PHYSICAL REVIEW B VOLUME 60, NUMBER 2 1 JULY 1999-II Surface/interface-roughness-induced demagnetizing effect in thin magnetic films Y.-P. Zhao Department of Physics, Applied Physics, and Astronomy, and Center for Integrated Electronics and Electronics Manufacturing, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 G. Palasantzas Department of Applied Physics, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands G.-C. Wang Department of Physics, Applied Physics, and Astronomy, and Center for Integrated Electronics and Electronics Manufacturing, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 J. Th. M. De Hosson Department of Applied Physics, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 3 December 1998; revised manuscript received 12 February 1999 We study the influence of surface/interface roughness on the demagnetizing factor of a thin magnetic film with a single or a double boundary of self-affine, mound or anisotropic roughness. For a film with a single self-affine rough boundary, the in-plane demagnetizing factor Nxx(yy) is proportional to the interface width w square and to the leading order is inversely proportional to the lateral correlation length . The roughness exponent is also shown to greatly affect Nxx(yy) . For a film with a single mound boundary, Nxx(yy) is inversely proportional to the apparent correlation length, and also depends on the ratio of the two different lateral lengths: the average mound separation and the randomness correlation length . It is also shown that an anisotropic surface morphology can induce anisotropic in-plane demagnetizing factors. The demagnetizing anisotropy can be magnified by a morphological anisotropy. Furthermore, we consider films with two rough boundaries. Besides a general formalism derived for the demagnetizing factor, we investigate how the cross correlation of the two rough boundaries affects the in-plane demagnetizing factors. Connections between the demagnetizing factor and thin-film growth mechanisms are also discussed. S0163-1829 99 12125-8 I. INTRODUCTION the coercivity increases with increasing surface roughness.5 Kim et al. studied the underlayer Si3N4 roughness on the Recently there has been a great interest on how surface coercivity of the Co/Pt multilayers.10 They also found that roughness will affect the properties of thin magnetic films, the coercivity enhanced with the increase of thickness such as coercivity, magnetic domain structure, magnetization roughness of the Si3N4 underlayer. reversal, and magnetoresistance.1­4 These magnetic proper- So far there are only a few theoretical examinations dis- ties greatly affect the applications of thin magnetic films in cussing the effects of surface/interface roughness on mag- magnetic recording industry, as well as other applications in netic films.11­13 This is probably due to the complicated na- magnetoelectronics. Many experiments have been performed ture of the problem. Physically, all magnetic properties are for thin magnetic films with two kinds of rough boundaries. related to the magnetic energy of a thin film. Besides surface/ One kind is a film with a single rough boundary.5­7 For interface roughness, many other factors such as film thick- example, Jiang et al. studied the relation of the coercivity ness, composition, crystalline structure of the magnetic film, versus surface roughness of Co ultrathin films deposited on magnetic domain distribution and correlation, contribute to an atomically flat Cu substrate.5 Vilain et al. investigated the the magnetic energy and determine the magnetization coercivity versus surface roughness of electrodeposited NiCo mechanism of a film. These are very important factors, and alloy films,6 and Malyutin, et al. showed that the coercivity cannot be neglected in practice. However, in order to distin- of chemically etched Ni-Fe-Co films increases with the sur- guish which factor dominates, each factor needs to be inves- face roughness.7 Very recently, Freeland et al. using the tigated individually. In this work, we concentrate on how x-ray resonant magnetic scattering studied hysteretic behav- surface/interface roughness affects the magnetic energy of a ior of CoFe thin films with varying roughness.8 They also thin film, or alternatively, how boundary roughness influ- found the coercivity increased with the surface roughness. ences the demagnetizing factor of a thin film. The other kind is a film with double rough boundaries.5,9,10 In general, the demagnetizing field of a magnetic material Recently Li et al. performed a detailed study of thin Co films is caused by the generation of ``magnetic poles'' near its deposited on plasma etched Si 100 films.9 They found that boundaries due to the finite shape of a material. The mag- the uniaxial magnetic anisotropy decreases with the increase netic poles give rise to a demagnetizing field Hd , which is of surface roughness. Jiang et al. also investigated ultrathin opposing the applied field. The strength of Hd depends on Co films on an Ar -sputtered Cu substrate, and found that the geometry and the magnetization of a material M:Hd 0163-1829/99/60 2 /1216 11 /$15.00 PRB 60 1216 ©1999 The American Physical Society PRB 60 SURFACE/INTERFACE-ROUGHNESS-INDUCED . . . 