermi surface of the spacer (and the cap material) in the direction normal to the layers. The wave vector is measured in units of d-', where d is the distance between the atomic layers. There may be several such vectors labelled by the index a. The first term in Eq. (4) gives the IEC for an infinitely thick cap and oscillates with respect to the spacer thickness with a period pa = 2n/qa and decays as N-*. The second one is found to oscillate with the same period pa but now with respect to both the spacer and the cap thickness, and to decay as (N + P)-*. Generally,!.he second term is smaller than the first one, so that for a given spacer thickness N, the oscillations of the IEC with the cap thickness P often do not lead to changes of the s:ign of the coupling. Let us now consider the more complicated case when th'e cap and the spacer are different materials. The expression for the IEC is then given by [9] H[ere, q, and q& are stationary spanning vectors of the bulk Fermi surface of the spacer and the cap, respectively, corresponding to the same in-plane wave vector kll(,). If there is no such vector, the second term in Eq. ( 5) has to ble dropped. Note, that unless kll(,) corresponds to a high-s,ymmetry point, a spanning vector q' of the cap material Fermi surface is unlikely to be stationary at one and the: same kll(a). A detailed discussion of this point is given in [4]. In the above equation. A and A' are related to the perpendicular Fermi velocity and the curvature radius of the Fermi surface of the spacer and of the cap, respectively. As before, the first term in Eq. (5) corresponds to the contribution of an infinitely thick cap, and the second one to the correction due to electron confinement in the cap. The first term is of the same form as before, but the second one turns now out to be more complicated: it is no longer an oscillatory function of N + P and the amplitude no longer decays as (N + P)-:. An approximate (N + P:)--2 decay may still be obtained provided that the Fermi surfaces of the spacer and the cap material are not too different Detuils of calculations Numerical studies were performed by assuming an ideal fcc( 1) stacking for the substrate, the spacer, the magnetic slab, and the cap corresponding to the experimental lattice spacing of fcc Cu. In each case, the magnetic slabs consist of CO layers, the spacer and the substrate are formed by Cu layers while the cap consists either of Cu or Rh layers, or of random substitutional Cu75Zn25 alloy layers. We employ the frozen potential approximation and align bulk fcc-cu, fcc-co, fcc-rh, and fcc-cu75zn25 to the Fermi energy of the substrate (for more details bee [ 141). The value of the dipole barrier between the cap and the vacuum was determined self-consistently for the case of a scmi-infinite fcc( 1) cap. Special care was devoted to the Brillouin zone and energy integrations (for more details

4 J. Kudronovskj et al./computational MatiTrials Science I (1998) see [lo]). Since temperatures can obscure an analysis of the oscillation amplitudes [4,1], all calculations reported here refer to T = K Ab initio formulation of the free-electron model The general formalism developed in Section 2.1 can be used to study a free-electron model which allows one to understand the effect of cap-layers on the properties of the IEC in a particularly transparent manner. We place empty spheres [ 151 at each lattice site of our ideal fcc( I) stack of layers and choose the system Fermi energy such that the corresponding Fermi vector is far enough from artificial zone boundaries introduced by the underlying lattice. The spacer-, cap-. and magnetic-bands differ from each other only by a rigid shift. There is only a single stationary vector at kll = that corresponds to the Fermi sphere diameter, but it can have a different value (leading to different periods) for the spacer and the cap. It should be noted that the solution of the free-electron model as obtained from an ab initio formulation is not limited to the asymptotic region as in other approaches but is valid also for systems &ith H thin spacer and/or a thin cap Analysis of results In order to analyze the set of values &,T(N, P) where pairs (N, P) typically include N, P E (1, 51, we employ two different methods. In the first approach a direct representation of &x(n, P) with respect to either N (spacer thickness) or P (cap thickness) is used by keeping the respective other variable implicit. The second method consists of a two-dimensional discrete Fourier transformation of I, (A'. P), N=Ni P=PI with respect to thz so-called biased value E1 (N), t:l(n) = &,(A', ) = lim &x(n. P), P+K (7) The prefactor (N + P)2 in Eq. (6) is consistent with the asymptotic behavior discussed in Section 2.2. Strictly speaking, the prefactor (N + PI2 In Eq. (6) is only correct when the spacer and cap are formed by the same material (see Elq. (4)), however, we shall deliberately use this prefactor also when the spacer and the cap correspond to different materials. The use of a large enough set of input data permits to exclude the preasymptotic region, e.g. by choosing N, Y E (I, 5). In this way the periods of oscillations and their amplitudes a5 a function of both the spacer and the cap thickness can clearly be extracted. 3. Results and discussion 3.1. Free-electron model Consider a system with a semi-infinite magnetic slab (left subsystem) and a five monolayer (ML) magnetic slab with a spacer and a cap (right subsystem) as motivated by comparison with a realistic Co/Cu/Co(O I ) system. In all cases considered, in the following the spacer and the magnetic majority bands coincide whereas the minority band is shifted upwards by.75 Ry. The Fermi energy is chosen to give the spacer spanning vector corresponding

