The stripe model of domain structure in multilayers is studied by micromagnetic simulation. The results indicate a strong reduction of the effective domain wall energy (by dipolar effects). Domain width measurements on sputtered Co/Pd multilayers are compared with the theory. The estimated exchange stiffness is comparable with that of bulk Co. The effects of interface roughness and of interlayer exchange are discussed

Gemperle, R.

Haan, P. de

Kambersky, V.

Lodder, J.C.

Porthun, S.

Simsova, J.

English

de Haan, P.

Simšová, J.

University of Twente Research Information

Journal of Magnetism and Magnetic Materials 157/158 (1996) 301-302 ~ l ~  journal of magnetism ~ a  and magnetic ELSEVIER ~ materials Domain wall theory and exchange stiffness in Co/Pd multilayers V. K a m b e r s k ~  a,*, p. de H a a n  b j. S im~ovf i  a S. Porthun b R. G e m p e r l e  a J.C. L o d d e r  b a Institute of Physics, Academy of Science, Cukrovarnickd 10, 16200 Prague 6, Czech Republic b MESA Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Abstract  The stripe model of domain structure in multilayers is studied by micromagnetic simulation. The results indicate a strong reduction of the effective domain wall energy (by dipolar effects). Domain width measurements on sputtered C o /P d  multilayers are compared with the theory. The estimated exchange stiffness is comparable with that of bulk Co. The effects of interface roughness and of interlayer exchange are discussed. Keywords: Multilayers; Exchange; Domain walls Domain structures in magnetic multilayers (MLs) with perpendicular anisotropy may attract interest for two main reasons: the tendency to spontaneous domain formation needs to be known and controlled in recording applica- tions, and comparisons of the observed structures with appropriate models may be used to study the exchange stiffness. This paper has the latter motive. We extend the stripe-domain model [1] including the real wall structure ~using fine finite elements), and compare the theory and experimental results on sputtered C o / P d  MLs [2,3] includ- ing new data; the estimated exchange stiffness fitting of the results is compared with bulk values [4]. The stripe domain model consisting of straight domains of alternating mean magnetization direction ( 'up '  and 'down' along the film normal) for MLs has so far been used in its 'classical' or 'rigid' version [1], where the magnetization is uniform in each domain and the walls are treated as ideal planar boundaries endowed with constant surface energy densities. The predicted dependence [3] of the equilibrium domain width D on the total ML thickness T closely follows a single 'universal'  curve if both D and T are scaled by the characteristic length I* = 4 ~  .2 / I . % M  s , (1) where Ms* =fMs, K~* = f K  u and A* = f A  are, respec- tively, the saturation magnetization, the uniaxial anisotropy constant and the (in-plane) exchange stiffness assigned to the ML as a whole, while M s, K u and A are the respec- * Corresponding author. Email: kamber@fzu.cz; fax: +42-2- 312-3184. tive constants assigned to the magnetic layers of thickness t m (separated by spacer layers of thickness ts), and f =  trn/(t  m + t s) is the filling factor. The 'universal' line (U in Fig. la,b) was obtained earlier [5] for uniform films ( f =  1). However, the Bloch wall structure assumed in Eq. (1) is not stable and the flux-closing (Nrel wall) structure has lower energy [6]. This effect is enhanced [7,8] by the lack of interlayer exchange in the ML. Thus realistic models should predict finer domains than the 'rigid' stripe model. We have now investigated the stripe model 'micromag- netically', simulating the entire magnetic structure by periodic arrays of fine, uniformly magnetized elements shaped as square rods, with sides equal to t m (1-4  atoms), and seeking equilibrium by iterative relaxation. The results are illustrated in Fig. l(a) as the scaled dependence of D / I  * on T / l  *, for three values of the quality factor of the magnetic layers, Q = 2 K u / t z o M  ~. These predictions are indeed far from those of the 'rigid' model (U line). As shown in Fig. l(b), almost universal scaling is approxi- mately restored when an effective characteristic length is used for the scaling: /eft" = 4 ~ / / X 0  Ms .2, (2) where K ~ * f f = f ( K , -  0.5/%M 2) is the effective (directly measured) anisotropy constant. The 'universality' breaks down when Q ~ l  or D--->T, and cannot be used for precise fitting. More specific model results were compared with our measurements of domain widths in demagnetized C o / P d  MLs prepared by rf sputtering in Ar, and characterized magnetically by VSM and torque measurements [2]. The average domain widths were measured by the colloid-SEM 0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSD1 0304-8853(95)01 162-5 302 V. Kambersl05 et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 301-302 40 D/I" 10 "~.3 " ~ ' ~ - - - v ~ ' ~  ~ e -  . . . .  