JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 2 15 JANUARY 2001 Effect of surface roughness on magnetic domain wall thickness, domain size, and coercivity Y.-P. Zhao,a) R. M. Gamache, G.-C. Wang, and T.-M. Lu Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 G. Palasantzas and J. Th. M. De Hosson Department of Applied Physics, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 17 December 1999; accepted for publication 11 October 2000 We study the effect of surface roughness on magnetic domain wall thickness, domain size, and coercivity of thin magnetic films. We show that the roughness increases decreases the domain wall thickness and domain size for Bloch walls NeŽel walls . The surface roughness affects the domain wall movement and causes the increase of coercivity for NeŽel walls. The coercivity due to domain rotation for Bloch walls decreases with the increase of roughness. The domain wall thickness, domain size, and coercivity are each related to the demagnetizing factor, which depends on the roughness and type of wall Bloch wall or NeŽel wall . The calculated coercivity versus thickness is compared with experimental data of ultrathin Co films, where the thickness dependent roughness parameters are available. © 2001 American Institute of Physics. DOI: 10.1063/1.1331065 I. INTRODUCTION hand Soohoo5 fitted a rather wide variety of coercivity data for thicknesses larger than 20 nm under the constraint that Magnetic properties of thin films are influenced by a dt/dx increases nearly linearly with film thickness. Such an variety of parameters such as film thickness, crystalline increase of the thickness fluctuations5 was attributed to structure, composition, and surface/interface roughness. Spe- roughness changes occurring at short roughness wave- cifically, surface/interface roughness influences magnetic lengths. properties such as magnetic moments, magnetic anisotropy, For ultrathin Co films deposited on rough Cu-buffered coercivity, magnetic domain structure, and motion, etc.1 Si 111 substrates the coercivity was shown to decay with Some examples are: the coercivity of chemically etched Ni- increasing film thickness t as t 0.4 0.1 for 12­44 monolayer FeCo films 20­100 nm thick was found to increase with equivalent MLE .8 In epitaxial ultrathin films studies, Co increasing film surface roughness.2 The coercivity of NiCo- films ranging from 2 to 30 ML deposited on a smooth alloy films 2 m thick first increases, then decreases, and Cu 001 substrate show that the coercivity increases from increases again as the surface roughness increases.3 Studies about 2 to 7­8 ML, followed by a slight decrease at higher in Co films 100 nm thick deposited on plasma etched thicknesses.9 The Hc even oscillates as a function of Co film Si 100 substrates showed that, by increasing surface rough- 4­14 ML deposited on Cu 001 . The oscillation period is 1 ness, the uniaxial anisotropy decreased and disappeared for ML; this corresponds to the layer-by-layer growth of Co af- the roughest films.4 Moreover, with increasing surface ter 2 ML thickness.10 The Hc for films deposited on rough- roughness the magnetization reversal changed gradually ened substrates are higher. Examples are Co/Cu 001 9 and from magnetization rotation dominant for smooth films to Ni/Cu 001 .11 domain wall motion dominant for the roughest films .4 Defining the relationship between the surface roughness The relation of the coercive field Hc for domain wall and the coercivity and determining the properties of mag- motion in thin films has been shown to be related by film netic domain change with surface roughness are questions of thickness fluctuations for zig-zag5 and straight6 domain interest when dealing with real films. Recently, we have ex- walls. NeŽel,7 based on the same concept, derived the well amined the effect of roughness on the demagnetizing factor known ``4/3'' law for the dependence of coercivity Hc on of thin magnetic films.12 In this work we add the energy the film thickness t, Hc t 4/3, which is valid under the as- minimization and extend the study of the demagnetizing fac- sumption that the thickness fluctuation dt/dx with x being tor in Ref. 12 to relate surface roughness with domain prop- the lateral direction along which the wall motion occurs is erty and coercivity. Our treatment is straightforward and can constant. However, in many cases e.g., in NiFe films6 such be applied to thin magnetic films. However, we can only find a law appeared to be invalid, in agreement with the fact that systematic experimental data of both surface roughness and a constant dt/dx cannot always be assumed. On the other magnetic properties for ultrathin films. When we apply our prediction for thin films to available ultrathin magnetic film a Electronic mail: zhaoy@rpi.edu data, we obtain a qualitative agreement. 