PHYSICAL REVIEW B VOLUME 59, NUMBER 18 1 MAY 1999-II Theory of roughness-induced anisotropy in ferromagnetic films: The dipolar mechanism Rodrigo Arias and D. L. Mills Department of Physics and Astronomy, University of California, Irvine, California, 92697 Received 7 January 1999 When ferromagnetic films are grown on stepped surfaces, or rough surfaces upon which there is a preferred direction, additional magnetic anisotropy associated with the presence of the roughness is found in experi- ments. This paper presents the theory of the contribution of one mechanism to this anisotropy, that is associ- ated with the roughness-induced increase in magnetic dipolar energy. When the film surface profiles are modulated, the magnetization of the film fluctuates in direction, thus generating stray dipolar fields. The energy stored in such fields depends on the angle between the mean magnetization, and the preferred axis of the modulated surface profile. We present explicit calculations for various models of films on stepped surfaces. S0163-1829 99 00218-0 I. INTRODUCTION thin film with perfectly flat surfaces, in the absence of per- pendicular anisotropy, the magnetization will be constant in There is currently great interest in the properties of very magnitude and direction, and parallel to the film surfaces. If thin ferromagnetic films, possibly incorporated into magnetic the surface profiles are modulated, the direction of the mag- multilayers or superlattice structures. Applications of such netization within the film will wander. A consequence is that materials to magnetic recording have been realized recently, fields of dipolar character are generated both by the effective and such structures may possibly lead to a generation of volume magnetic charge density *M , and also surface magnetic memories. More generally, we have here a fasci- charges. There is an energy density associated with these nating class of magnetic materials, with unique properties fields that is a function of the angle between the average which range from the well-known giant magnetoresistance, magnetization, and the preferred direction associated with to magnetic phase diagrams that are rich and subject to con- the modulated surface profile. Thus, we have a contribution trol through variations in the microstructure of the multilay- to the anisotropy energy from this dipolar mechanism. To ers. initiate such studies, we consider the simplest physical pic- Most theoretical studies explore the properties of ideal- ture. We have a ferromagnetic film placed in an external dc ized films, whose surfaces are perfectly smooth and flat. In magnetic field H fact, the ultrathin films of current interest are grown on sub- 0 parallel to its nominal surface. We con- sider the Zeeman, dipolar, and exchange energies in our strates which themselves are not smooth. Even the highest analysis. quality substrate has steps, for example. One thus must in- In Sec. II, we derive the roughness-induced anisotropy quire about the influence of steps, or more generally, of sur- energy, for the case where the amplitude of the roughness face roughness, on the magnetic properties of such films. We may be assumed to be small. In Sec. III we consider a simple note that Slonczewski1 has argued that the presence of steps example: a semi infinite medium with a surface roughness plays a critical role in the biquadratic coupling found in corresponding to a single Fourier component. Section IV Fe/Cr multilayers. It appears to be the case that this mecha- presents a sequence of numerical studies, and Sec. V con- nism indeed dominates, for some samples.2 cluding remarks. In the recent literature, attention has been directed toward experimental studies of roughness-induced anisotropy, for ultrathin films grown on surfaces whose profile has been II. THEORETICAL DISCUSSION modulated in a unidirectional manner; a stepped substrate The geometry we consider is illustrated in Fig. 1. We provides an example of such a surface.3­8 Several mecha- have a ferromagnetic film, of nominal thickness D. By this nisms have been invoked to explain such data. For example, we mean we have an upper surface given by y D/2 magnetic ions which reside very close to a step reside in sites (x,z) where the average of (x,z) over the entire sur- of low symmetry, and thus experience anisotropy whose face is zero, i.e., 0. Similarly, the lowest surface is character and strength differs from ions which sit on a flat y D/2 (x,z), where 0. Thus, y D/2 is the region. Such ions transmit information about the anisotropy nominal upper surface, y D/2 the nominal lower surface, they experience by virtue of their exchange coupling to the and D is the average thickness of the film. magnetic species elsewhere in the film. Also, the presence of steps will lead to strain within the magnetic film, and through An external dc magnetic field H 0 is applied parallel to the magnetoelastic coupling this can generate magnetic anisotro- z axis, located in the plane y 0, parallel to the two nominal pies in the film. surfaces. If (x,z) (x,z) 0 everywhere, then the In this paper we present the theory of a third contribution