PHYSICAL REVIEW B VOLUME 60, NUMBER 21 1 DECEMBER 1999-I Magnetic reversal of ultrathin films with planar magnetization R. A. Hyman* and A. Zangwill School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 M. D. Stiles Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 Received 26 July 1999 Classical spin simulations are used to study magnetic reversal in ultrathin 1­6 monolayers films with planar magnetization and surface roughness typical of epitaxially grown samples. Reduced site symmetry at surface steps leads to strong, local anisotropies that both nucleate reversal and pin domain wall motion. The results we obtain from realistic models with periodic roughness are interpreted using a much simpler model with a single, finite-length step. These models show how growth induced roughness can lead to oscillations in the coercive field as the film thickness is increased, as seen in some experiments. They also demonstrate explicitly how local step anisotropies become less important and magnetostatic interactions become more important as the film thickness increases. S0163-1829 99 00946-7 I. INTRODUCTION planar magnetized films as well. Indeed, Kolesik et al. have studied its effect on the magnetic reversal of an Fe/W 110 The potential for novel physics and exciting applications sesquilayer film using a planar Ising simulation.11 They did has motivated many studies of ultrathin magnetic films.1 To not investigate the effect of step-induced anisotropy. In this explore new physics, it is usual to focus on simple model work, we take account of both effects. systems and equilibrium properties such as exchange, anisot- Our work was motivated by experiments such as those ropy, and the thermodynamic phase diagram in the space of published recently for the ultra-thin Co/Cu 001 system. The temperature and thickness. To exploit new applications, it is measured coercivity shows submonolayer oscillations super- typical to study more complex systems and nonequilibrium imposed on a monotonic increase with thickness for 2­15 properties such as hysteresis, domain wall motion, and mag- monolayers of deposited cobalt.12 Similarly, the coercivity of netotransport. Common to both is the observation that varia- Cu/Co/Cu 001 is strongly nonmonotonic for 0­2 monolay- tions in surface roughness and film morphology often have a ers of deposited copper.13 significant effect on magnetic structure. We use a simple but realistic simulation model of mag- Magnetometry2 and the surface magneto-optic Kerr netic reversal to show how typical epitaxial growth surface effect3 are widely used to probe magnetization reversal in morphologies can lead to oscillatory and other nonmonotonic ultrathin films. A typical experiment reports representative behaviors for the coercive field of planar magnetized ultra- hysteresis loops and the coercive field as a function of total thin films. This work extends to the multilayer regime previ- deposited material. It is generally appreciated that the films ous theoretical and simulation work by the authors for the in question exhibit surface roughness, but the consequences case of monolayer-height islands on a single complete of this fact are not often addressed explicitly. One exception layer.14,15 It also exploits a new conceptual framework devel- is a theoretical argument presented a few years ago by oped by the authors to understand magnetization reversal in Bruno.4 Making simple assumptions regarding thickness vicinal samples.16 fluctuations and the nature of domain wall pinning in films We are interested in film morphologies typical of ultrathin with perpendicular magnetization, he derived a coercive magnets grown epitaxially on a nonmagnetic substrate. In the field HC t 5/2 where t is the film thickness. Experimental cartoon version of such a film shown in Fig. 1 a , several tests of this prediction for both the Co/Pd 111 Ref. 5 and completed magnetic layers lie beneath multilevel roughness Co/Pt 111 Ref. 6 systems are not consistent with each in the form of irregular pits and islands. Needless to say, other. some simplification is required in order to perform a trend Our interest here is hysteresis and coercivity in ultra-thin study as we wish to do. films with planar magnetization. Roughness is very impor- Figures 1 c and 1 d show the morphologies we have tant in such systems because surface steps break translational chosen to study in detail. They consist of several completed invariance and thereby induce local, in-plane anisotropies magnetic layers with one incomplete layer in the form of a that differ from the intrinsic in-plane anisotropy of the flat regular array of square islands or pits. These two limiting film.7,8 This is significant because, as stressed by Arrott,9 cases turn out to be sufficient to capture most of the physics local anisotropies can nucleate and pin domain walls during of the more realistic morphology. In fact, the essential roles the magnetization process. The corresponding hysteresis can of step length and step anisotropy in magnetic reversal are be very complex indeed.10 captured already by the even simpler ``isolated step'' model The phenomenon considered by Bruno-the variation of shown in Fig. 1 b . The islands and pits add the effects of domain wall energy with local film thickness-is present for step separation and magnetostatics. 