5 1 <92 J. Kirdronovsk?; et 1. /Coniputrrtional Materials Science 1 (1998) E:.2 1 Model A Model B E3 Fig. 1. Absolute values of the discrete two-dimensional Fourier trmiformation of (N + P)?f2(N. P) (1 5 N, P 15 SO) with respect to the spacer and the cap thickness for the free-electron case described iin the text: (a) Model A, and (b) Model B. to oscillations with a period of about 2.8 MLs. The vacuum is again represented by a free-electron band shifted upwards by I Ry as compared to the spacer band and represents a perfectly reflecting barrier. Three different geometries are assumed: (i) Model A: the cap band coincides with the spacer band, and the same is, true for their spanning vectors; (ii) Model B: the cap band is shifted rigidly upwards by.375 Ry with respect to the spacer band and its spanning vector is reduced in its size (the opposite is true for a downward shift); and (iii) Model C: the cap band is shifted upwards by.15 Ry such that its bottom lies above the Fermi level (the band is ulnoccupied and has thus no spanning vector at ki, = ). The absolute values of the discrete two-dimensional Fourier transformation of ( N + PI2E2(N, P) are presented in Fig. 1. The following conclusions can be drawn: (i) Model A (Fig. l(a)): we observe a clearly pronounced peak situated at q~ = qp 2.25 which corresponds to oscilla~.ions with respect to the spacer and cap thickness with the same period p, = ~ pp 2.8 MLs. and, in accordance with the asymptotic expression, Eq. (4), with the functional dependence & 2(N, P) = &2(N + P), The amplitude of oscillations (the height of the peak) remains unchanged for drfferent subsets (N, P) of 2( N. P) used for the Fourier transformation which confirms the predicted (N + P)- decay of the oscillations; (ii) Model B (Fig. 1 (b)): the most important difference in comparison with the Model A is the shift of the peak of the discrete Fourier transformation to yp 1.87, however, with an unchanged value of qn. This is consistent with a smaller Fermi sphere radius of the cap (the smaller spanning vector) and, consequently, with a weaker confinement. As a result, the amplitude of oscillations also decreases. It should be noted that for a downward shift of the cap-band relative to the spacer leads to a stronger confinement which in turn shifts the peak tcl larger values of qp; (iii) Model C (not shown): no peak is seen because there is no spanning vector (the cap-band is not occupied). Consequently, the values of E,(N, P) are identical for P > Co/Cu/Co(O 1 )-triluyrr with the Cu cap Consider now a realistic Co/Cu/Co(O 1) system with two different geometrical arrangements for the Cu-cap, namely (i) magnetic slabs of different thicknesses (M =, M = 5), and (ii) the case when both magnetic slabs have the same finite thickness (M = M = I). In the limit of an infinite cap [lo] in the former case the so-called long-period oscillations (LPO) with respect to the spacer thickness are almost suppressed, while in the latter case both, the SPO and the LPO, show up with comparable weights in the infinite cap limit [lo]. The corresponding plots of two-dimensional discrete Fourier transformations of (N + P)2 2(N, P) are presented in Figs. 2 and 3,