magnetization M 0 will lie in plane, uniform in magnitude to the anisotropy, and evaluate its magnitude for several and direction, and also parallel to H 0. We neglect anisot- models of thin films with modulated surface profiles. In a ropy perpendicular to the x z plane, save for the dipolar 0163-1829/99/59 18 /11871 11 /$15.00 PRB 59 11 871 ©1999 The American Physical Society 11 872 RODRIGO ARIAS AND D. L. MILLS PRB 59 the second is the dipolar energy, and the third is the ex- change energy. The integrals are over the actual volume of the rough film. We ignore surface anisotropy, so when we generate expressions for mx(x ) and my(x ), we use the boundary conditions n * mx 0, and n * my 0 on each surface, where n is a normal to the surface. Since, in fact, mx and my are first order in the roughness amplitude, so long as we require these quantities only to lowest order, we may replace the exact boundary conditions by the four statements mx,y y 0. 3 y D/2 FIG. 1. Schematics of a rough film, with the applied field H 0 H0z in the plane of the nominal film of thickness D. The ampli- We write the total energy of the system as tudes , (x,z) describe the upper and lower surfaces roughness. E H0M0V E, 4 anisotropy built into our analysis. If either (x,z) or (x,z) are nonzero, or both are nonzero, there will be spa- where H0M0V is the Zeeman energy of the uniformly tial variations of the magnetization direction in the film, thus magnetized film, and E is the change induced by the pres- ence of the roughness. For the Zeeman term, M 0 becomes a function of position, M 0(x ). These spatial variations will clearly increase the Zeeman and exchange H energies. In addition, there will be dipolar fields generated by E 0 2 2 Z 2M d3x mx my , 5 the effective magnetic charge density 0 V¯ M(x ) * M 0(x ) within the film, and also by magnetic surface charges with where, since mx and my are first order in the roughness, the origin in those areas of the surface where M 0(x ) has a non- integral is confined to the volume V¯ between the nominal zero perpendicular component. These dipolar fields increase surfaces at D/2. We write for the change in dipolar energy, the energy of the system as well. If the surface roughness has with H a directional character, say the film is grown on a stepped d(x ) hx(x )x hy(x )y hz(x )z , surface, then this energy will clearly depend on the angle M0 1 between the preferred direction, and that of the applied mag- E D 2 d3xhz x d3x mxhx myhy . 6 V 2 V¯ netic field H 0. In this section, we obtain expressions for these roughness- In the second term in this expression, both mx and hx are first induced energy changes, within the framework of a pertur- order in the roughness amplitude, so we may integrate only bation theoretic scheme. We assume the deviation in the over the ``nominal film'' volume V¯. In the first term, since hz magnetization M 0(r ) from the nominal value M 0 M0z is is first order, we must take due account of the actual rough- small, and may be calculated to first order in and . ened surfaces, as we shall see shortly. With this information in hand, we may calculate the energy In regard to the exchange energy, note that change of the system to second order in these quantities. This section is devoted to the basic formulation of the theory, and subsequent sections to applications. d3x M x 2 d3x * M M M 2M V V In the presence of roughness, we write the magnetization in the form d3xM 2M dSM n * M 1 V S M x M 2 2 0 2M mx x my x z mx x x my x y , 0 1 d3xM 2M . 7 V where the quantities mx(x ) and my(x ) are of first order in the The integral over the film surfaces vanishes to all orders in amplitudes of the roughness. The energy of the system is the roughness, by virtue of our boundary condition n * M then 0 on the surface. We then have, with 1 E H 0 d3xMz x 2A 2 d3xH d x *M x h(ex) x,y 2m M2 x,y , 8 0 A d3x M M2 2. 2 1 A 0 E (ex) (ex) 2 2 ex 2 d3x mxhx myhy 2 m m , 2 x y In these expressions, H V¯ M0 d(x ) is the dipolar field generated by nonuniformities in the magnetization, produced by the 9 roughness. The first term in Eq. 2 is the Zeeman energy, where the last term has its origin in Mz 2Mz . We have PRB 59 THEORY OF ROUGHNESS-INDUCED ANISOTROPY IN . . . 11 873 d3x 2 m2 2 x,y dSn * mx,y 2 dSmx,yn * mx,y 0 V¯ S¯ S¯ 10 by virtue of the boundary condition. Hence, the last term vanishes. Thus, 1 E (ex) (ex) ex 2 d3x mxhx myhy . 11 V¯ Relations between various contributions noted above follow when one realizes that for the system to be in equilibrium, we must have the zero torque condition FIG. 2. A path of integration in the rough surface region. M x H eff 0, 12 to a finite domain of the variable z, from L/2 to L/2. In the where M (x ) is given in Eq. 1 , and the effective field sensed end we let L . But for finite L, 0 as z , and z by the magnetization is the sum of the Zeeman field, the . Now suppose has the maximum positive value dipole field, and the exchange field. The contributions to the M , and a maximum negative value m . Similarly, local torque to first order in the roughness amplitude read varies from m to M . Then in the region (D/2) m y (D/2) m . There is zero contribution to the inte- M x H (ex) gral, since we integrate continuously from to , and eff x H0my M 0 hy hy vanishes at the two limits for finite L. y M (ex) 0 hx h We then concentrate on the regime where y lies between x H0mx . 