VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 Magnetization Reversal in Ultrathin Films with Monolayer-Scale Surface Roughness A. Moschel,* R. A. Hyman, and A. Zangwill School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 M. D. Stiles Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Received 15 April 1996) The intrinsic anisotropy of nominally flat, ultrathin ferromagnetic films typically is augmented by a uniaxial anisotropy at step edges. We report model calculations of hysteresis for such systems with in- plane magnetization and monolayer-scale roughness. The reversal process is a combination of domain nucleation at step edges, expansion of these domains through morphological constrictions, and coherent rotation within domains. The initiation of reversal at well separated step edges can explain the very small coercive fields measured for real ultrathin magnetic films. [S0031-9007(96)01544-X] PACS numbers: 75.70.­i, 75.60.­d Ultrathin film magnetism has evolved into a mature field the average surface plane for this geometry. Surface of study over the past fifteen years [1]. Two-dimensional magnetocrystalline anisotropy [10] can either support or critical phenomena provided much early motivation, but oppose this orientational tendency, and both cases are considerable interest and activity now focuses on the com- observed frequently in the laboratory [11]. For the present plexities of exchange coupling and anisotropy in order to study, we focus exclusively on the case of in-plane understand the unusual hysteresis loops commonplace in magnetization. We also choose a particularly simple, magnetic multilayers. Insight has been gained from even high symmetry, surface morphology. In detail, the film the simplest models of magnetization reversal [2], where is taken to be a continuous single crystal composed of one one assumes perfectly flat interfaces, and the coherent ro- complete magnetic layer on a nonmagnetic substrate with tation model of Stoner and Wohlfarth [3]. Typical gen- a periodic array of square monolayer-height magnetic eralizations examine some effects of interface roughness islands with side length L and center-to-center separation within the coherent rotation model [4] or consider the ef- D placed on top (Fig. 1). fects of inhomogeneous magnetization reversal within the Exchange coupling guarantees that atomic moments flat interface model [5]. But as Arrott [6] has pointed out, remain aligned over microscopic distances. For this there is an intimate connection between roughness and re- reason, a two-dimensional classical XY model with spin versal that has been insufficiently explored for both simple lengths Si proportional to the film thickness at lateral and multilayer films. atomic site i will be sufficient for our purposes. The Our calculations are motivated by the steadily accumu- magnetic energy is lating experimental evidence that growth-induced surface X X roughness can profoundly affect magnetization reversal E 2 Jij cos qi 2 qj 2 a2 Ki2Si cos2 qi and coercivity in ultrathin films [7]. Scanning tunnel- i,j i X X ing microscopy graphically demonstrates that roughness 2 a2 Ki S at the monolayer scale is the best that can be achieved for 4Si cos2 2qi 2 mH i cos qi 2 w , i i any coverage of deposited material [8]. For this reason, (1) even the best as-grown or annealed ultrathin films have where the angles qi denote the directions of the vector some step edges associated either with the perimeter of spins Si relative to 100 , Jij J min Si, Sj 2 is the monolayer-height islands that nucleate during growth or with the steps of an intentionally miscut substrate. This is significant because the magnetic anisotropy at sites of reduced crystallographic symmetry can compete success- fully with the intrinsic anisotropy of the flat surface and thereby control coercivity and magnetization reversal [6]. Measurements for magnetic films grown on vicinal sub- strates add force to this general argument [9]. In this paper, we study magnetization reversal at T 0 for a model ultrathin ferromagnetic film with simple cubic FIG. 1. Schematic view of the rough ultrathin film morphol- crystal structure and monolayer-scale surface roughness. ogy used in this work. The indicated island geometry is re- Magnetostatic shape anisotropy favors magnetization in peated periodically. Arrows indicate local anisotropy axes. 0031-9007 96 77(17) 3653(4)$10.00 © 1996 The American Physical Society 3653 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 exchange energy between nearest neighbor sites i and j, to a large negative value. At each new field, the conjugate Ki2 and Ki4 specify the strength of twofold and fourfold gradient method is used to find a local minimum close magnetic surface anisotropies at site i, a is the lattice to the previous local minimum. But when jumps in the constant, and m m0m, where m is the atomic magnetic magnetization occur, i.e., the minima are not close, the moment. The in-plane magnetic field H is oriented at an