0021-8979/2001/89(2)/1325/6/$18.00 1325 © 2001 American Institute of Physics Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 1326 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 Zhao et al. typical plot of nxx as a function of . nxx decreases as increases i.e., as the surface roughness becomes less wig- gly . III. DOMAIN WALL THICKNESS The surface energy w for a domain wall can be written as19­21 w ex an mag , 6 where ex is the exchange energy, an is the anisotropy energy, and mag is the magnetostatic energy. In general, for 180° domain, the exchange energy ex can be expressed as A FIG. 1. Demagnetizing factor ratio nxx as a function of roughness ex 2 ex D , 7 exponents . where Aex is the exchange constant defined as Aex JS2/a0 , J is the exchange integral, S is the spin, and a0 is II. SURFACE ROUGHNESS MODEL AND the atomic length scale. D is the thickness of the domain DEMAGNETIZING FACTOR wall.The anisotropy energy an can be written as A wide variety of surfaces/interfaces occurring in nature are well described by a kind of roughness associated with K 1D self-affine fractal scaling.13­17 For self-affine fractals the an 2 , 8 roughness spectrum h(k) 2 scales as18 where the in-plane anisotropy constant K1 Kv 2Ks /t, Kv is the in-plane volume anisotropy constant, Ks is the surface h k 2 k 2 2 , for k 1, 1 anisotropy constant.22 Here we only consider the uniaxial const, for k 1 anisotropy contribution. with the roughness exponent (0 1) being a measure The magnetostatic energy mag is of the degree of surface irregularity, such that small values of 2 characterize more jagged or irregular surfaces at short mag NDMs, 9 roughness wavelengths . Here is the lateral correlation where N is the demagnetizing factor for the magnetic domain length. The scaling behavior depicted by Eq. 1 can be de- wall and Ms is the saturation magnetization. scribed by the simple K-correlation model:18 Therefore, Eq. 6 can be rewritten as F w2 2 A K h k 2 ex 2 1D 2. 10 2 4 1 ak2 2 1 , 2 w D 2 NDMs where F is the surface area we consider, w is the root-mean- The minimization of surface energy for the domain wall square rms roughness, and a (1/2 ) 1 (1 w / D requires that aQ2c 2) , with Qc being the upper spatial frequency A K N cutoff. The rms local slope defined as ex 2 1 2 2 rms h 2 can be DMs expressed as D2 2 NMs D 0. 11 There are two kinds of magnetic domain walls: Bloch wall rms 2 4 for thick films in which the magnetization rotates out of the F k2 h k 2 dk 1/2. 3 film plane when crossing the wall and NeŽel wall for thin The demagnetizing factors satisfy films in which the magnetization rotates within the film plane N when crossing the wall. For both Bloch and NeŽel walls, ex xx Nyy Nzz 1. 4 and an are the same. However, the magnetostatic energy Under the small slope assumption rms 1 and for an isotro- mag is different due to the difference in the demagnetizing pic surface, Nxx Nyy , we have an in-plane demagnetizing factor N. For a Bloch wall with a perfectly flat surface: factor:12 D 0 N 2 4 2 NBloch n xx kx t D . 12 xx N0 zz 2tF k h k 2 dk. 5 For a NeŽel wall with a perfectly flat surface Here N0zz is the demagnetizing factor for a smooth film in the t z direction and t is the film thickness. n 0 xx is proportional to NNeŽel w2, while its relationship with and is more complicated t D . 13 as seen in Eqs. 4 and 5 , and in Ref. 12. Figure 1 shows a These are well-known results.19­21 Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 Zhao et al. 1327 1/2 w M s 4 Aex 1 2nxx 2 3/2Aex M s 1 nxx , 18 i.e., the surface energy of the domain wall will decrease. For a NeŽel wall, if t D, Eq. 11 gives A 1 D Aex 1/2 ex 1 , 19 1 n 2 xx M s Ms 2 nxx and 1 1/2 w M s 4 Aex 1 nxx 2 3/2Aex M s 1 2 nxx . 