13 (D/2) , and (D/2) , and similarly for the lower sur- We then require the relation M m face. We examine this contribution with the aid of Fig. 2. For M fixed x and y, we imagine the contribution by integrating in z m 0 (ex) x,y along the solid line in the figure. We have contributions from H hx,y hx,y . 14 0 only those positions of the solid line which lie within the Then notice film. We focus attention on the contribution from the particu- lar line segment AB. The contribution to the integral in Eq. M 18 from this segment is (x,y,z E 0 B) (x,y ,zA) dxdy . D Eex 2 d3xhz x V Now let dS be an element of vector surface area on the real film, using the usual convention that dS points outward from 1 (B) (ex) (ex) the volume bounded. Then at point B, dS dxdy, while at 2 mx hx hx my hy hy z V¯ point A, dS(A) z dxdy . Hence the contribution to the inte- (B) M H gral from this line segment is (x,y,zB)dSz 0 0 2 2 (A) 2 d3xhz x d3x mx my . (x,y,zA)dSz . From this argument, one concludes that V 2M0 V¯ Eq. 18 may be written as 15 Notice that the second term precisely cancels the Zeeman M0 energy, so that we have quite simply E 2 dSz , 19 S M E 0 where the integral is over both surfaces of the film, upper and 2 d3xhz x , 16 V lower. Indeed the result of Eq. 19 follows directly from Eq. where as we have emphasized earlier, the integration on the 18 by use of a general version of the divergence theorem: right-hand side of Eq. 16 is over the volume of the real dV / xi dSi . film, with its rough surfaces. If the surfaces of the film are perfectly flat, dSz 0, and Now in the magnetostatic approximation, E 0. The quantity dSz is thus nonzero only when (x,z) and/or (x,z) are nonzero. To lowest order in the rough- ness, it is an elementary exercise to show that on the upper or hz x z 17 lower surface with the magnetic potential, a quantity first order in the , roughness amplitude, to leading order. Thus, dS , z dxdz z . 20 M E 0 2 When this is inserted into Eq. 19 , we may simply calculate V z dzdy dx. 18 the magnetic potential to first order in the roughness am- To evaluate the integration in Eq. 18 , we must consider plitude, evaluate it on the nominal surfaces y D/2, and various regions. Suppose first that the roughness is confined integrate over x and z. 11 874 RODRIGO ARIAS AND D. L. MILLS PRB 59 We thus arrive at a remarkably simple expression for the We thus have fringing fields outside the film generat- total change in energy of the system, to second order in the ed from a magnetic potential with the spatial variation roughness amplitude: exp(iQ * )exp( Qy) above the film, and exp(iQ * )exp(Qy) below the film. M E 0 Once we find the most general solution of the magnetic 2 dxdz x,D/2,z z x,z potential and m inside the film, we must match these to the magnetic field outside the film, through appropriate bound- x, D/2,z ary conditions. Four boundary conditions are stated already z x,z . 21 in Eq. 3 . In addition, we must insure continuity of tangen- We shall Fourier transform the various quantities which en- tial components of h , and the normal component of B across ter the above expression. For example, if lies in the xz the actual surface of the film. It is well known that conser- plane, and Q Q vation of tangential h is assured if the magnetic potentials xx Qzz , we write inside and outside the film are continuous. The magnetic potentials should be matched across the actual rough sur- x,y,z Q ;y eiQ * , 22 faces. But is nonzero only by virtue of the roughness. If Qx ,Qz we are interested only in the contribution to first order in and similarly for , (x,z). Then the change in energy per and , it suffices to match the magnetic potentials inside surface area is and outside the film at the nominal surfaces y D/2. E iM The requirement that normal components of b be con- 0 served requires a bit of discussion. Consider for the moment A 2 Qz Q * Q ;D/2 Qx ,Qz the upper surface, and let n be the unit normal, erected at a point. We have, in the coordinate system of Fig. 1 Q * Q ; D/2 , 23 where A is the quantization area , ( Q ) , (Q )* . 1 n y Our task is now to find the magnetic potential (x ), for the 1 / x 2 / z 2 1/2 x x z z film with rough surfaces. For this purpose, we consider sinu- soidally modulated surfaces for which y x x z z , 29 , x,z , Q eiQ * c.c. 24 where the last expression is to lowest order in . Then just c.c. represents complex conjugate . If b (x ) h (x ) inside the film, if only first order terms are retained, 4 m (x ) (x ) 4 m (x ), we require *b (x ) 0 everywhere within the film, or when we Fourier transform all n *b quantities, y 4 M0 z 4 my, 30 where the magnetic potential is evaluated at the nominal 2 m boundary y D/2. If the magnetic potential above the film Q2 Q ,y 4 iQ y Q ,y y2 xmx Q ,y 4 y 0. is , then we conserve normal b to first order by requiring 25 Here Q2 Q2 2 4 m x Qz . Two additional relations follow from y y y D/2 4 M 0 z . Eq. 14 . These take the form D/2 