6 Discrete Fourier transformation en :? UN W ' 5g ;g -5 zs GF Discrete Fourier transformation 9 e F? E> 73 h 3 b IA VI v

7 194.I. KudronovsG et al. /Computational Materials Science 1 (1998) 18&197 I I I n 2.5 E W 5 G a.r( d 5 9 e N= 13 N= 12 N= ; Cu-cap thickness (MLs) Fig. 4. Dependence of the exchange coupling (in mry) on the thicknms of the Cu-cap (diamonds) for a fixed spacer thickness of N = 12, 13, and 14 in the case of M = 3. M' = 5. The lines serve as a guide 11 the eye. The dashed lines refer to the corresponding value for an infinite Cu-cap. piirely ferromagnetic or antiferromagnetic depending on the spacer thickness N. At a given spacer thickness N the oscillations with respect to the cap thickness are generally around a non-vanishing value, the so-called biased value. As illustrated in Fig. 5, another important feature of the IEC oscillations relative to the cap thickness is the decrease of their amplitudes with both the spacer- and cap-thickness. In particular, the amplitudes corresponding to one and thle same cap thickness, progressively decrease with increasing spacer thickness N. Both features are in agreement with a recent experiment on a Co/Cu/Co(OO 1) cap system [5] Co/Cu/Co(O I )-triluyer with U direrent spacer and a cap The case with a cap and a spacer formed by different materials is more complicated. As an example the case of a Cu-spacer and a Cu75Zn25-cap is considered for which one expects that the spanning vectors of the spacer and cap materials are stationary approximately at the same kll(cr) because the Fermi surfaces are similar. This situation is related to Model B of the free-electron case with a downward shift of the cap-band because the alloying of Cu with Zn atoms increases the number of valence electrons and hence the size of the alloy Fermi surface. The IEC of Co/Cu+n25/Co trilayers has been studied in a previous paper [ 121, in which oscillations with a period of about 3.1 MLs (i.e., q x 2.) were found, that corresponds to an in-plane wave vector close to the one giving rise to the SPO oscillation for a pure Cu spacer. From the two-dimensional Fourier transform of (N + P)*E*(N. P), shown in Fig. 6, a peak located at ( 4 x ~ 2.5. qp x 2.) is clearly identified, which is in complete agreement with the piedictions of the electron confinement model (Section 2.2).

8 J. Kudronovslj et al. /Computational Materials Science 1 (1998) I I I I I I I 1.2 I, N=5. If N=lO. I N=1S.5 N= Cu-cap thickness (MLs) Fig. 5. Dependence of the exchange coupling (In mry) on the thickness of the Cu-cap (diamonds) for a fixed spacer thickness of N = 5, 1, 15, and 2ML in the case of M = M' = 5 as a function of the thickness of the Cu-cap. The lines serve as a guide to the eye. The dashed lines reft:r to the corresponding value for an infinite Cu-cap. An arbitrary shift is applied to each curve in order to prevent overlapping. The other example is the case of a Cu-spacer and a Rh-cap for which one expects that the spanning vectors are stationary at different kllta). Indeed, inspection of the Rh Fermi surface [3] reveals that there is no stationary spanning vector in the vicinity of the klila) that corresponds to the short period oscillation for a Cu spacer. One thus expecls no oscillations with respect to the cap thickness in the asymptotic regime. This is indeed the case as can be seen from the two-dimensional Fourier transform of (N + P)2E2(N, P) in Fig. 7 which in turn has to be compared to the case of a Cu-cap in Fig. 2. We see that the Rh-cap essentially suppresses the oscillations with respect to the cap-thickness.