20 Thus the NeŽel wall thickness decreases with the roughness but the wall energy increases. We notice that in this case D cannot always decrease according to Eq. 19 , because as D approaches , nxx also becomes D dependent. IV. DOMAIN SIZE Next we consider the effect of surface roughness on the domain size for a closure domain. The domain energy Ed can be written as a sum of wall energy Ew , anisotropy energy Ean , and magnetostatic energy Emag :21 Ed Ew Ean Emag FIG. 2. a Bloch wall, and b the NeŽel wall. D and t are wall thickness and film thickness, respectively. K w t 8 1 L 1L 2 L 2 NLMs, 21 Now we consider the surface of a magnetic film to be where L is the domain size. Usually, for L , N is indepen- rough. For a Bloch wall, as shown in Fig. 2 a , the surface dent on L. Thus the energy minimization process gives the roughness will decrease the demagnetizing factor perpen- size of the domain wall: dicular to the film surface, according to Eq. 4 : L 2 wt 1/2 2 . 22 N 0 K1 NMs Bloch NBloch 1 2nxx . 14 If we assume the effect of roughness on the surface energy Here n 0 xx is the ratio Nxx /Nzz, which is greater than zero for a rough surface. However, for a NeŽel wall, as shown in Fig. w of a domain wall is small, then w is almost a constant. Therefore, for a Bloch wall, we have 2 b where the magnetization, the surface roughness will in- crease the demagnetizing factor parallel to the film surface: L 1/2 Bloch 2 wt , 23 K 2 N 0 1 1 2nxx M s NeŽel NNeŽel 1 nxx . 15 i.e., the increase of surface roughness will increase the size Now if we assume nxx is not a function of domain wall of the Bloch domain. However, for a NeŽel wall thickness D which means D , we can estimate the effect of roughness on the D. For a Bloch wall, in the bulk limit L 1/2 NeŽel 2 wt , 24 t D, the magnetostatic energy term K 2 mag can be 1 nxxM s neglected:20,21 i.e., the increase of surface roughness will decrease the size D & A of NeŽel domain. ex /K1 1/2, 16 i.e., the domain wall thickness almost does not change with surface roughness. However, if K 2 1 2 M s , the anisotropy V. COERCIVITY energy an can be neglected, and A. Wall movement A D Aex 1/2 ex The coercivity of a thin magnetic film caused by domain 1 n 1 2n 2 xx , 17 xx M s Ms wall movement can be written as6,20 i.e., the domain wall thickness will increase. The surface 1 dh dh dl mov w w w energy of the domain wall becomes Hc 2M , 25 s t dx t dx l dx Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 1328 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 Zhao et al. where h is the surface height and l is the length of the wall. 1 KvD Dt 2D2 According to Soohoo5 one can assume that the length of wall Hmov Aex 2 2 c 2M Dt 2t t D 2 Ms rms , l remains unchanged during the wall movement, we have s 31 1 i.e., for the same film thickness, the rougher the surface, the Hmov w w dh c 2M larger the coercivity. This result is quite consistent with the s t t dx . 26 experimental results obtained by, for example, Malyutin From a statistical point of view, for a rough surface, we et al.2 and Li et al.4 replace (dh/dx) by rms local slope rms . The rms local slope is unitless and yields a measure of the average local surface slope. Thus, Eq. 26 becomes B. Wall rotation 1 Hmov w w If a magnetic field H is applied to a thin film and causes c 2M rms . 27 s t t the domain to rotate coherently, then the energy of a domain Therefore, we learn that the coercivity is closely related to can be expressed as6,20,21 surface roughness. According to Eq. 10 , for a NeŽel wall, we E E have w Ean Emag EH A K tD w 1 cos 2 ex 2 s 2 w D Kv 2 t D t D Ms . 28 1 LK1 sin2 2 LNeM2 cos2 Therefore, the coercivity caused by domain wall movement can be written explicitly as 1 2 LNhM2 sin2 LHM cos , 32 1 2 2 DtM M A Hmov D2Ms s s ex 2 c 2M where is the angle between the magnetization M and the s t D 2 t D t Dt easy axis, and Ne(Nh) is the demagnetizing factor in the easy K DM2 hard direction. is the angle between the magnetic field H vD s 2t t D rms . 