D/2 31 2A 2 A similar statement applies at the lower surface. M 0 y Q ,y H0 M Q2 my Q ,y 0 We thus have a total of eight boundary conditions. Six of 0 y2 26 these, the four exchange boundary conditions, and the re- quirement that be continuous across each surface, are ho- and mogeneous equations. In contrast, Eq. 31 and its analog on the lower surface are inhomogeneous equations. These allow 2A 2 us to obtain all quantities, to first order in and . iQ xM 0 Q ,y H0 M Q2 mx Q ,y 0. We can now make one general observation, before pro- 0 y2 27 ceeding with an explicit calculation. Suppose, say, the sur- face has steps of linear grooves, parallel to the applied mag- The three statements in Eq. 25 , and Eqs. 26 and 27 , netic field H allow us to determine the most general form of , m 0 and nominal magnetization M 0. Then and x and will depend only on x, so the right-hand side of Eq. 31 my within the film. Outside the film, of course, mx my 0, and its partner on the lower surface vanish. All eight bound- and also 2 0, or ary conditions then become homogeneous equations. In this circumstance m 2 x my 0. The nominal magnetization Q2 Q ,y 0. 28 M 0 must be aligned so it has a nonzero projection along the y2 line perpendicular to the steps or grooves for stray fields to PRB 59 THEORY OF ROUGHNESS-INDUCED ANISOTROPY IN . . . 11 875 be generated. Since these stray fields necessarily increase the vectors Q QN , we must incorporate exchange, to obtain an energy of the system, it follows that so far as the dipolar accurate description. We have, changing notation, mechanism is concerned, the easy direction will always be parallel to edges of steps or grooves. Q2 2 N H0 The structure of the analysis is now complete. We pro- 1 Q2 2 4 M , 36 ceed by seeking solutions for m 0 x , my , and within the film, through use of Eqs. 25 ­ 27 . We seek solutions with and , mx , and my each proportional to exp( y). There are then three values found for 2. The first, 2 Q2 2 1 we may call a 2 N B0 2Qz pure exchange root: 2,3 Q2 4 4 M 1 1 8 4 M0 . 0 B0 Q2N M 37 2 0 1 Q2 2A H0. 32 In the next section, we explore two simple limiting ex- amples. When 2 21 , 0 so no magnetic fields are generated by the spins. We have my (iQx / )mx ( is now consid- III. AN EXAMINATION OF A SIMPLE LIMITING CASE ered as positive , for this case. Then We have carried out a series of numerical studies of step- M induced anisotropy, through use of the theory developed in 2 0 2 2 2,3 Q2 4A B0 B0 32 AQz , 33 Sec. II. These results will be presented in Sec. IV. Before we turn to these, it is useful to explore a simple limiting case, where B where analytic expressions for various quantities which enter 0 H0 4 M 0. The most general solution for the magnetic potential in the film is then the theory may be obtained. This provides one with insight into the role of the various interactions contained within it. 3 Consider a semi-infinite ferromagnet, which resides in the Q ,y ( ) ( ) i e iy i e iy , 34 lower half space y 0. The external dc magnetic field H i 1 0 is parallel to the z direction, so if the surface is perfectly and expressions for mx and my follow from Eqs. 26 and smooth the magnetization M 0 is constant in direction every- 27 . Above the film, we have (Q ,y) exp( Qy), and where, and parallel to H 0. We have M 0 M0z . Now suppose below we have (Q ,y) exp(Qy). The eight constants (x,z) depends only on z, so if we imagine the surface , , ( ) ( ) i , and i follow from submitting the so- contains steps, M 0 and H 0 are perpendicular to the step lution just described to the boundary conditions, which we edges. Here, we confine our attention to the response of the see are in the form of eight inhomogeneous equations. system to a single Fourier component in the modulated sur- We inquire into the role of the exchange in what follows. face profile, and furthermore the profile is ``perpendicular'' The limit A 0 describes the limit where the exchange is to the applied magnetic field, so we let (x,z) ignored, and only Zeeman and dipolar energies enter. As A 0 exp(iQz) c.c. If desired, Eq. 23 applied to this cir- 0, the roots 2 2 cumstance may be used to synthesize an expression for E 1 and 2 both approach infinity, and their contribution becomes vanishingly small. Then for actual profiles. One sees easily that for this case, mx 0 so only and H m lim 2 0 2 2 y are nonzero. If we seek solutions where both my and 3 Qz Qx. 35 have the spatial variation exp(iQz y), then we find two A 0 B0 roots for . We refer to these as 0 and x , respectively, for The ``dipole only'' problem may be addressed by setting reasons that will be clear shortly. One has 1 and 2 aside, including only the terms exp( 3y) in the analysis, and employing only the boundary conditions which M 2 0 2 0 Q2 32 AQ2 38 describe the conservation of tangential h and normal b . 4A B0 B0 We conclude by arranging some results above in a form and where various limiting behaviors may be perceived more readily. In the ferromagnet, a fundamental length is LN , the M0 width of a domain wall of Ne´el character. In zero external 2 2 x Q2 4A B0 B0 32 AQ2 . 