1217 N *M, where N is the demagnetizing tensor, depending on the shape of a magnetic object. For a smooth and infinite large thin magnetic film, its boundary in the film plane ex- tends to infinity, and the demagnetizing factor in the film plane should be zero, but the demagnetizing factor along the out-of-plane direction of a thin film is nonzero. However, if the film is rough, the local roughness features will induce local in-plane ``magnetic poles,'' which may result in a non- zero in-plane demagnetizing factor. This problem was ini- tially treated by Schlo¨mann in the 1970s for a single sinu- soidal rough boundary.11 He found that the in-plane demagnetizing factor Nxx(yy) 2/ , where and are the amplitude and the wavelength of the sinusoidal boundary, FIG. 1. A cross section of a rough film lying in the x-y plane. respectively. Recently, one of us G.P. extended this treat- The growth front is in the z direction. The film thickness is d, with ment to some special statistically rough self-affine surfaces13 the boundaries L1 : d/2 h1(r) and L2 : d/2 h2(r). and found that Nxx(yy) w2/ with w being the surface width and being the in-plane roughness correlation length, as well n *M x as a strong dependence on the roughness exponent of a M x x x da surface. In general, surface roughness is determined by the s thin-film deposition methods and conditions as well as the h initial substrate roughness. Furthermore, the growth front of dr 1 / x M0x h1 / y M0y M0z a thin film and the substrate roughness are closely related by r r 2 z d/2 h1 r 2 the thin-film growth mechanism. A different surface mor- phology such as self-affine, mound or anisotropic surface can h dr 2 / x M0x h2 / y M0y M0z . be formed from a different growth mechanism. Previous the- r r 2 z d/2 h2 r 2 oretical works11,13 did not consider how these different kinds of morphology and especially the dynamics of growth 1 mechanism will affect the demagnetizing factors. Experi- mentally it has been shown that substrate roughness can con- Note that n is the surface normal pointing away from the tribute strongly to the magnetic properties,5,9,10 but the quan- surface, and da is the differential surface area. According to titative connection with theoretical result was not made.11,13 Schlo¨mann,11 the self-energy can be written as The organization of this paper is as follows: In Sec. II we derive a general formalism for the demagnetizing factor of 1 thin films with rough boundaries. In Sec. III we consider the W 2 dr dzH*M, 2 demagnetizing factor of a single rough boundary, where we investigate thoroughly how different surface morphologies with the magnetic-field strength H M . Substituting Eq. affect this factor. In Sec. IV we investigate the cross- 1 into Eq. 2 see Appendix A we obtain the final expres- correlation effect of double rough boundaries by taking into sions for the demagnetizing factors Nxx , Nyy , and Nzz in account dynamic growth effects through linear Langevin real space. Here Nxx , Nyy , and Nzz are the diagonal compo- growth models. In the end, we conclude our results. nents of the demagnetizing tensor N . A similar calculation can be applied to the nondiagonal components Nxy , Nyz , II. DEMAGNETIZING FACTORS FOR MAGNETIC FILMS and Nzx . To evaluate further the average in-plane demagne- WITH ROUGH BOUNDARIES tizing factor, we consider the Fourier transform The basic assumptions made here are that the film is uni- 1 form and single domain with a homogeneous magnetization h i k hi r eik*rdr, 3 M 2 2 0 throughout the film. We assume the general case where the two interfaces of the magnetic film as shown in Fig. 1 are rough. These interfaces are described by the boundaries d/2 h h 1(r) and d/2 h2(r), respectively, with hi(r)(i 1,2) i r h i k e ik*rdk, 4 being single-valued random surface height fluctuations. Here r (x,y) is the in-plane position vector, and d is the average and assume a translation invariant surface/interface: film thickness. The magnetization in a film can be written as 2 4 d d h i k h j k M x M A h i k h j k k k , 5 0 z 2 h2 r z 2 h1 r where i, j 1, 2; and denotes an average over all possible with x (r,z) and (z) is a step function. According to choices of origins and an ensemble average over all possible Jackson,14 for a uniform magnetization, the magnetic scalar surface configurations. Upon substitution, we obtain the en- potential can be written as semble average which finally yields 1218 ZHAO, PALASANTZAS, WANG, AND DE HOSSON PRB 60 2 4 k2 N x xx 2dA dk k h 1 k 2 h 2 k 2 2e dk h 1 k h 2 k . 