0163-1829/99/60 21 /14830 7 /$15.00 PRB 60 14 830 ©1999 The American Physical Society PRB 60 MAGNETIC REVERSAL OF ULTRATHIN FILMS WITH . . . 14 831 FIG. 2. Hysteresis loops for the isolated step model. The char- acteristic fields HN , HS , and HT are discussed in the text. The Roman numeral labels correspond to different parts of the phase diagram of Fig. 3. FIG. 1. Schematic view of a a ``real'' ultrathin film, b a flat film with a single step, c a period island morphology, and d a tive, but HN can have either sign. The rationale for the Ro- periodic pit morphology. Arrows indicate the easy axes of the in- man numeral loop labels will be explained below. plane anisotropy. We focus first on loop III a because every feature of the reversal mechanism is reflected separately in its structure. The plan of our paper is as follows. Section II focuses on The first deviation of the magnetization from saturation oc- the isolated step model. We catalog the various hysteresis curs at the ``nucleation field'' HN . Spins within a few ex- loop topologies that occur and correlate them with the di- change lengths of the step rotate coherently away from the mensionless control parameters of the model in the form of a saturation direction (180°) because of the torque exerted by phase diagram. Section III is a discussion of the island model K2. At the end of this interval of smoothly changing magne- including the effect of magnetostatics. Results are presented tization, a lens-shaped domain has formed with for the hysteresis loops and the coverage dependence of the approximately-90° domain walls interposed between the step coercive field. The isolated step model is used to rationalize spins and the terrace spins.14 At the ``step instability field'' the global behavior. Section IV briefly discusses the pit HS , the domain walls depin from the step and sweep across model, a synthetic island pit model, and compares our the film. This leaves the system in a ``90° state'' with nearly results with experiment. Section V is a summary. zero magnetization along the direction of H. Another regime of coherent rotation follows until HT when a II. THE ISOLATED STEP MODEL Stoner-Wohlfarth18 type instability occurs on the terrace far from the step. When this happens, the terrace spins coher- The model film of Fig. 1 b is one monolayer thick and ently jump from the 90° state to the 0° reversed state. The lies on a flat nonmagnetic substrate. Every atomic site in the step spins lag slightly behind because they feel the pinning film carries a spin that is constrained to lie in the plane of the effect of K2. Smooth coherent rotation completes the rever- surface. Ferromagnetic exchange J couples nearest neighbor sal process. spins and a fourfold planar anisotropy K4 acts on every spin. The IV loops generically have one jump because the ter- The effect of an isolated step is modeled by adding an addi- race instability formally occurs before the step instability tional twofold planar anisotropy K2 along a line segment of domain wall depinning in these cases. Of course, the spins width W. Thicker films can be modeled by varying the ex- cannot jump coherently from 90° to 0° until they get to 90° change and anisotropy constants because we assume that the in the first place. As a result, the spins rotate all the way to magnetization does not vary in the direction perpendicular to near reversal as soon the domain walls depin from the step. the film surface. The direction of the twofold axis is perpen- The difference between loops III a and III b and be- dicular to the step and parallel to one of the fourfold axes. tween loops IV a and IV b depends on whether or not K2 is An external magnetic field H is applied parallel to the step. large enough to prevent the terrace jump from ``dragging'' Magnetostatics is ignored. the step spins all the way to saturation at HT . The loops The hysteresis curves for this model were found from III c and IV c differ from their b counterparts because the numerical simulations of a classical spin Hamiltonian see step instability formally occurs before nucleation. When this below that incorporates all the features outlined above. We is the case, the domain walls depin from the step as soon as find only ``one-jump'' and ``two-jump'' hysteresis loops nucleation occurs. over the entire range of parameters. Subtle