39 magnetic field, in our notation, L 2 N (A/4 M 0)1/2. We intro- duce the wave vector Q 2 N 1/LN (4 M 0/A)1/2. When we Various limiting behaviors of 0 and x are of interest. First, are considering spatial modulations whose length scale is suppose we ignore the influence of exchange, and we con- very long compared to LN , we expect exchange to be quite struct a theory where only Zeeman and dipolar energies en- unimportant in describing the spatial modulation in the mag- ter. We may do this by taking the limit A 0 in all quanti- netization, and the ``dipole only'' theory should suffice. We ties. When we do this, as A 0, 2x M0B0 /2A . We are then in the regime where the important wave vectors Q shall see below that in this limit, the root x vanishes from are small compared to QN . We shall see, however, that this the problem. This is thus an ``exchange root,'' that enters the expectation is only correct when the applied field is not too analysis by virtue of the presence of exchange. Only the weak. Conversely, when we examine the response to wave ``dipole root'' 0 remains. One has 11 876 RODRIGO ARIAS AND D. L. MILLS PRB 59 That is, we overlook the fact that the magnetization direction lim 1/2 0 H0 Q, 40 varies in the substrate, after the surface profile is modulated. A 0 B0 Then stray magnetic fields are generated only by the mag- a special case of Eq. 35 . Here we have Q netic charges on the surface. If we refer to the magnetic x 0, Qz Q. Notice that in zero external magnetic field, H potential in this picture as (0)(y,z), and the energy change 0 0, in fact 0 vanishes. If we ignore exchange, and retain only the dipolar as E(0), then for all values of y we have and Zeeman energies, then in weak applied magnetic fields, the disturbance produced by modulating the surface profile (0) y,z i2 M0 0 exp Q y exp iQz , 48 penetrates very deep into the material. and one finds However, when exchange is present, as H0 0, in fact 0 remains quite well behaved and finite. One has E(0) 2Q 0 2. 49 M2 AQ2 2AQ2 A 2 M0 lim 2 0 1/2 0 1 1 . 41 2 2 Of interest is the ratio E/ E(0); this provides us with the H A M M 0 0 0 0 error we make if we assume simply that the magnetization is Thus, in weak applied magnetic fields, exchange enters criti- fixed rigidly, with fields generated by the surface magnetic cally in the discussion of the response of the system to poles. We have modulations in the surface profile. Now suppose we consider the limit Q , or in the lan- E 2 0 x 0 x guage used at the end of Sec. II, the regime Q Q . 50 N . One sees easily that E(0) 0 Q x Q 0 x Q lim On physical grounds, it is the case always that E/ E(0) 0 lim x Q. 42 Q Q 1. That is, the ``rigid magnetization'' picture always over- estimates the step-induced anisotropy. This follows because We shall see implications of these limiting behaviors shortly. E is the change in energy produced by a magnetization It is a straightforward matter to find expressions for distribution that minimizes the total energy of the system. and my , regarding each as superpositions of exp( 0y) and Hence, E E(0) always. Let us suppose that we ignore exp( xy), in the substrate. We have the influence of the exchange by allowing A to vanish. Then as we have seen, m x , and if we refer to the energy y y ,z my Q,y eiQz c.c., 43 change in this case as E(DIP), we have and similarly for (y,z). One finds, for y 0, E(DIP) 2 0 m 0 Q x Q y Q,y iM 0 0 E(0) 0 Q , 51 0 x Q x 0 or with 0 (H0 /B0)1/2Q in this limit, we have xe 0y 0e xy , 44 and if we refer to the potential in the medium as (Q,y), E(DIP) 2H1/2 0 then . 52 E(0) B1/2 1/2 0 H0 4 iM Q,y 0 0 0 x In the weak-field regime, H0 4 M0, we have a strongly 0 x 0 x Q field-dependent step-induced anisotropy energy. When H0 4 M0 , E(DIP) does approach the rigid magnetization x Q limit. e 0y 0 Q e xy . 45 0 Q x Q Now we explore the full theory, with exchange included. Outside the material, in the region y 0, We then encounter a characteristic length scale, the width LN of the Ne´el wall, discussion in Sec. II. We have the charac- i4 M teristic wave vector Q 2/A)1/2. When Q Q Q,y 0 0 x 0 x 0 exp N (4 M 0 N , we Qy . are considering a surface with features on the length scale 0 x Q 0 Q x Q 46 small compared to LN , and when Q QN , the length scale is Notice that in all the expressions above, if we ignore ex- very long compared to QN . change by taking the limit A 0, When Q QN , we have seen above that 0 x Q. In x indeed drops out of all the expressions, and only the ``dipole root'' remains. this regime, we find Finally, the energy change per unit area E/A produced by modulating the surface profile is E 1. 53 E(0) E 4 M20Q 0 2 0 x 0 x A . 47 As one would expect, the magnetization cannot follow fea- 0 Q x Q 0 x Q tures on such a small length scale, and the rigid magnetiza- Suppose now that we keep the magnetization pinned rigidly tion picture works well. in place, in the presence of the modulated surface profile. The regime Q QN is a bit more complex. In this limit PRB 59 THEORY OF ROUGHNESS-INDUCED ANISOTROPY IN . . . 11 877 Q N B0 1/2 x , 54 2 4 M0 while 0 is well approximated by 2Q2 1/2 0 Q H0 B 2 4 M0 . 