6 A similar expression for Nyy can be obtained. Nzz can be calculated from the orthogonality condition Nzz 1 Nyy Nxx .11 Equation 6 is the basic formula that we will study in this paper. However, we should emphasize that the as- sumption for Eq. 6 is that w d, and the average local slope is also much less than one. FIG. 2. Log-log plot of the in-plane demagnetizing factor III. DEMAGNETIZING FACTORS Nxx /N0 as a function of the roughness exponent for an isotropic FOR MAGNETIC FILMS self-affine surface. Here N0 w2/d . WITH SINGLE ROUGH BOUNDARY A. Isotropic self-affine surface Magnetic thin-film growth usually commences on a very smooth surface. In this case h In this case, one has19­21 2 0, we may simplify Eq. 6 to the form11,13 A 2w2 h k 2 2 5 1 2k2 1 . 8 2 4 k2 N x Here, w is the interface width describing the fluctuation of xx 2dA dk k h 1 k 2 , 7 the surface height. is the lateral correlation length within which the surface heights of any two points are correlated. The roughness exponent (0 1) describes how wiggly which is actually the formula obtained by Schlo¨mann.11 the surface is. Due to the isotropy N However, Schlo¨mann only considered the case for a sinu- xx Nyy . Substituting Eq. 8 into Eq. 7 we can obtain N soidal rough interface, which may not occur in reality. In xx as22 fact, due to the inherent noise during growth, the growth w2 front of a thin film is statistically rough for the majority of N xx 2 1/2 1 kc 1/2 H1/2 cases. Under different preparation conditions substrate tem- 2d kc perature, pressure, growth rate , or different growth methods physical vapor evaporation, sputtering, chemical vapor Y 1/2 k 2 1/2 kc 1/2 deposition, etc. , one may obtain a wide variety of different c surface morphologies which are inherently related to differ- 1 1 ent growth mechanisms. H 1/2 k Y 1/2 k , 9 So far, there are three kinds of statistical rough surfaces c c obtained in thin-film growth 1 Self-affine surface: This where Hv(x) is the Struve function and Yv(x) is the Neu- kind of surface usually results from the pure noise driven mann function, kc is the upper spatial frequency boundary. mechanism,15 and one needs three parameters to characterize Note that we assume the statistics is enough that it covers the the surface, the interface width w, the lateral correlation entire scaling region. This assumption is made through out length , and the roughness exponent . 2 Mound surface: the whole paper and will not be stated again. Obviously for If the surface has a diffusion barrier or has both smoothening any self-affine surface, the in-plane demagnetizing factor to and roughening mechanisms, then a mound surface is its leading order scales as Nxx(yy) w2 /d . This result is obtained.16­18 For this kind of surface, there are two lateral similar to that obtained in Refs. 11 and 13. Figure 2 shows length scales that characterize the morphology, namely the how the roughness exponent affects the demagnetizing average mound separation and the randomness correlation factor. As increases from 0.001 to 1, the in-plane demag- length .18 3 Anisotropic surface: if the substrate has an netizing factor decreases almost three orders of magnitude. anisotropy, the growth front can be anisotropic. Recently The dependence of the demagnetizing factor on the rough- Zhao et al. showed that when growth starts from a smooth ness exponent can be understood in the following: the substrate, if different growth mechanisms govern different roughness exponent essentially represents how much high growth directions, one could also obtain an anisotropic spatial frequency surface components are included in the sur- surface.19,20 An intuitive question to ask is to what extent a face. As approaches from 1 to 0 from the smooth facet to morphological anisotropy will induce a magnetic anisotropy. the more wiggly local slope variation , more and more high- Since there are two different kinds of anisotropy: lateral frequency components are included in the power spectrum, length anisotropy and scaling anisotropy,20 it is important to which means that the surface has more small features of local investigate how they would affect the demagnetizing factor. variations. This will generate more ``magnetic poles'' on the In the following, we shall consider the effects of these sta- surface parallel to the film plane, and will give rise to a tistical rough surfaces on the demagnetizing factors. stronger in-plane demagnetizing field. In fact, this result is PRB 60 SURFACE/INTERFACE-ROUGHNESS-INDUCED . . . 1219 consistent with Schlo¨mann's derivation that higher fre- quency component will contribute more to the demagnetiz- ing factor.11 Therefore, as decreases, Nxx should increase. For a self-affine growth on flat substrates, the dynamic scaling hypothesis can be assumed which states that w t , and tl/z, with being the growth exponent and z being the dynamic exponent such that z / .15 Thus, for linear growth (d t) we obtain to the leading order the temporal variation of the lateral demagnetizing factor Nxx t2 / 1. For example, if the growth is governed by the surface diffusion, 1 and 1/4 which gives Nxx t 3/4, i.e., with the increase of the growth time the demagnetizing factor caused by surface roughness diminishes. FIG. 3. In-plane demagnetizing factor N B. Isotropic mound surface xx /N0 as a function of the lateral ratio / for a mound surface with a fixed apparent In this case, one has18 lateral correlation length . A 2w2 4 2 k2 2 the early stage equation of growth that describes the mound h k 2 2k , 2 5 2 exp 4 2 2 I0 formation.18 On the other hand, the average local slope w/ 10 may remain unchanged due to the slope selection.16 There- fore, the in-plane demagnetizing factor may increase with where is the average mound separation, is the random- growth time as N ness correlation length, and I xx w2/d e pt/t, or at least maintains a 0(x) is the zeroth-order modi- constant if w t. In addition, as time increases, the ratio fied Bessel function. The in-plane demagnetizing factor can becomes smaller and the dominated correlation length will be written as be the average mound separation . Note, however that, if w2 4 2 k2 2 the interface width grows exponentially, the condition w/d N 2w2k2 2k 1 required for the validity of the in-plane demagnetizing xx 2d 2 exp 4 2 2 I0 dk factor expansion may not be satisfied for any film thickness 4w2 2 2 2 2 d. d 3/2 1 exp 2 M 32 ;1; 2 , 11 C. Anisotropic self-affine surface with the Kummer's function M(p;q;x). In this case, the Correlation length anisotropy: First we consider the lat- apparent lateral correlation length is a function of both eral anisotropy case where one has20 and :18 1/ 2 (1/ 2) ( 2/ 2). If we assume a fixed value of the apparent correlation length , then one can introduce a A 2 x yw2 dummy angle such that 1/ 2 (1/ 2)cos2 , 2/ 2 h k 2 2 5 1 2k2 2k2 1 , 13 (1/ 2)sin2 , and the demagnetizing factor can be rewrit- x x y y ten as with x and y being the correlation lengths in the x and y axes, respectively. From Eq. 13 we obtain the demagnetiz- 8w2 ing factor N 1 xx d 3/2 1 exp 2 M 3 1 2 2 ;1; 2 , 12 2 kc N x yw2 xx d with tan / representing the ratio of the random- d 0 0 ness correlation length to the average mound separation. k2 cos2 Equation 12 clearly states that the demagnetizing factor is 2 2 still inversely proportional to the lateral correlation length, 1 xk2 cos2 yk2 sin2 1 dk. 14 and obeys the relation Nxx w2 /d . Moreover, from Eq. 12 Here we have changed the coordinates to cylindrical. If we we can see that for the mound surface, even for the same consider k lateral correlation, the demagnetizing factor depends also on c and 0.5 1, and consider the integral I(x,y) 2 (x cos2 y sin2 ) 1/2d , we obtain the ratio . 0 Figure 3 shows the demagnetizing factor Nxx /N0 as a x yw2 1/2 function of the ratio for a fixed 10, and N 2N 2N I 2, 2 , 15 0 x xx y yy 4 d x y (8w2/d ) 3/21 . Nxx increases with increasing which means that will contribute significantly to the demagnetiz- ing factor. In general the formation of a mound surface is the N x yw2 1/2 2 2 xx Nyy I x, y . result of the competition between roughening and smoothen- 2 d x y ing growth mechanisms. Eventually, for a long time, the 16 roughening mechanism will dominate, which suggests that Where I(x,y) can be reduced to an elliptic integral. Equa- the interface width w may increase exponentially with time tions 15 and 16 show that the demagnetizing factor has while the film thickness still grows linearly. One example is the same relation for the roughness exponent as for the iso- 1220 ZHAO, PALASANTZAS, WANG, AND DE HOSSON PRB 60 FIG. 4. Log-log plot of the ratio of in-plane demagnetizing fac- tors Nxx /Nyy as a function of the lateral correlation length ratio x / y for a lateral length anisotropic surface. Note that in this case the ratio Nxx /Nyy does not depend on the roughness exponent . FIG. 5. Semilog-log plot of the in-plane demagnetizing factors tropic self-affine surface with the relation Nxx(yy) w2/d still N valid. Figure 4 shows N xx and Nyy as functions of x for a scaling anisotropic surface. xx /Nyy as a function of the ratio Here x y 50 nm and y is fixed for a y 0.5, and b y x / y where the anisotropy of in-plane demagnetizing effect 1.0. appears to rotate by 90° with respect to the surface morphol- ogy anisotropy. The in-plane demagnetizing factor anisot- x y , while for x y we have Nxx Nyy and vice ropy and the lateral length scale anisotropy obey the relation versa. Moreover, Fig. 6 shows how the general anisotropy Nxx /Nyy ( x / y) 1.7. This result implies that the slight an- affects the in-plane demagnetizing factors for x 150 nm, isotropy of surface morphology will amplify the in-plane de- y 50 nm, y 0.5; and x 50 nm, y 150 nm, y 0.5. magnetizing effect. Therefore, such a result indicates that Obviously, for x y , the intersection of Nxx and Nyy surface morphology anisotropy will have a great impact on curves shifts to smaller x , while for x y , the intersec- the anisotropy of the magnetic properties. We demonstrated tion shifts to larger recently that the lateral length anisotropic surface is caused x . by the same growth mechanism but with different strength in IV. MAGNETIC FILMS WITH DOUBLE the x and y directions.19 Therefore, during growth, although ROUGH BOUNDARIES both w and x(y) are functions of growth time, the anisotropy ratio does not change temporally. As a result the anisotropy In this section we concentrate on how the cross correla- of the demagnetizing factor will not change