differences divide Figure 3 is a phase diagram derived from our simulations each of these into three subclasses. The six typical loops that that connects the loop topologies ``the phases'' to the occur Fig. 2 are characterized by the fields HN , HS , and model parameters. As the diagram axes show, the latter are HT where characteristic changes in the magnetization best organized into the dimensionless parameters K occur.17 In the discussion to follow, we consider external K2/2 and W W/ . The first of these is the step anisot- fields increasing from large negative values to large positive ropy scaled by the domain wall energy 2JK4. The sec- values. The characteristic fields HS and HT are always posi- ond is the step width scaled by the exchange length 14 832 R. A. HYMAN, A. ZANGWILL, AND M. D. STILES PRB 60 FIG. 3. Loop structure phase diagram for the isolated step model. K is the scaled step anisotropy and W is the scaled step width. FIG. 5. Characteristic fields as a function of K for fixed W. J/2K4. The particular numerical range of K and W used 4 eventually cross as W increases and the III phases supplant in Fig. 3 arises when realistic values are chosen for the the IV phases. The field HS decreases with W for the follow- physical parameters see below . ing reason.15 As H increases, the lens domain expands to The phases have been labeled for consistency with our gain Zeeman energy. This is opposed by the domain wall previous vicinal surface results16 where the characteristic energy which is proportional to the domain perimeter. The fields HN , HS , and HT occur also.19 Indeed, the phase dia- domain wall depins when these two balance. One gets HS gram here and the one derived in Ref. 16 are simply two- 1/W because the domain wall is pinned at opposite ends of dimensional slices of a three-dimensional phase diagram the step. with axes: scaled step anisotropy K, scaled step width W and The (a) (b) and (b) (c) transitions in Fig. 3 occur scaled terrace length L. The phase diagram for the vicinal with decreasing K. The first of these agrees with the ``drag- film is the two-dimensional slice at W . The phase dia- ging'' argument given above. That is, when the step anisot- gram for the single step limit is the perpendicular slice at ropy is large, the Stoner-Wohlfarth instability of the terrace L . There is a phase boundary between these two slices spins does not produce enough torque to rotate the step spins at L and W where HS 0) that separates phases IIa to complete reversal. But when K is small enough, a direct and IIb in the vicinal film from phases IIIa and IIIb in the jump to 0° is possible. The (b) (c) transition occurs be- finite step film. cause H The general placement of the various phases in Fig. 3 can N increases in magnitude past HS as the step an- isotropy is reduced. This happens because, as noted earlier, be understood from the variation of the characteristic fields easy small magnitude nucleation is encouraged by the with W and K. These are shown in Figs. 4 and 5, respec- torque exerted by K2 on the step spins. tively, scaled by the Stoner-Wohlfarth field HSW These results are sufficient to qualitatively explain the ex- 8a2K4 / . The latter is the external field value at which perimental observation noted earlier12 that the coercive field easy-axis reversal occurs for a single domain system with HC rises rapidly with deposited material with a small ampli- fourfold anisotropy.18 From Fig. 3, we see that the III phases tude oscillation superimposed. We need only recall that K appear at larger values of W than the IV phases. HT K2/2 where 2JK4 is the domain wall energy. The 6/9HSW Ref. 16 is independent of step width it is latter is proportional to the film thickness so K t 1. Figure driven by the terrace spins while HS smoothly decreases 5 then shows that HC should indeed increase rapidly as t with W. This guarantees that the curves of HS and HT in Fig. increases, at least for larger thicknesses. On the other hand, it is well understood that the step density oscillates as growth proceeds.20 This translates into oscillations in W in the present model so Fig. 4 implies that oscillations will occur in the coercive field as well. When HC HN small K), the oscillation amplitude is smaller or comparable to the overall change in HC . When HC HS large K), the oscillation am- plitude is comparable or larger than the change in the coer- cive field. III. THE ISLAND MODEL The periodic surface morphologies shown in Fig. 1 permit us to model the variations in step length and step separation that occur during growth in a fairly realistic manner. In both cases, the square structures have center-to-center separation D and side length L. If the flat surface has t complete mag- FIG. 4. Characteristic fields as a function of W for fixed K. netic layers, Fig. 1 c and Fig. 1 d will be called the island PRB 60 MAGNETIC REVERSAL OF ULTRATHIN FILMS WITH . . . 