O - - - e "  . . . .  - O ' ' "  2 .2  ' ' . . . .  i 2 r ) , "  100 20 I0 ~ "~~._-.o 5.5 . . . .  i 2 5 'T/i:f Fig. 1. Dependence of the domain width D on the multilayer thickness T predicted by the stripe model with realistic wall structure for 3 parameter sets: (1) Q = 2.2, f = 0.55, (2) Q = 1.2, f =  0.55, (3) Q = 1.1, f =  0.275, compared with the curve (U) obtained earlier for uniform films and 'rigid' magnetization (in- finite Q). D and T are scaled by factors (a) l * (Eq. (1)), and (b) l~f (Eq. (2)). method and by MFM. Table 1 presents our earlier data [3], and new results for two thin MLs with very flat domains. The micromagnetic model was used for each set of ML structure and magnetic parameters. With D oscillating around the experimental D O , the exchange stiffness con- stant A was changed until D o corresponded to the mini- mum of total energy. The resulting A values, also pre- sented in Table 1, are much larger than our earlier ( ' r igid '  Table 1 Data on the Co/Pd ML structure, magnetization and anisotropy (per Co volume), domain width and in-plane exchange stiffness from the model fitting. The numbers of bilayers are 25 and 7 in samples (rows) 1-4 and 5-6, respectively tCo tvd M s Kef f D A (nm) (nm) (kA/m) ( k J / m  3 ) (nm) (p J /m)  0.85 1.4 1850 .40 90 31 0.43 2.9 1580 1550 145 2.3 0.45 1.4 2026 1006 135 24 0.23 1.4 2445 1615 185 29 0.40 1.1 2100 360 220 25 0.40 1.1 1780 300 280 15 model) estimates [3]. The value from neutron diffraction on bulk Co [4] is 30 p J / m .  In the last four samples (rows) in Table 1, the variation of A correlates with that of M~, and the latter exceeds that of bulk Co, which indicates distribution of both the mag- netic moments and the lateral exchange stiffness over magnetic thickness t m exceeding the nominal tco. This view is supported by X-ray diffraction data [2] and also by recent pattern-recognition analysis of XPS data [9], which indicate considerable interface roughness in our MLs. Such broadening of the magnetic layer profiles was also simu- lated in the present model. As long as the macroscopic (measured) T, K~}f and Ms* (and thus leaf) are kept constant (which is the case for curves 2 and 3 in Fig. 1), the predicted domain size does not practically change (as seen in Fig. lb). The remarkably low value of A obtained for the sample with the largest Pd thickness, exceeding the margin of 2 nm which seems to be the limit for interlayer exchange coupling (IEC) in C o / P d  MLs [10], raises the question of the possible role of the IEC in the domain structures. A large IEC would inhibit the flux closure reducing the wall energy. This was also verified in our model: we varied the ratio of the IEC to the in-plane exchange coupling and found that the effective wall energy (or characteristic length [6]) increases rapidly between the values 0 and 0.1 for this ratio, and remains constant thereafter. Thus a lower in-plane exchange coupling would be needed to fit the experiments with a large IEC added to the model. The present one-parameter fitting thus calls for extension to include variations in the domain width in MLs with vary- ing IEC (i.e. spacer layer thickness or composition). Acknowledgements: The computations on the Cray Y-MP EL at the Institute of Physics in Prague were supported financially by Westinghouse Electric Co. R e f e r e n c e s  [1] A. Suna, J. Appl. Phys. 59 (1987) 313. [2] P. de Haan, Q. Meng, T. Katayarna and J.C. Lodder, J. Magn. Magn. Mater. 113 (1992) 29. [3] J. Sim~ovfi, R. Gemperle, V. Kambersk~, S. Porthun, P. de Haan and J.C. Lodder, IEEE Trans. Magn. 30 (1994) 784. [4] H.A. Alperin, O. Steinsvoll, G. Shirane and R. Nathans, J. Appl. Phys. 37 (1966) 1052. [5] Z. Mfilek and V. Kambersk2), Czech. J. Phys. 8 (195"8)416. [6] T.G.W. Blake, C.C. Shir, Y.O. Tu and E. Della Torre, IEEE Trans. Magn. 18 (1982) 985. [7] R. Ploessl, J.N. Chapman, M.R. Scheinfein, J.L. Blue, M. Mansttripur and H. Hoffmann, J. Appl. Phys. 74 (1993) 7431. [8] V. Kambersk2?, J. Magn. Soc. Jpn. 19 Suppl. S-1 (1995) 37. [9] B. Lesiak, J. Zemek, P. de Haan and A. Jozwik, Surf. Sci. 346 (1995) 79. [10] W.R. Bennett, C.D. England, D.C. Person and C.M. Falco, J. Appl. Phys. 69 (1991) 4384. 

Domain wall theory and exchange stiffness in Co/Pd multilayers

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