during the depo- tion of the two rough boundaries affects the demagnetizing sition process. factors. For simplicity we will consider only isotropic rough Correlation length and scaling exponent anisotropy: Fi- boundaries. In this case Nxx Nyy , and Nxx can be expressed nally, in the case of the additional scaling anisotropy in as roughness exponents, we have20 A 2 h k 2 x yw2 1/2 x 1/2 y , 17 2 5 1 2 2 2 2 x kx 1/2 x 1 yky 1/2 y with x and y being the roughness exponents along the x and y axes, respectively. Therefore, we obtain 2 kc N x yw2 1/2 x 1/2 y xx d d 0 0 k2 cos2 dk. 1 2 2 x k2 cos2 1/2 x 1 yk2 sin2 1/2 y 18 It was discussed in Ref. 20, for a scaling anisotropy surface, the anisotropy is determined by the lateral correlation lengths x , y , and also the roughness exponents x , y . Figure 5 shows the numerically calculated Nxx and Nyy as functions FIG. 6. Semilog-log plot of in-plane demagnetizing factors N of xx x for fixed y: y 1 and y 0.5. Here x y and Nyy as functions of x for a scaling anisotropic surface. Here 50 nm, w 1.0 nm, and d 40 nm, respectively. The inter- y 0.5 is fixed for a x 3 y 150 nm, and b 3 x y section of Nxx and Nyy curves show that Nxx Nyy only at 150 nm. PRB 60 SURFACE/INTERFACE-ROUGHNESS-INDUCED . . . 1221 FIG. 8. Log-log plot of the Nxx as a function of growth time for FIG. 7. Log-log plot of the demagnetizing factor Nxx as a func- a self-affine rough substrate with 1, w 5, 20 for different F tion of a the lateral correlation length with w 1.0 nm, d and D values: a F 1.0, D 1.0; b F 5.0, D 1.0; and c F 10 nm, and b film thickness d with w 1.0 nm, 20 nm for an 5.0, D 5.0. in-phase cross-correlation and an out-of-phase cross-correlation. Dynamic growth effects: In the following we consider 2 5 N how the dynamic growth will affect the demagnetizing fac- xx 2dA dkk2 h 1 k 2 h 2 k 2 tor. Since the growth starts from a rough substrate initially, the cross-correlation coefficient between the growth front 2e 2dk h 1 k h 2 k . 19 and the substrate is positive, but less than 1, as shown in the Appendix B. Therefore, the ultimate effect of the cross cor- Effects of in-phase and out-of-phase boundaries: First, we relation is to reduce the demagnetizing factor. A simple case consider a simple case in which h1 h2 , i.e., the two is to consider the linear dynamic growth as shown in Eqs. rough boundaries are totally correlated: the positive sign and B8 and B9 in Appendix B, where the in-plane demagne- the negative sign represent the surfaces which are exactly tizing factor can be written as in-phase and exactly out-of-phase, respectively. The in-plane demagnetizing factor can be written as 1 w2 N dkk2 h k xx 2 k 2 1 2e Fkte k2 k4 t c 2k2 1 e 2dk 2Ft N xx 2d 0 1 2k2 1 dk. 20 e2 k2 k4 t 1 In Fig. 7 we plot the demagnetizing factor N e2 k2 k4 t D . 21 xx as functions k2 k4 of both the lateral correlation length with w 1.0 nm, d 10 nm, and film thickness d with w 1 nm, 20 nm for For the linear dynamic growth equation shown in Appendix an in-phase cross correlation and an out-of-phase cross cor- B, d Ft with F being the film growth rate. Here we have relation, respectively. Clearly the demagnetizing factor of the Ft w1(t) in order to satisfy the perturbation condition. in-phase boundaries is less than that of the out-of-phase Equation 21 shows that the substrate effect decreases at boundaries. Nxx still decreases monotonically with increas- least according to t 1, but the effect of the growth front is ing both the lateral correlation length and film thickness d. determined by the growth mechanism. According to the dis- However, for large roughness exponents 1 , the demag- cussion in Sec. III A, for a dynamic scaling growth front, the netizing factor Nxx for the in-phase boundaries is signifi- change of Nxx caused by this front evolved as t2 / 1 cantly smaller than that for the out-of-phase boundaries as where usually 1 / 2 1. Therefore, after a certain increases for a fixed film thickness, see Fig. 7 a or as d time, the change of demagnetizing factor Nxx is dominated decreases for a fixed , see Fig. 7 b . Moreover, both the by the surface roughness contribution. Figure 8 plots the Nxx behaviors of Nxx versus and Nxx versus d for the in-phase as a function of the growth time t for the Mullin's diffusion boundaries obviously deviates from the inversely propor- growth mechanism starting from a rough surface with 1, tional behavior with film thickness. Quantitatively, for the w 5, 20 for different F and D values: a F 1.0, D out-of-phase boundaries Nxx 0.85 and Nxx d 1.1 . As the 1.0; b F 5.0, D 1.0; and c F 5.0, D 5.0. Under value of the roughness exponent decreases, the Nxx vs the same growth mechanism we also plot the Nxx as a func- and Nxx vs d behaviors for both the in-phase and out-of- tion of the growth time t starting from a smooth surface. One phase boundaries becomes similar. Nxx overlap with each can see that as F increases, Nxx decreases, and the difference other for small exponents and the inverse dependence over of Nxx between the rough substrate and smooth substrate the lateral