14 833 and pit models, respectively. In this section, we focus on the island model exclusively. Except for the addition of magnetostatics, the magnetic energy we used to study hysteresis in the island model is the same as the one we used to analyze the isolated step model. In detail, the substrate is taken as centered cubic so the thick- ness t is measured in units of t0 a/2 where a is the in-plane lattice constant. Classical Heisenberg spins at each surface site i point in the direction S i . We get a two-dimensional model because the spins are forced to be parallel within each atomic column. This is acceptable because we limit our- selves to values of t that are much less than the exchange length. FIG. 6. Scaled coercive field as a function of coverage for the Each spin is subject to nearest-neighbor ferromagnetic ex- island model. The solid curves give the results without magnetostat- ics. The heavy dashed curve and light dashed curve give the results change J, a large twofold perpendicular surface anisotropy with magnetostatics for the cases of step anisotropy parallel and KZ 0, a fourfold planar anisotropy K4 0, an external field perpendicular to the steps, respectively. H, and magnetic dipole-dipole interactions from all other spins. The saturation magnetization is MS and a two-fold Figure 6 illustrates the variation of the computed coercive anisotropy K2 is present at step edge sites only. The mag- field HC with total coverage t in monolayers for netic energy is the island model. The islands contribute the partial coverage (L/D)2 to the total. To mimic a typical growth scenario, the island separation was fixed at D 64 and the coverage EM JijS i*S j a2K2 S i*b i 2 was increased by increasing the island dimension L. The i,j i step heavy light dashed curves are for step anisotropies that fa- vors spin alignment parallel perpendicular to the step. The 2a2K x y 4 ti S i 4 S i 4 H* tiS i solid curve was computed without magnetostatics. There is i i only one such curve because the magnetic energy Eq. 1 is invariant if K2 K2 and we rotate the Cartesian axes a2K z Z S i 2 tiS i*Hint 1 from which the spin angles are measured by 90° in the i 2 i plane. For simplicity, we discuss the case of no magnetostatics where ti is the film height at site i in units of t0 , b i is a unit first. The most striking features in Fig. 6 are the periodic vector parallel to the local step edge, Jij J min ti ,tj , and maxima where H C HSW . These are actually artifacts of the 0a2t0M S where 0 is the magnetic constant (a k a growth scenario sketched above because they correspond to permeability of free space in SI units. perfect layer completion and perfect Stoner-Wohlfarth rever- The internal magnetostatic field Hint was calculated by sal at integer values of thickness. We will correct for this solving the discretized Maxwell equations *Hint *M artifact qualitatively in the next section. Another major trend and Hint 0. For this calculation an effective surface in Fig. 6 is that H layer with uniform thickness was used to exploit a two- C decreases very rapidly as each layer begins to grow.14 This may be understood immediately from dimensional fast Fourier transform algorithm designed for a the lower panel of Fig. 4 for the isolated step model where thin film with no surface roughness.21 H Typical values of the material parameters, J 10 21 J C is given by either HN or HS . Both decrease rapidly, especially when W, i.e., is small. Notice that our param- and K4 1 10 3 mJ/m2,22 imply, in the absence of mag- eter choices for the island model are such that the full range netostatics, a domain wall energy per unit length of W in this figure corresponds to about 1/5. t 2JK4 t 10 14 J/m and an exchange length Another trend seen in Fig. 6 is that the coercive field J/2K4 20 nm. The numerical results reported below all averaged over each partial monolayer apparently decreases use the values K2 KZ 1 mJ/m,22 a 0.3 nm, and MS slowly until about 3 and then increases rapidly. In fact, 1.44 106 A/m (1440 emu/cm3), in addition to those the behavior of H quoted above for J and K C as t increases by integer amounts for 4. The lengths L and D are mea- fixed values of the partial coverage depends very strongly sured in spin block units of 10a /7, a distance over which on . Figure 7 and Fig. 8 illustrate this for 0.06 and no appreciable spin rotation occurs. Positive step anisotropy 0.76, respectively. The 0.06 curves shown in Fig. 7 are K2 0 corresponds to a preferred spin axis parallel to the directly interpretable using the isolated step model. The se- step edges while negative step anisotropy K2 0 corresponds quence shown illustrates the transition from phase IV b to to a preferred spin axis perpendicular to the step edges. phase IV c loops. The coercive