29 and the magnetization M, , where is the angle between the magnetic field and the easy axis. At equilibrium Here we consider the possibility of the thickness dependent ( E/ ) 0, one has Ms . In general, for an ultrathin film, the saturation magne- tization M 1 1 s depends on the film thickness. According to K Glass and Klein,23 for a face-centered cubic fcc film: 1 2 NeM2 2 NhM2 sin 2 2HM sin M G3 s t 1 kT w 1 cos k M0 1 3 1 L 2 sin sin 2 0. 33 s 16 S2G3 J 3 0 ln 1 e B ln 1 e A , 30 Furthermore, ( 2E/ 2) 0 implies where M0 K1 NeM2 NhM2 cos 2 HM cos s is the bulk saturation magnetization, Gi is the number of cubic cells in the ith direction of the crystal, G is w a large number 107 or more , and L cos cos2 0. 34 16JS A If 0, we obtain the coercivity in the easy axis as kT 1 2/4G2 1 2/4G2 cos k3 , K Hrot 1 16JS c M Ne Nh M. 35 B kT 1 /4 1 /4 cosk3 , For the NeŽel wall, since N rot e Nh , Hc is independent on 2 surface roughness. However, for the Bloch wall since Ne k 3 N 3 G . zz 1 2Nxx , Nh Nxx , we have 3 K In fact, G rot 1 3 t/a0 , where a0 is the lattice constant and 3 is Hc an index. Note that in comparing with experiments, it is not M 1 3Nxx M. 36 the local spin S that is determined by the effective magnetic Clearly, as the roughness increases, the rotational coercivity moments per atom m and g values. Thus, in Eq. 30 the for the Bloch wall decreases. This conclusion is partially spin S should be effectively replaced by S m/gmB with mB consistent with the experimental result, for example the the Bohr magneton. At any rate, the coercivity has a compli- NiCo film where the coercivity only decreases within a cer- cated relationship as a function of the film thickness t. tain roughness regime.3 The reason can be partly attributed For a thin film, when the saturation magnetization Ms to that the actual domain rotation may not be coherent.20,21 In becomes a constant, Eq. 29 can be reduced as fact, the coercivity­roughness relationship for a thick film is Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 Zhao et al. 1329 FIG. 3. rms local slope rms as a function of the Co film thickness t for Co/Cu 001 system obtained from Ref. 9. The solid curve is a guide to the FIG. 4. Coercivity Hc as a function of the Co film thickness t for Co/ eyes. Cu 001 system. The filled circles are the experimental coercivity obtained from Ref. 9. The represents the calculated coercivity using the experi- mental thickness dependent Kerr intensity data in Ref. 9. The open squares, more complicated than a simple monotonic relationship, and circles, and diamonds are calculated coercivities for average spins 0.2, 0.1, the magnetization reversal process may change with the and 0.05, respectively. The solid curves are guides for the eyes. change of surface roughness. C. Comparison with experiments tal data showed that at t 8 ML the coercivity starts to de- crease. There are at least four possible reasons that may con- In order to see how the roughness affects the coercivity tribute to this discrepancy, besides the calculation being for caused by domain wall movement, we calculate the coerciv- thin films instead of ultrathin films: 1 The absolute thick- ity of ultrathin Co film grown on Cu 001 using the rough- ness of ultrathin Co films could be off by 2­3 ML. The ness data from Table I in Ref. 9. Note that during interdiffu- uncertainty comes from the uncertainty in Auger electron sion at the Co/Cu interface the Co moment is not quenched escape depth if Auger electron spectroscopy is used. How- in the Cu matrix, but only diluted. Here we assume that the ever, the relative monolayer thickness change as observed in ultrathin Co film has a fcc structure, and that the surface is a HRLEED diffraction peak intensity oscillation in epitaxial self-affine rough surface this assumption may not be valid; layer growth is relatively reliable after the first two layers see later discussion . Figure 3 shows the rms slope rms of because the degree of Co and Cu intermixing and the forma- the Co film as a function of the film thickness t. The rms tion of bilayer islands reduce.9,24,25 More precise thickness slope rms increases as t increases. When we calculate the determination would require the use of transmission electron coercivity, we use the following constants for a bulk Co microscopy TEM