correlation length , Nxx 1, and film thickness becomes large. As D increases, the difference of Nxx be- d, Nxx d 1 recover. tween the rough substrate and smooth substrate becomes 1222 ZHAO, PALASANTZAS, WANG, AND DE HOSSON PRB 60 small, and Nxx increases. Initially, the behavior of Nxx versus netizing factor depends not only on w but also on the film growth time t is greatly influenced by the substrate rough- thickness d, the lateral correlation length , and the rough- ness, as shown in Fig. 8 the Nxx versus time for a rough ness exponent if the surface is self-affine . Take an ex- substrate does not parallel that of a smooth substrate. How- ample of the noise-driven growth discussed in Sec. III A. We ever, after a long time, the surface growth dominates Nxx . know from the dynamic scaling theory that the interface The change of substrate morphology also affects the deter- width grows as w t , where in general 0.15 That is, with mination of the relationship Nxx versus t. If the substrate has the increase of growth time, the surface becomes rougher. a very long correlation length compared to its interface However, from our discussion in Sec. III A, the in-plane de- width, then the substrate almost has no effect on the time magnetizing factor scales with the growth time as N behavior of N xx(yy) xx . The change of the substrate roughness ex- t2 / 1, where the exponent usually is negative. That is, ponent also affects the absolute value of Nxx , but not as the demagnetizing factor decreases with the growth time. dramatic as the affect of the surface roughness exponent as This demonstrates that the increase of surface roughness w discussed in Sec. III A. does not mean the increase of the demagnetizing factor. However, if the growth is in mound formation, as discussed V. DISCUSSIONS OF THE RESULTS IN CONNECTION in Sec. III B, the increase of roughness can indeed increase WITH EXPIREMENTS the demagnetizing factor. Therefore, the relation between the roughness and the demagnetizing factor is more dependent As we discussed above, the demagnetizing field changes on the growth mechanism, or the detailed morphology of the the field strength inside the magnetic material. The magnetic surface; so does the apparent coercivity, given that the film is field inside the material can be written as H Happ Hd a single domain. Nonetheless, how does the detailed mor- Happ N *M. For an isotropic surface, one can prove that phology of the surface affects the apparent coercivity was the nondiagonal components Nxy , Nyz , and Nzx of the de- not considered in most experiments.6­8,10 Only a few experi- magnetizing tensor N are equal to zero. The actual demag- ments relate the change in coercivity to the change of the netizing field depends on the diagonal components Nxx , interface width w. Recently, some detailed works have been N performed.5,9 yy , and Nzz of N , that is Experiments of Co films: In the following we shall discuss H the connection of our theory and our experimental work. x Happ,x NxxM x0 , Hy Happ,y NyyMy0 ; First, we examine the single rough boundary. For Co ultra- H thin films deposited on atomically smooth Cu substrate, z Happ,z NzzM z0 Happ,z 1 Nxx Nyy M z0 . 22 Jiang et al. found that the apparent coercivity increases when the Co thickness increases from 1 to about 7 ML, and de- Therefore, if the in-plane demagnetizing factor increases, in creases slightly when Co grows beyond 7 ML thick.5 In ad- order to achieve the same magnetic field inside the material, dition, they measured the detail surface morphology param- one needs to increase the applied field. In the same time, the eters Table I in Ref. 5a using high-resolution low-energy out-of-plane demagnetizing field will decrease, which results electron diffraction. One thing quite obvious is that the in- in the decrease of the applied out-of-plane field. Immedi- terface width w almost does not change for the thickness ately, one can connect this with the coercivity measurement measured, but both the lateral correlation length and the of rough thin magnetic films. If we assume that during the roughness exponent have more dramatic change. From 3 to thin-film formation, the film remains as a single domain 25 ML, decreases from 0.95 to 0.54, and decreases from structure and the magnetic energy is dominated by the 285 to 94 Å. They used Schlo¨mann's theory to estimate the magnet-static energy, then the actual coercivity of the film is in-plane demagnetizing factor, and concluded that the de- fixed. Under this assumption, if the roughness of the film is magnetizing factor decreases as the film thickness increases.5 changed, then the applied field corresponding to the real co- However, the absolute value of the demagnetizing factor is ercivity field also changes. According to Eq. 22 , for the quite small, which cannot contribute to the change of the in-plane coercivity measurement, the apparent coercivity will coercivity. Using roughness data in Table I of Ref. 5 a , we change linearly with the in-plane demagnetizing factor. In calculated the demagnetizing factor of the