field is set by H The simulation technique was the same as described N and its increase in magnitude as t increases so K decreases is previously.14 Beginning with a large value of H clear from Fig. 5. In this regime, the coercive field is a Hx 100 , the local minimum of Eq. 1 was followed as monotonic function of t. the field was reversed adiabatically by a combination of con- Technically, neither the upper panel (t 1) nor middle jugate gradient minimization and relaxational spin dynamics. panel (t 3) of Fig. 8 belong to the phase diagram of Fig. 3 The magnetization parallel to H was computed directly from because both have additional magnetization jumps associated the corresponding spin configurations. with the presence of multiple steps. The associated changes 14 834 R. A. HYMAN, A. ZANGWILL, AND M. D. STILES PRB 60 cumstance both because the steps are long and because they are close together. The long length simply generates more torque. The step proximity effectively doubles K if the step separation is less than an exchange length16 and therefore enhances nucleation as well. Of course, this effect is also absent from the single step model. We turn now to the influence of magnetostatics reflected in Fig. 6. This contribution to the energy is extensive and so has little effect until the film begins to thicken. Dipole inter- actions then generally increase HC although the effect is far more pronounced when the step anisotropy is perpendicular to the steps than when it is parallel to the steps. Because HC HN in the relevant regime, we can understand this by rewriting the magnetic energy in the form *M r *M r E 0 D-D 2 d3r d3r . 2 r r FIG. 7. Hysteresis loops for the island model for 0.06 and t 1,3,5. This term clearly disfavors the creation of ``magnetic charge'' with density *M(r). This has two consequences, in the magnetization affects the coercive field H both of which break the symmetry between parallel and per- C HT for these two cases; this never occurs for the single step model pendicular anisotropy. First, the magnetization is larger on much more than they affect the characteristic fields H the islands than on the terraces. This means that *M(r) N , HS , and H will be unequal to zero at the island edges even if the spin T . Moreover, the values of K used in Fig. 4 are nearly the same as the ones for the cases under discussion. Notice configuration is completely uniform unless the magnetiza- that the step instabilities in Fig. 8 (H tion points along the edges . Magnetostatics therefore in- S for t 1 and HN for t 3) occur at practically the same value of external field as creases the parallel step anisotropy and decreases the perpen- they do in Fig. 4. H dicular step anisotropy. C is larger at t 1 than t 3 because the step edges parallel to the field inhibit the coherent rotation of A second effect is more important. At saturation, Mx is a terrace spins to the 90° state more effectively when K is constant and My 0. To lowest order, Mx remains constant larger. This effect is absent from the isolated step model. By at nucleation so only the variations of My in the y direction the time we reach t 5 lower panel of Fig. 8 , K is so small contribute to ED-D . The magnetization pattern at nucleation that H consists of lens-shaped domains centered on those step edges N determines the coercivity in phase IV c at the large value implied by Fig. 4. Evidently, H where the local anisotropy axis points in the y direction.23 At C is not always a mono- tonic function of film thickness. the smallest coverages when the islands are very small, My Finally, we remark on the very rapid decrease in the co- My(x,y) is a function of both x and y so a finite magne- ercive field that occurs near layer completion ( 1) when tostatic effect is expected. This effect increases as the island is large in Fig. 6. In this regime, H size increases initially because more step edge contributes to C HN as we have discussed and the film consists of large islands that are nucleation. But for large enough island size, the magnetiza- closely spaced. Nucleation occurs very readily in this cir- tion pattern at nucleation is nearly one dimensional with My My(y) for perpendicular step anisotropy and My My(x) for parallel step anisotropy. The effect of magneto- statics thus increases in the former case and decreases in the latter case as the islands grow larger. This is the trend seen in Fig. 6. IV. COMPARISON TO EXPERIMENT A direct comparison between experiment and the island model results of Fig. 6 is not really justified because, for just less than one, the model morphology consists of large disconnected islands separated by long narrow troughs. In reality, island coalescence occurs at these coverages and the morphology is better approximated by a collection of small pits. The pit model of Fig. 1 d is an idealization of this morphology. We performed simulations for this model also using the Hamiltonian Eq. 1 . Figure 9 compares the com- puted coercive field for the island model dashed line and the pit model light solid line for the case when the step FIG. 8. Hysteresis loops for the island model for 0.76 and anisotropy is parallel to the steps. Notice that the coercivity t 1,3,5. at t for the island model is similar to the coercivity at PRB 60 MAGNETIC REVERSAL OF ULTRATHIN FILMS WITH . . . 