imaging.26 2 The roughness parameters crystal:19 measured from HRLEED may not be accurate. Comparing J 155 K Figs. 3 and 4, we can see that the increase of the coercivity value from 11 to 15 ML is due to the increment of the rms a0 2.5 10 8 cm local slope from 11 to 15 ML. In fact, the surface morphol- M0 ogy of the ultrathin Co film may not be treated as a self- s 1425 G affine surface. Therefore, the method used for a self-affine D 5 10 7 cm surface to extract the roughness parameters may not give the and the anisotropy constant for a thin Co film:22 true values. Even if the surface is self-affine, the height dis- tribution may not be a Gaussian function. According to Zhao Kv 2.3 106 erg/cm3 et al.,27 the roughness parameters extracted from the theory of a Gaussian surface are not the same from that extracted Ks 0.034 erg/cm3 from a non-Gaussian surface. 3 According to Soohoo5 the According to Soohoo,5 the average spin S for a Co film is average spin S is actually a function of the film thickness. less than 0.65 for t 100 Ć. However, there is no experi- However, in our calculation we did not take this thickness mental measurement so far for the S value in the ultrathin dependent S into account. 4 The surface magnetization of film regime. Figure 4 shows the calculated coercivity of the Co layer grown on Cu 001 has been observed by magneti- ultrathin Co films as a function of the film thickness t for zation induced second harmonic generation MSHG to have different average spin S. For various S values, the coercivity one ML period.28 The Co has a layer-by-layer growth mode increases from 3 to 15 ML, and then gradually decreases and the step density is expected to change periodically with from 15 to 25 ML. The overall behavior is qualitatively simi- one ML period. The magnetic moments of edge atoms in the lar to the experimental data of Jiang et al.9 However, the two-dimensional islands are not the same as those of non- quantitative values are not exactly the same: the experimen- edge atoms. This contributes to the change in surface mag- Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 1330 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 Zhao et al. netization. Extensive studies of the ultrathin Co film growth thickness is in qualitative agreement with experimental data on Cu 100 by various techniques,9,24,25 show that Co and Cu of Co ultrathin films grown on Cu 001 surfaces. For the interdiffusion exists for the first 2­3 ML. For Co film thick- same film thickness, the coercivity of thin magnetic films ness greater than a few monolayers the interdiffusion is sup- increases with surface roughness, which is qualitatively con- pressed and one can ignore the effect of interdiffusion. A sistent with many experimental results. We also find the co- more realistic calculation needs the use of experimentally ercivity for coherent domain rotation for the Bloch wall de- measured magnetization. Jiang et al. measured the relative creases as the roughness increases. change of Kerr intensity as a function of Co film thickness,9 where the Kerr intensity is proportional to magnetization. ACKNOWLEDGMENTS Since the squareness of measured hysteresis loops (MR /Ms) This work was supported by the NSF. The authors thank is close to one, one can assume that the remnant magnetiza- J. B. Wedding for reading the manuscript. tion is equal to the saturation magnetization. Also we must neglect the third and fourth terms containing exchange con- 1 Ultrathin Magnetic Structures I and II, edited by J. A. C. Bland and B. stant and in-plane volume anisotropy constant, respectively Heinrich Springer, New York, 1994 ; C. H. Chang and M. K. Kryder, J. Appl. Phys. 75, 6864 1994 ; P. Bruno, G. Bayureuther, P. Beauvillain, C. on the right-hand side of Eq. 29 , and rescale the calculated Chappert, G. Lugert, D. Renard, J. P. Renard, and J. Seiden, ibid. 68, 5759 result. Figure 4 plots the calculated coercivity using the ex- 1990 . 2 perimentally measured relative Kerr intensity versus thick- V. I. Malyutin, V. E. Osukhovskii, Yu. D. Vorobiev, A. G. Shishkov, and V. V. Yudin, Phys. Status Solidi A 65, 45 1981 . ness. We see that the calculated coercivity data are closer to 3 S. Vilain, J. Ebothe, and M. Troyon, J. Magn. Magn. Mater. 