simulation backs up to the configuration before the jump, angle w from 100 . and relaxation dynamics is used to find the new minimum We restrict attention here to the case of w 0 [12] energy configuration. and choose the material parameters as J 10221 J, a For surfaces with no steps, or when the island sepa- 0.3 nm, and m 10223 J T. All sites are assigned a ration D is small, our model reproduces the Stoner- small fourfold anisotropy K4 1022 mJ m2, and, as Wohlfarth result that magnetization reversal occurs by suggested by the phenomenological Néel model [10], step coherent rotation with a coercive field HC equal to edges are assigned a uniaxial anisotropy. The latter is HSW 8a2K4 m. The magnitude of HSW 5 3 chosen here to lay perpendicular [13] to the local step 105 A m 2p103 Oe is about 100 times larger than edge with strength K2 1 mJ m2. All these numerical typical measured coercivities for ultrathin films [7,9,20]. values are consistent with recent experiments [14,15]. Such a discrepancy between experiment and theory for Unlike most micromagnetic calculations [16], the en- the coercive field is known as Brown's paradox [21]. It ergy expression Eq. (1) does not include an explicit con- is resolved for bulk samples by invoking the presence tribution from magnetostatics. We suggest that this is of crystalline defects, where local magnetic properties acceptable in the present case because (i) the effect of may differ considerably from the average and thus serve shape anisotropy is already included when the planar mag- as nucleation centers for reversal or pinning sites for netization we assume is uniform in space, (ii) the mag- pre-existing domain walls [16]. netostatic contribution to the energy of the nonuniform Our calculations support the view that monoatomic magnetization distribution within a Néel domain wall is steps of single crystal ultrathin films both nucleate rotated negligible in the ultrathin film limit [6,17], and (iii) it is domains [6] and impede the motion of domain walls. The useful to analyze the effect of competing anisotropies alone reversal process is a combination of nucleation, expansion so that the effect of reintroducing the dipolar interactions of domains through morphological constrictions, and can be appreciated more readily. For example, the atomic- coherent rotation within domains. For the geometry scale discontinuity of the surface height (and hence of the studied here, the competition between these processes magnetization) at a step edge yields a magnetostatically leads to many types of complex hysteresis loops as a induced contribution to K2 [18] which breaks the symme- function of island size and separation. Figure 2 shows one try between parallel and perpendicular anisotropy at the characteristic of these loops-the coercive field scaled by step edges. HSW -for three choices of the island separation D. The The material parameters imply a domain wall width p numerical results accord surprisingly well with simple W 8 J 2K4 200a, indicating that the magnetiza- energy balance arguments [22] that predict four regimes tion changes exceedingly slowly on the atomic scale. For where HC HSW varies successively as W L, LW D2, this reason, large system sizes can be studied by trans- W D 2 L , and D 2 L 1 2W D as L D increases. formation to a representation where the sum over atomic sites in Eq. (1) is replaced by a sum over blocks of aligned spins. We choose square blocks with b W 20a atomic spins per side so that the magnetization changes very little even from block to block. Each block-spin length is rescaled to reflect the local average surface height as before, and the parameters in Eq. (1) are renormalized to guarantee that the new, coarse-grained representation ac- curately reproduces the original atomic spin representa- tion. One easily checks that the fourfold anisotropy and Zeeman terms each acquire a factor of b2 (all b2 spins per block contribute) while the twofold anisotropy term acquires a factor of b (only the b spins per step that run through the block contribute). The exchange term is FIG. 2. Coercive field HC of hysteresis loops obtained upon unaltered by the blocking transformation [19] as is the quasistatic reversal of the external field in (1) with w 0 for anisotropy associated with the island corners. the film geometry of Fig. 1 as a function of L D for different Zero temperature magnetization reversal is studied by system sizes as labeled. Not shown are the coercivities for L 0, 2, D, for which H following the local minimum of Eq. (1) as the external C HSW in this model. The vertical lines divide the D 128 curve into four regions that are field is reversed in small steps from a large positive value discussed in the text. 3654 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 As long as D is not so small that HC HSW, reversal islands are pinned at the island corners (see Fig. 4). They begins with 90± domains nucleated at the edges of each is- cannot expand laterally due to the energy cost to create