ultrathin Co film other words, if the change of apparent coercivity has no re- as a function of the thickness using Eq. 6 and the result is lation to the demagnetizing factor, then the magnetization plotted in Fig. 9. Except for thickness d 10 ML, the demag- mechanism of the thin film may be different, i.e., the as- netizing factor increases with the film thickness. The slow sumptions for a single domain and a dominate magnet-static change of the demagnetizing factor in the small thickness energy are broken. regime is probably due to that the small roughness approxi- Connections with experiment: Experimentally the general mation (w d) does not apply here. This trend of the demag- trends for the magnetic thin films are that the apparent coer- netizing factor as a function of film thickness is opposite to civity increases with film roughness,5­10 which seems to the behavior of the apparent coercivity, which suggests that agree with our above simple argument. In fact, the situation the magnet-static energy may not play an important role in is more complicated: the increase of surface roughness does these ultrathin films. In fact, the absolute value of the demag- not guarantee the increase of the demagnetizing factor. In netizing factor is also quite small, which supports this point. general one tends to use the interface width root-mean Finally, we discuss the double rough boundary case. A square roughness w to measure how rough the surface is: if detailed experiment of Co films deposited on plasma etched w is large, then the surface is rougher. However, through the Si 100 substrate was performed by Li et al.9 The substrates discussion in Sec. III, we have demonstrated that the demag- were first plasma etched for various time periods, then about PRB 60 SURFACE/INTERFACE-ROUGHNESS-INDUCED . . . 1223 FIG. 9. The in-plane demagnetizing factor of the Co film as a FIG. 10. Log-log plot of the in-plane demagnetizing factor as a function of thickness calculated using Eq. 6 from the data in Table function of the substrate etching time for the Co film using Eq. 20 I of Ref. 5 a . and Schlo¨mann's approximation. 970 Å Co films were deposited simultaneously on those sub- the average mound separation and the randomness correla- strates. Both substrate morphology before film deposition and the Co film morphology after deposition were measured tion length. by atomic-force microscopy, and the roughness parameters c An anisotropic surface morphology can induce aniso- analyzed from height-height correlation functions were tropic in-plane demagnetizing factors in such a manner that found to be correlated in-phase approximately. Therefore, the demagnetizing anisotropy can be magnified drastically by Eq. 20 can be applied for this case. In Fig. 10 we plot the morphological anisotropy. More precisely, the ratio of lateral demagnetizing factors of the Co film as a function of the demagnetizing factors Nxx /Nyy as a function of the lateral substrate etching time using both Eq. 20 and Schlo¨mann's correlation length ratio x / y appeared to rotate by 90 de- approximation. Clearly Schlo¨mann's approximation gives a grees with respect to surface morphology anisotropy. The much larger demagnetizing factor and the demagnetizing in-plane demagnetizing factor anisotropy and the lateral factor increases with the etching time. However, Eq. 20 length scale anisotropy were found to obey the relation shows that the demagnetizing factor increases from 1 to 20 Nxx /Nyy ( x / y) 1.7, implying that the slight anisotropy of min, then decreases after that. In fact, for t 30 min, where surface morphology will be enlarged in the in-plane demag- the interface width w 440 Å, the small roughness perturba- netizing effect. tion is not valid because w is comparable to the thickness d. d Finally, we considered the case of films with two Going back to Eq. 5 , we can actually expect a smaller N rough boundaries where we investigated how the cross cor- than the value calculated in Fig. 10. This result is consistent relation of the two rough boundaries affects the in-plane de- with the measured apparent coercivity, which shows a maxi- magnetizing factors. The demagnetizing factor of in-phase mum around 20 min, and then decreases later in Ref. 9 b . boundaries is less than that of the out-of-phase boundaries. However, the roughness-dependent demagnetizing factor The thickness and correlation length dependence of the lat- cannot explain the behavior after 40 min. eral demagnetizing factor Nxx(yy) depends strongly on the corresponding roughness exponent. Indeed, for large rough- ness exponents 1 , the demagnetizing factor Nxx(yy) for VI. CONCLUSIONS the in-phase boundaries is significantly smaller than that for the out-of-phase boundaries. Moreover, the behavior of In conclusion, we studied in detail the influence of Nxx(yy) versus d and for the in-phase boundaries obviously surface/interface roughness on the demagnetizing factor of a deviates from the inversely proportional behavior with film thin magnetic film for a wide range of rough morphologies. thickness, instead