9 196 J. Kudronovsb et al. /Compututionul Muterials Science 1 (1998) Fig. 6. Absolute values of the discrete two-dimensional Fourier transformation of (N + P )2&2(P, N) (1 5 N, P 5 5) with respect to the spacer and the cap thickness for the case of Co/Cu/Co(O 1) trillayer ( M = x, M = 5) with a Cu75Zn25-cap. 231 Fig. 7. Absolute values of the discrete two-dimensional Fourier transformation of (N + P)2&2( P, N) (1 5 N. P 2 5) with respect to the spacer and the cap thickness for the case of Co/Cu/Co(OO 1 ) trilayer (M = M, M = 5) with a Cu-cap. 4. Conclusions We have investigated systeniatically the effect of a finite cap on the interlayer exchange coupling using a freeeliectron model as well as ab initio calculations. The results confirm the basic predictions of the electron confinement model as well as available experimental data, namely (i) the periods of the oscillations of the IEC with respect to the

10 J. et al. /Computational Materials Science 1 (1 998) cap thickness can be related to the Fermi surface of the cap material; (ii) the oscillations are around a biased value which is generally non-zero and depends on the spacer thickness; (iii) the coupling energies decrease asymptotically with the thickness of the spacer ( N) and the cap ( P) a5 (N f P)-2, and (iv) the oscillations with respect to the cap thickness can be strongly suppressed if the spanning vectors of the spacer and cap material are stationary at different kit-vectors. Acknowledgements This work is a part of activities of the Center for Computational Material Science sponsored by the Academy of Sciences of the Czech Republic. Financial support for this work was provided by the Grant Agency of the CLech Republic (Project No. 22/97/598), the Project Scientific and Technological Cooperation between German) and the Cicech Republic, the Center for the Computational Materials Science in Vienna (GZ and GZ 45.42). and the TMR Network Interface Magnetism of the European Commission (Contract No. EMRX-CT96-81)). References I 1 I D.M. Edwards, J. Mathon. R.B. Munir and M.S. Phan. Phys. Rev. Lett. 67 (1991) 493. [2] P. Bruno, J. Magn. Magn. Mat. 121 (1993) 248. [3] M.D. Stiles, Phys. Rev. B 48 (1993) P. Bruno, Phys. Rev. B 52 (1995) [5] S.N. Okuno and K. Inomata, J. Phys. Soc. Jpn. 64 (1995) [h] J.J. de Vries, A.A. Schuldelaro. R. Jugblut. P.J.H. Bloemen. A. Reinders, J. Kohlhepp, R. Coehoorn and W.J.M. de Jonge, Phys. Rev. Lett. 75 (1995) 436. [ 71 A Bounouh, P. Beauvillain, P. Bruno. C. Chappert. R. Mtgy and I? Veillet, Europhys. Lett. 33 (1996) 315. [d] J. BarnaS. Phys. Rev. B 54 (1996) [Cl] P. Bruno, J. Magn. Magn. Mater. 164 (1996) 27. [ lol J. Kudrnovskf, V. Drchal, I. Turek and P. Weinherger, Phyh. Rev. R 5 (1994) 16 15: V. Drchal. J. Kudmovsklj, I. Tureh and P. Weinberger, Phys. Rev. B 53 ( 1996) I I I 1 J. Kudrnovskf, B. Wenzien, V. Drchal and P. Weinberger, Phys. Rev. I3 44 (1991) J. Kudrnovskf, V. Drchal, P. Bruno. I. Turek and P. Weinberger, Phys. Rev. B 54 (1996) R P. Bruno, J. Kudrnovskf, V. Drchal and 1. Turek. Phys. Rev. Lett. 76 ( 1996) J. Kudrnovskf. V. Drchal, I. Turek, M. Sob and P. Weinbergel; Phys. Rev. B 53 (1996) O.K. Andersen.. Jepsen and D. Gliitzel, in: Highlights of Condensed-Matter Theory, eds. E Bassani. E Fumi and M.P. Tosi (horth-holland. New York. 1985).