land where the torque, due to the local twofold anisotropy, more wall length. Only when the field reaches the value is largest [23]. Figure 3 shows a stable configuration at HD ~ HSWW D 2 L does a jump in magnetization remanence where these lens-shaped domains are pinned signal that the domains have squeezed through this to the island edges by the energy cost to increase the morphological constriction. But when HD , HR, this domain wall length. For small L D, the 90± domains jump does not affect the coercive field because the expand as the field is reduced from positive to negative postjump magnetization is still positive. Not until HD $ until they "burst" at a field HL ~ HSWW L. As long HR does HD becomes the coercive field. Finally, near as D 2 L . L, the domains expand freely through the layer completion L D the positive magnetization of channels between the islands until nearly the entire sur- the unrotated edge spins can be compensated by rotation face is covered. The remaining unrotated spins are con- of the magnetization in the 90± domains that now do not fined to regions of area A ~ LW that surround the island include the narrow channels. The coercive field in this edges with anisotropy parallel to the applied field. For p final regime is HF ~ HSW D 2 L 1 2W D. very small L D, HL exceeds the field H 2HSW 3 6 Magnetization reversal that appears to be closely at which the 90± state is unstable to complete reversal into related to the results reported here has been observed the 180± state. Accordingly, there is an additional jump in Kerr microscopy and vector magnetometry experi- in the hysteresis loop when H . HL. But HC HL ments reported by Cowburn et al. for an ultrathin nonetheless because the magnetization in the domains is Ag Fe Ag(001) multilayer system [24]. These authors not precisely perpendicular to the H axis. There is a small discussed their results using a model that combines negative component of the magnetization (Mx H HSW the domain wall pinning mechanism of coercivity with for small H) along the reversed field direction that over- a Stoner-Wohlfarth model that features both fourfold compensates the positive contribution from the unrotated and twofold anisotropies. The origin of the uniaxial spins noted above. anisotropy was not specified, but a rather small value of For fixed D, this scenario remains correct as L in- K2 1024 mJ m2 was found to produce the best fit to creases until HL becomes so small that the negative mag- experiment. Note, however, that this value is assigned netization of the 90± domains cannot compensate the (by necessity) to every site of the surface in their spatially unrotated edge spins. The hysteresis loop magnetization uniform theory. By contrast, the Néel model [10] and thus remains positive after its jump at H HL. The experiments [15] for stepped surfaces suggest a value of magnetization smoothly passes through zero as the spins K2 that is 104 times larger at step edge sites. We use in the 90± domains coherently rotate with the applied this larger value but assign it only to step edge sites. field. The magnetization is zero when the magnetic field Systematic experiments where the step density is varied reaches the value HR ~ LW D2. The crossover from by changing deposition conditions, coverage, or vicinal the first to the second scaling regime in Fig. 2 occurs miscut will help resolve this matter. when HL HR. The generalization of the results presented here to the When L . D 2, the channel width is smaller than the case of an external magnetic field applied at an angle island edge length, and the 90± domains between the w fi 0, islands with nonsquare shapes and multilevel FIG. 3. Spin configuration for L 32, D 64 at the rema- FIG. 4. Spin configuration for L 32, D 64 for nent state H 0 . Only every fourth spin block (in each H HSW 20.077. Only every fourth spin block (in each di- direction) is shown for clarity. The lines indicate the island rection) is shown for clarity. The lines indicate the boundaries boundary. This configuration is reproduced periodically in of two neighboring islands. This configuration is reproduced the plane. periodically in the plane. 3655 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 roughness, is straightforward. Our investigation of these [9] B. Heinrich, S. T. Purcell, J. R. Dutcher, K. B. Urquhart, situations, including the effects of magnetostatics, will be J. F. Cochran, and A. S. Arrott, Phys. Rev. B 38, reported in full elsewhere [22]. 12 879 (1988); A. Berger, U. Linke, and H. P. Oepen, A. M. and R. A. H. were supported, respectively, by De- Phys. Rev. Lett. 68, 839 (1992); J. Chen and J. L. partment of Energy Grant No. DE-FG05-88ER45369 and Erskine, Phys. Rev. Lett. 68, 1212 (1992); W. Weber, National Science Foundation Grant No. DMR-9531115. C. H. Back, and R. Allenspach, Phys. Rev. B 52, R14 400 A. M., R. A. H., and A. Z. thank the Electron Physics (1995); W. Weber, C. H. Back, A. Bischof, C. 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