55 0 B0 Q2N Clearly x 0 , so to excellent approximation E 2 0 FIG. 3. Geometry of a film with stepped surfaces. The upper and E(0) 0 Q , 56 lower steps are displaced by a distance C, the steps height is H, the as in Eq. 51 . If the applied external field H resultant angle of ``descent'' or ``ascent'' is H/L, with L the 0 is very weak, we find an additional length scale long compared to L period of the steps. N en- ters the problem. This is Lc (B0 /H0)1/2LN . Associated with this is the wave vector Qc QN(H0 /B0)1/2. Thus, 0 are perpendicular to the magnetic field, we show the geom- can be written as etry we have employed in Fig. 3. Unless otherwise specified, the nominal film thickness D has been chosen to be 10 Å, the 2 step height H 2 Å, appropriate to monatomic steps, and the 2Q2 1/2 0 Q Qc 2 4 M0 . 57 Q2 B 2 applied field as H0 0.1 4 M0. We also have chosen the N 0 QN offset C 2 Å, in the initial set of results to be shown below. When H It should be remarked that we have explored the influence of 0 4 M 0 or Qc QN , we have two regimes: i Q C on the anisotropy, to find its influence rather weak. If C c Q QN : Then 0 2Q2/QN , and we have varies from 0 to 20 Å, for the angle H/L of one degree, E 2Q Q the step-induced anisotropy changes by less than 10%. 2 2 H0 1/2 2 c . 58 The first question is the angular variation of the stepped- E(0) QN B0 QN induced anisotropy. We have seen in Sec. II that within our While one's first thought is that for length scales long com- perturbation theoretic treatment, the easy axis is always par- pared to L allel to the step edges. Our numerical studies show E( ) to N , exchange effects can be set aside, and the ``di- pole only'' theory should be appropriate, we see exchange vary quite accurately as cos2( ). The deviations, for the full still enters importantly. The theory with exchange ignored theory with exchange included, are in the range of 1%, or underestimates the roughness-induced anisotropy substan- less. Thus, we have simple uniaxial anisotropy, to an excel- tially. lent approximation, so far as we can see. Because of this, in ii 0 Q Q what follows, we shall confine our attention to the step- c QN : Here 0 (H0 /B0)1/2Q, the ``di- pole only'' result, and one may safely ignore the influence of induced energy change for the case where the step edges are exchange. perpendicular to H 0, where the magnetization is parallel to In any real material, of course, anisotropy will be present. the hard direction. The reader may assume the cos2 varia- If the external field H0 is applied parallel to the easy axis, tion applies. For this special case, it is possible to derive then one may include anisotropy by replacing H0 by (H0 relatively simple expressions for the various quantities, Ha), with Ha the strength of the effective anisotropy field. through a suitable extension of the discussion presented in For Fe, as an example, Ha 550 G, while 4 M0 21 kG. Sec. III. We summarize the expressions in the Appendix. Thus this material, in zero external magnetic field, can be We first consider the variation of the step-induced anisot- characterized by the ratio H(eff) ropy with magnetic field H 0 /4 M0 0.02. The weak- 0. For the case where the angle field limit just discussed thus applies to Fe in zero external H/L is one degree, we show the field dependence in Fig. magnetic field, with anisotropy treated in this manner. 4. We show this calculated for the full theory with exchange included, and for the case where we ignore exchange and IV. NUMERICAL STUDIES OF DIPOLAR ANISOTROPY include only the Zeeman and dipolar energies. In the latter INDUCED BY UNIDIRECTIONAL SURFACE case, we see a very strong dependence on H0 similar quali- PROFILE MODULATION tatively to that contained in Eq. 52 for the semi-infinite case. This very strong field dependence is suppressed when We have carried out numerical studies of the dipolar an- exchange is included; the field dependence is then very isotropy induced by unidirectional modulation of the surface weak. profile, with emphasis on the case where one has a film de- In Fig. 5, we show the dependence of the anisotropy on posited on a stepped surface. We have ultrathin few atomic the angle . For very small angles, we have a linear varia- layer films in mind for these studies, because of the recent tion, and with increasing angle the strength of the anisotropy experimental interest in such films grown on substrates with increases somewhat more slowly than linearly. Something steps. All of our calculations use parameters characteristic of close to linear behavior has been observed experimentally Fe, for which M0 1.7 103 G, and A 2.1 10 6 erg/cm. for a Co film on a curved Cu 001 substrate. In that case the For a stepped surface, and the case where the step edges easy axis is also parallel to the step edges. The authors of 11 878 RODRIGO ARIAS AND D. L. MILLS PRB 59 FIG. 4. Change of magnetic energy per surface area as a func- FIG. 6. Change of magnetic energy per surface area as a func- tion of applied field, for a geometry with stepped surfaces. The tion of film thickness, for a geometry with stepped surfaces. The energy change is plotted for the applied field perpendicular to the energy change is plotted for the applied field