N Moreover, the formalism was extended to account for films xx(yy) varies as Nxx d 1.1, and Nxx 0.85. However, as roughness exponent decreases, the with film/substrate and film/vacuum rough interfaces by tak- N ing into account interface cross correlation effects. The fol- xx(yy) vs d and behaviors for both in-phase and out-of- phase boundaries become similar and overlap with each lowing concludes our findings. other and the inverse dependence over d and resumes. Con- a For a film with a single self-affine rough boundary, the nections with thin-film growth mechanisms were also ex- in-plane demagnetizing factor Nxx(yy) is proportional to the plored and strongly influence roughness induced demagne- interface width w square and to leading order is inversely tizing effect. proportional to the lateral correlation length . The roughness exponent is also shown to greatly affect Nxx(yy) in such a manner that the demagnetizing factor can increase two orders ACKNOWLEDGMENTS of magnitude as decreases from 1 to 0. b For a film with a single mound boundary, Nxx(yy) is This work was supported by the NSF. G.P. would like to inversely proportional to the apparent correlation length, and acknowledge support from the Netherlands Institute for also depends on the ratio of the two different lateral lengths: Metal Research. 1224 ZHAO, PALASANTZAS, WANG, AND DE HOSSON PRB 60 APPENDIX A Calculation of demagnetizing factors: In this section we explain briefly the algebra that leads to the final expressions for the demagnetizing factors; in-plane and out-of-plane. From Eqs. 1 and 2 we obtain the self-energy 1 d d W M x 2 M0x dr dz x z 2 h2 r z 2 h1 r 1 d d h1 h2 2 M0x dr M r, 2 h1 x M r, 2 h2 x 1 h 2 / x M 0x h2 / y M 0y M 0z h1 2 M0x dr dr h1 / x M0x h1 / y M0y M0z r r 2 h x 1 r h1 r 2 r r 2 d h1 r h2 r 2 1 h 2 / x M 0x h2 / y M 0y M 0z h2 2 M0x dr dr h1 / x M0x h1 / y M0y M0z r r 2 h x . 2 r h1 r d 2 r r 2 h2 r h2 r 2 A1 The expression for Wy is similar to Wx . For Wz , we have 1 d/2 h1 1 d d W M z 2 M0z dr dz dr M r, M r, d/2 h z 2 M0z 2 h1 2 h2 2 1 h/ x M 0x h1 / y M 0y M 0z 2 M0z dr dr h2 / x M0x h2 / y M0y M0z r r 2 d h 1 r h2 r 2 r r 2 h1 r h1 r 2 1 h 2 / x M 0x h2 / y M 0y M 0z 2 M0z dr dr h1 / x M0x h1 / y M0y M0z . A2 r r 2 h 2 r h1 r d 2 r r 2 h2 r h2 r 2 Since W (4 dA/2)M*N *M,11 where A is the average flat surface area, we have 1 h N 2 / x h1 / x xx 4 dA dr dr h1 / x h1 / x r r 2 h 1 r h1 r 2 r r 2 d h1 r h2 r 2 h h 1 / x h2 / x 2 / x h2 / x , A3 r r 2 h 2 r h1 r d 2 r r 2 h2 r h2 r 2 1 1 N zz 2 dA dr dr 1 r r 2 h 1 r h1 r 2 r r 2 d h1 r h2 r 2 1 1 . A4 r r 2 h 2 r h1 r d 2 r r 2 h2 r h2 r 2 Here Nxx , Nyy , and Nzz are the diagonal components of the demagnetizing tensor N . A similar calculation can be applied to the nondiagonal components Nxy , Nyz , and Nzx . If we assume the surface roughness w is much smaller than the film thickness d(w d), then the roughness can be treated as a small perturbation. In this limit the in-plane demagnetizing factor Nxx can be approximated as 1 2 h N 2 / x h1 / x xx 4 dA dr dr h1 / x h1 / x h2 / x h2 / x r r . A5 r r 2 d2 Upon substitution of the Fourier transforms from Eqs. 2 ­ 4 we obtain PRB 60 SURFACE/INTERFACE-ROUGHNESS-INDUCED . . . 1225 h hi i x x k2 dr dr x r r dr dr dk dk r r h i k h i k exp ik*r ik *r 2 4 k2 x A dr dr dk r r h i k 2 exp ik* r r k2 2 5 dk xk h i k 2 A6 and h 2 4 k2 h dr dr 1 / x h2 / x x 1 k h 2 k exp ik* r r dr dr dk r r 2 d2 A r r 2 d2 RJ 2 5 dkk2 0 kR x h 1 k h 2 k dR 0 R2 d2 k2 2 5 dk xk e dk h 1 k h 2 k . A7 Substituting Eq. A6 and Eq. A7 into Eq. A5 , we obtain In addition, the expression for Eq. 6 . A APPENDIX B h 1 k,t h 1 k ,t e2L k t h 2 5 2 k 2 In this appendix we consider a general case of the cross correlation between h1 and h2 due to dynamic roughening. e2L k t 1 We assume h D k k . B5 2 to be the initial height in a rough substrate and h L k 1 the growth front following certain growth mecha- nisms. The simplest case is to assume that the growth mecha- For simplicity, we adapt the linear model discussed in Ref. nism is linear. Then the equation of the growth front rough- 18, the linear operator L has the form ening can be written as L 2 4, B6 h1 t Lh1 , B1 or alternatively in k space L k k2 k4. B7 where L is the linear operator, and is Gaussian white noise, satisfying the relations is proportional to the surface diffusion coefficient. For we have the case of stable growth noise induced roughen- ing with proportional to the surface tension coefficient. r,t 0, For we have the case of unstable growth unstable mound formation due to the diffusion Schwoebel barrier. r1 ,t1 r2 ,t2 2D r1 r2 t1 t2 . B2 Therefore, we obtain A Performing a Fourier transform of Eq. B1 , one obtains the h 1 k,t h 2 k solution for h 2 5 e k2 k4 t h 2 k 2 k k , 1 in Fourier space18 B8 and t h 1 k,t k,t eL k t t dt h 2 k eL k t. B3 0 A h 1 k,t h 1 k ,t e2 k2 k4 t h 2 k 2 Since h 2 5 2(k) (k ,t) 0, the cross correlation in k space can be written as e2 k2 xk4 t 1 D k k . A k2 k4 h 1 k,t h 2 k 2 5 eL k t h 2 k 2 k k . 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