14 835 dicts that a twofold anisotropy will be induced for those cobalt atoms that lie just beneath the copper step edges. For 2, we compare our results with the measurements of Weber et al.12 who deposited Co on a surface slightly vicinal to Cu 001 . The presence of steps produces split loop hysteresis curves whose origin we have discussed elsewhere.16 Here, we focus on the thickness dependence of the width of the shifted loop which, as the authors note, corresponds to the coercive field. The basic observation is that HC exhibits small amplitude, submonolayer oscillations superposed on a monotonic rise with increasing coverage. This is in qualitative agreement with the bold curve in Fig. 9. FIG. 9. Scaled coercive field as a function of coverage. The light The authors of Ref. 12 attributed the existence of HC solid line is the island model. The dashed line is the pit model. The oscillations to periodic oscillations in the surface morphol- bold solid line synthesizes these two into a curve more suitable for ogy as we do. However, owing to the uniaxial anisotropy comparison with experiment. induced by the steps, they supposed that the presence of t for the pit model. This is because the magnetic rectangular islands was the key to the effect. Our results reversal is almost the same for islands and depressions of the show that oscillations arise even with square islands. Indeed, same size. Small differences occur because the magnetic do- the isolated step model shows that it is only necessary that main wall energy is not exactly the same in the two cases. there be a periodic change in the length of the steps in one Of course, neither the island model nor the pit model is direction. correct when the coverage is exactly or very near t com- plete layers. A better model would exhibit both small islands V. SUMMARY and small pits. Based on the foregoing, we would expect the coercivity to be determined by the pits or islands with the We have used several classical spin models to study zero longest sides. On the other hand, no large pits form during temperature hysteresis in ultrathin films with planar magne- growth and islands with long sides coalesce to form pits with tization and surface roughness characteristic of the epitaxial relatively small step lengths. For this reason, the coercivity growth process. To lend insight into the complicated situa- never becomes very small. The bold curve in Fig. 9 ``syn- tion of many steps, we first discussed a simplified model of a thesizes'' this behavior from our island and pit results by smooth surface with a single finite-length step. The magnetic removing the unphysical jumps at layer completion and oth- interactions were taken to be a fourfold anisotropy at all erwise tracking whichever of the models has the larger co- sites, a twofold anisotropy at step edge sites, and the Zeeman ercivity. interaction with an external magnetic field. The hysteresis The single step model explains almost all of the structure loops that occur were presented in the form of a phase dia- gram with axes labeled by scaled step anisotropy and step that appears in our synthetic HC( ) curve. When is small, length. the scale of the oscillations dominates the variations in the We then considered a periodic island and periodic pit mean value of HC . In the isolated step model, this is the model that allowed for one layer of incomplete coverage. large K limit when HC HS and is nearly independent of K Magnetostatic interactions were added at this stage. The but varies with W. Conversely, when is large (K small , characteristic hysteresis fields H the scale of the oscillations is comparable to the change in N , HS , and HT were found to vary as a function of both film thickness and partial cov- the mean value of HC , which is increasing. This is the small erage of the surface layer in an explicable manner. As a K limit of the isolated step model where HC HN . HC in- result, the coercivity exhibited monolayer-scale oscillations creases as K decreases and varies with W on a comparable on a background that varies nonmonotonically with thick- scale. ness. These results are all in qualitative agreement with re- We now are in a position to compare with experiment. cent experiments for the planar Co/Cu 001 system. Our lower coverage results (0 2) are relevant to the results of Buckley et al. for the Cu/Co/Cu 001 