perpendicular to the steps and including full theory and excluding the exchange term steps and including an exchange term corresponding to Fe. In this dipolar only . In this case H C 2 Å, D 10 Å, and 10. case H C 2 Å, 10, and H0 0.1 4 M0. When the exchange term is included, the exchange constant corre- sponds to Fe. well, the contribution explored here should play an important role in real materials, possibly for the Co film on Cu 001 Ref. 8 propose a mechanism different than that exposed here, studied in Ref. 8. in their discussion of the anisotropy. In Fig. 6, we show the dependence of E on film thick- It is the case, however, that the magnitude of the anisot- ness, for thicknesses in the range from 10 to 100 Å. ropy is in the range of that found experimentally, for the Throughout this range, we see that E exhibits a very weak ultrathin films studied. We find, for in the range of a few dependence on D. It is common to divide the strength of degrees, that E 0.02 0.04 ergs/cm2. If we express E anisotropies measured in ultrathin films by D, and then plot in terms of an effective magnetic field acting on the magne- the result as a function of D itself. Volume anisotropies are tization in the film, then E M0DH(eff), for a film of thick- independent of D, while surface anisotropies provide a con- ness D. For Fe, M0 1.7 103 G, so if D 10 Å and E tribution inversely proportional to D, when the data is dis- 0.04 ergs/cm2, then H(eff) 240 G, in the range found ex- played in this manner. From Fig. 6, we see that despite the perimentally for stepped-induced anisotropy in ultrathin long-range nature of the dipolar fields, the step-induced an- films. Thus, while other mechanisms surely contribute as isotropy behaves very much like a surface anisotropy. We conclude with information on the spatial distribution and magnitude of both the stray fields and magnetization, for a particular profile illustrated in Fig. 7. We have a film whose nominal thickness is 40 Å, with upper surface at y 20 Å, and lower surface at y 20 Å. Steps are located at z 40 Å and z 60 Å. The terrace length is L 100 Å, and the step height has been adjusted so that H/L 10. In Fig. 8, we show the variation of hz with z evaluated at the center plane (y 0 Å of films of thickness D 40 Å corresponding to case a and D 10 Å to case b . We see that in the center of the film the field can be as large as a half of 4 M0 for the thinner film recall that for Fe, 4 M0 21 kG). However, if one evaluates hz close to the surfaces of the films, one sees that near the steps located at z 40 Å and z 60 Å hz assumes very large values indeed, so within a few Angstroms of a step hz assumes values large compared to 4 M0. We show the variation with the coordinate y of the com- ponent h FIG. 5. Change of magnetic energy per surface area as a func- y in Fig. 9. Once again, near the steps, very large tion of angle of ``descent'' or ``ascent'' , for a geometry with stray fields are generated. One can perceive one aspect of the stepped surfaces. The energy change is plotted for the applied field field distribution, illustrated more clearly below, from these perpendicular to the steps and including an exchange term corre- curves. The step on the upper surface behaves as a positive sponding to Fe. In this case H C 2 Å, D 10 Å, and H0 0.1 line charge, out of which magnetic-field lines diverge. In 4 M0. contrast, the step on the lower surface acts as a negative PRB 59 THEORY OF ROUGHNESS-INDUCED ANISOTROPY IN . . . 11 879 FIG. 7. Geometry of a stepped surface of period L 100 Å, with steps separated by a distance C 20 Å, and angle of ``descent'' or FIG. 9. Plot of the y component of the stray field due to the ``ascent'' 10. The field H roughness of a geometry with stepped surfaces. This component of 0 is applied in the film plane (H0 0.1 4 M the field is plotted along vertical lines that pass close to the lower 0), and the film thickness is 40 Å. This geometry cor- responds to the plots of Figs. 8­12. The exchange constant for step at Z 40 Å and to the upper step at Z 60 Å, and along a those plots corresponds to that of Fe. vertical line separated from the steps at Z 80 Å . If one realizes that n *M acts as an effective magnetic surface magnetic line charge. Field lines diverge outward from the charge density, and notes that the outward normal n is op- former, and inward from the latter. The two steps thus act as positely directed on the upper and lower surface, one sees the a magnetic dipole in two dimensions, i.e., we have a positive origin of the magnetic dipole discussed in the previous para- line charge in near proximity to a negative line charge. In Fig. 10, we show the variation of m graphs. The piece n *M 0 evaluated on the step edges is the y throughout the film, within the framework of a calculation which sets the dominant source of surface charge. strength of the exchange stiffness A to zero. We see that the We conclude in Fig. 12 with a figure which shows the perturbation of the magnetization is confined to the near vi- spatial variation of the magnetic-field lines, along with equi- cinity of the steps. However, inclusion of exchange alters potential surfaces for the