157, 274 the measured coercivity data. 1996 . We should point out that the upper limit of film rough- 4 M. Li, G.-C. Wang, and H.-G. Min, J. Appl. Phys. 83, 5313 1998 ; M. Li, ness is the film thickness, i.e., the rms surface roughness Y.-P. Zhao, G.-C. Wang, and H.-G. Min, ibid. 83, 6287 1998 . 5 R. F. Soohoo, J. Appl. Phys. 52, 2459 1981 . amplitude should be smaller than the film thickness. In gen- 6 S. Middelhoeok, Ferromagnetic Domains in Thin Ni­Fe Films Drukkerij eral, the thickness dependent roughness is strongly related to Wed. G. Van Soest, Amsterdam, 1961 . the film growth modes and the type of material being grown. 7 L. NeŽel, C. R. Acad. Sci. Paris 241, 533 1955 . 8 An example of roughness evolution is depicted in Fig. 3 H.-G. Min, S.-H. Kim, M. Li, J. B. Wedding, and G.-C. Wang, Surf. Sci. 400, 19 1998 . which shows the local slope rms versus film thickness t. The 9 Q. Jiang, H.-N. Yang, and G.-C. Wang, Surf. Sci. 373, 181 1997 . growth of Co film on Cu 100 surface has a layer by layer 10 W. Weber, C. H. Back, A. Bischof, Ch. Wursch, and R. Allenspach, Phys. growth mode in the first few monolayers.29 Rev. Lett. 76, 1940 1996 . 11 S. Z. Wu, G. J. Mankey, F. Huang, and R. F. Willis, J. Appl. Phys. 76, 6434 1994 . VI. CONCLUSIONS 12 Y.-P. Zhao, G. Palasantzas, G.-C. Wang, and J. Th. M. De Hosson, Phys. We have shown that the roughness of an isotropic self- Rev. B 60, 1216 1999 . 13 Dynamics of Fractal Surfaces, edited by F. Family and T. Vicsek World affine surface changes the demagnetizing factors in magnetic Scientific Singapore, 1990 . thin films. The roughness induced demagnetizing factors 14 A.-L. BarabaŽsi and H. E. Stanley, Fractal Concepts in Surface Growth change values differently for Bloch walls and NeŽel walls. Cambridge University Press, New York, 1995 . 15 H.-N. Yang, G.-C. Wang, and T.-M. Lu, Diffraction from Rough Surfaces The demagnetizing factor decreases in the direction perpen- and Dynamic Growth Fronts World Scientific, Singapore, 1993 . dicular to the film surface in the case of Bloch walls whereas 16 P. Meakin, Phys. Rep. 235, 1991 1993 . the demagnetizing factor increases in the direction parallel to 17 J. Krim and G. Palasantzas, Int. J. Mod. Phys. B 9, 599 1995 . 18 the film surface in the case of NeŽel walls. Since the coerciv- G. Palasantzas, Phys. Rev. B 48, 14472 1993 ; 49, 5785 E 1994 . 19 C. Kittel, Rev. Mod. Phys. 21, 541 1949 . ity, magnetic domain wall thickness, and domain sizes are 20 R. F. Soohoo, Magnetic Thin Films Harper and Row, New York, 1965 . each a function of the demagnetizing factor, one can calcu- 21 M. Prutton, Thin Ferromagnetic Films Butterworths, Washington, 1964 . late the change in these magnetic properties as the surface 22 P. Krams, F. Lauks, R. L. Stamps, B. Hillerbrands, and G. GuŻntherodt, roughness changes. If we neglect anisotropy energy, then the Phys. Rev. Lett. 69, 3674 1994 ; P. Krams, F. Lauks, R. L. Stamps, B. Hillerbrands, G. GuŻntherodt, and H. P. Oepen, J. Magn. Magn. Mater. domain wall thickness increases decreases as the demagne- 121, 483 1993 . tizing factor increases for Bloch walls NeŽel walls . For a 23 S. J. Glass and M. J. Klein, Phys. Rev. 109, 288 1958 . closure domain, the domain size increases decreases as the 24 M. T. Kief and W. F. Egelhoff, Jr., Phys. Rev. B 47, 10785 1993 . 25 demagnetizing factor increases for Bloch walls NeŽel walls . J. Fassbender, R. Allenspach, and U. Dušrig, Surf. Sci. 383, L742 1997 . 26 S. C. Ma, C. K. Lo, Y. D. Yao, D. Y. Chiang, T. F. Ying, and D. R. If we replace the local film thickness variation by the rms Huang, J. Magn. Magn. Mater. 209, 131 2000 . local slope and assume that the saturation magnetization de- 27 Y.-P. Zhao, G.-C. Wang, and T.-M. Lu, Phys. Rev. B 55, 13938 1997 . pends on film thickness, then the calculated coercivity from 28 Q. Y. Jin, H. Regensburger, R. Vollmer, and J. Kirschner, Phys. Rev. Lett. domain wall movement for a fcc film increases as the local 80, 4056 1998 . 29 Q. Jiang, H.-N. Yang, and G.-C. Wang, J. Vac. Sci. Technol. B 14, 3180 slope increases. We found the calculated coercivity versus 1996 . Downloaded 20 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html