system.13 There, the observed coercivity decreases very rapidly for ACKNOWLEDGMENTS submonolayer deposition of copper and then shows a local R.A.H. acknowledges support from National Science maximum near 1. In some cases, additional oscillations Foundation Grant No. DMR-9531115, and hospitality from in HC were detected. Copper is nonmagnetic but our results the Department of Physics and Astronomy of the University are still germane because the copper islands break the trans- of Georgia where some of this work was completed. M.D.S. lational invariance of the surface. The NeŽel model7 then pre- acknowledges useful conversations with R. D. McMichael. *Permanent address: Department of Physics, DePaul University, K.H.J. Buschow Elsevier, Amsterdam, 1993 , Vol. 7, Chap. 1. Chicago IL, 60614. 3 S. D. Bader and J. L. Erskine, in Ultrathin Magnetic Structures II 1 Ultrathin Magnetic Structures I, edited by J.A.C. Bland and B. Ref. 1 , p. 297. Heinrich Springer-Verlag, Berlin, 1994 ; Ultrathin Magnetic 4 P. Bruno, G. Bayreuther, P. Beauvillain, C. Chappert, G. Lugert, Structures II, edited by B. Heinrich and J. A. C. Bland D. Renard, J. P. Renard, and J. Seiden, J. Appl. Phys. 68, 5759 Springer-Verlag, Berlin, 1994 . 1990 . 2 U. Gradmann, in Handbook of Magnetic Materials, edited by 5 S. T. Purcell, M. T. Johnson, N. W. E. McGee, J. J. de Vries, W. 14 836 R. A. HYMAN, A. ZANGWILL, AND M. D. STILES PRB 60 B. Zeper, and W. Hoving, J. Appl. Phys. 73, 1360 1993 ; T. 20 S. Clarke and D. D. Vvedensky, J. Appl. Phys. 63, 2272 1988 . Kingetsu, Jpn. J. Appl. Phys., Part 1 33, 1890 1994 . 21 We have generalized a fast Fourier transform FFT technique 6 N. W. E. McGee, M. T. Johnson, J. J. de Vries, and J. aan de used for flat films see, e.g., by M. Mansuripur, The Physical Stegge, J. Appl. Phys. 73, 3418 1993 ; T. Kingetsu, Jpn. J. Principles of Magneto-optical Recording Cambridge University Appl. Phys., Part 2 33, L1406 1994 . Press, Cambridge, 1995 , Sec. 13.2 to the case of rough films. 7 L. NeŽel, J. Phys. Radium 15, 225 1954 . By construction, the magnetization is uniform within each spin 8 M. Albrecht, T. Furubayashi, M. Przybylski, J. Korcki, and U. block. The magnetostatic energy contains an intrablock term and Gradmann, J. Magn. Magn. Mater. 113, 207 1992 . an interblock term. The intrablock energy accounts for the shape 9 A. S. Arrott, J. Appl. Phys. 69, 5212 1991 ; A. S. Arrott and B. anisotropy of the block. This part of the magnetostatic energy is Heinrich, J. Magn. Magn. Mater. 93, 571 1991 . 10 functionally identical to a bulk crystalline anisotropy that favors A. S. Arrott, T. L. Templeton, and Y. Yoshida, IEEE Trans. in-plane spins. Surface roughness is easily incorporated in this Magn. 29, 2622 1993 ; A. S. Arrott, in Nanomagnetism, edited term because it is local and explicitly linear in the thickness. A by A. Hernando Kluwer, Amsterdam, 1993 , pp. 73­85. 11 M. Kolesik, M. A. Novotny, and P. A. Rikvold, Phys. Rev. B 56, 2D FFT magnetostatic routine is used to calculate the interblock 11 791 1997 . magnetostatic energy. The routine has been derived for uni- 12 W. Weber, C. H. Back, A. Bischof, Ch. Wursch, and R. Allens- formly flat films. However, surface roughness can still be incor- pach, Phys. Rev. Lett. 76, 1940 1996 . porated. To lowest order in the thickness, the interblock energy 13 M. E. Buckley, F. O. Schumann, and J. A. C. Bland, Phys. Rev. B is dependent only on the magnetic moments of the blocks. 52, 6596 1995 ; M. E. Buckley, F. O. Schumann, and J. A. C. Therefore, surface roughness is included by performing the FFT Bland, J. Phys.: Condens. Matter 8, L147 1996 . on an effective flat film in which the magnetization magnitude is 14 A. Moschel, R. A. Hyman, A. Zangwill, and M. D. Stiles, Phys. adjusted so that the block magnetic moments of the two films Rev. Lett. 77, 3653 1996 . are identical. The magnetization in the effective film is Mi 15 R. A. Hyman, M. D. Stiles, L.-H. Tang, and A. Zangwill, J. Appl. tiMSS i per site, chosen so that the block magnetic moments of Phys. 81, 3911 1997 . the flat and rough films are identical. 16 R. A. Hyman, A. Zangwill, and M. D. Stiles, Phys. Rev. B 58, 22 B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 1993 . These 9276 1998 . are the values used in Refs. 14,15. The size of the fourfold 17 The fields here called HS and HT were called HL and H in Refs. anisotropy was misstated there as K4 1 10 2 mJ/m2. 14 and 15. The notation in this paper is the same as Ref. 16. 23 The magnetization pattern at nucleation discussed here is unstable 18 E. C. Stoner and E. P. Wohlfarth, Philos. Trans. A240, 74 1948 . if a jump in magnetization accompanies nucleation. But this 19 The phases labeled III and IV in Ref. 16 should now be called IIIc pattern is still used to determine the energy barrier to nucleation. and IVc. See Ref. 16 for more details.