magnetic potential. We see field this picture qualitatively, as we see from the full calculation lines diverging from the step on the upper surface, and con- with exchange included, presented in Fig. 11. The perturba- verging into that on the lower surface. tion in the magnetization produced by the steps now extends V. FINAL REMARKS throughout the film. For any choice of the coordinate y, we see a nearly sinusoidal spatial variation in the magnetization. Within a continuum theory, we have presented a descrip- tion of the influence of surface roughness on the distribution FIG. 8. Plots of the z component of the stray field due to the roughness of a geometry with stepped surfaces. This component of FIG. 10. Schematics of the y component of the magnetization the field is plotted along a line that goes through the middle plane of induced by the roughness of a geometry with stepped surfaces. A the film (Y 0 Å , for films of thicknesses D 40 Å for case a relative magnitude is plotted at different points of the film. This and D 10 Å for case b . case corresponds to the dipolar only theory exchange excluded . 11 880 RODRIGO ARIAS AND D. L. MILLS PRB 59 I, other mechanisms surely contribute as well. Experiments show this is the case as well, since for some samples one sees a quadratic variation of the step-induced anisotropy with . Also, the easy axis may be normal to, and not parallel to the step edges in some cases. Further study is clearly re- quired to establish the conditions under which a given mechanism may dominate. The theory presented here should enable one to address other influences of interface roughness on the properties of ultrathin ferromagnetic films, and with suitable extensions, in the properties of magnetic multilayers. ACKNOWLEDGMENTS This research was completed with support from the Army Research Office, under Contract No. CS0001028. R.A. was given partial financial support by the Organization of Ameri- can States. FIG. 11. Schematics of the y component of the magnetization induced by the roughness of a geometry with stepped surfaces. A relative magnitude is plotted at different points of the film. This APPENDIX case corresponds to the full theory exchange included . Here we summarize the expressions for the perturbation of magnetization within a ferromagnetic film. Stray magnetic of the magnetization, magnetic potential, and energy of a fields are generated, and the magnetic energy per unit area is thin ferromagnetic film whose upper and lower surfaces have increased by these effects. When the roughness has a unidi- arbitrary unidirectional roughnesses. It is also assumed that rectional character, as is the case for a film grown on a the applied field H stepped surface, the increase in the magnetic energy per unit 0 H0z in the plane of the film is per- pendicular to the roughness features. area depends on the direction between the nominal magneti- The roughnesses of the upper and lower surfaces of the zation, and the step edges. We thus have a mechanism for film are written as step-induced anisotropy. Within our perturbation theoretic description of the dipo- lar contribution to the anisotropy energy, the easy axis will u,d z u,d Q eiQnz A1 always be parallel to the step edges. Its strength varies lin- Q n n early with the vicinal angle H/L for small , according to our numerical studies, and increases more slowly for with Qn 2 n/L, n , . . . , , and L a quantization larger values. For the model films explored, we find anisot- length. For simplicity we will concentrate only on one wave- ropy energies fall in the range of those found experimentally. length, i.e., we assume Thus, the mechanism explored here should be an important source of step-induced anisotropy. However, as noted in Sec. u,d z u,d Q eiQz c.c. A2 with u,d u,d u,d Q Q ei Q the results for an arbitrary roughness, as that of Eq. A1 , follow by simple superposition . Due to the symmetry of this geometry, mQx(y,z) 0. The following forms of mQy(y,z) and Q(y,z), valid inside the film, solve to first order the equilibrium equation M H eff 0 and the Maxwell equation *B 0: mQ 2 2 y y ,z Ax x Q2 cosh xy A0 0 Q2 cosh 0y Sx 2x Q2 sinh xy S0 20 Q2 sinh 0y eiQz c.c., A3 Q y,z 4 Ax x sinh xy A0 0 sinh 0y Sx x cosh xy S0 0 cosh 0y eiQz c.c., FIG. 12. Graphic representation of the stray fields and magnetic A4 potential induced by the roughness of a geometry with stepped sur- faces. Lines represent equipotentials, and arrows the direction and and outside the film, in the upper and lower regions, respec- relative magnitude of the (y,z) components of the stray fields. tively, the potential reads as PRB 59 THEORY OF ROUGHNESS-INDUCED ANISOTROPY IN . . . 11 881 u 2 Q y ,z 4 Ax x sinh xD/2 A0 0 sinh 0D/2 a0 0 0 Q2 cosh 0D/2 , Sx 2 x cosh xD/2 ax x x Q2 cosh xD/2 , S0 0 cosh 0D/2 e Q(y D/2) iQz c.c., d 2 0 0 0 Q2 sinh 0D/2 , A5 d 2 x x x Q2 sinh xD/2 , A8 dQ y,z 4 Ax x sinh xD/2 A0 0 sinh 0D/2 and Sx 2 2 x cosh xD/2 BQ 0 x x 0 cosh xD/2 cosh 0D/2 S0 0 cosh 0D/2 eQ(y D/2) iQz c.c. 2 x x Q2 Q cosh xD/2 sinh 0D/2 A6 2 Q2 Q cosh In these expressions the ``decay constants'' 0 0 0D/2 sinh xD/2 , 0 and x are the same as the exchange and dipolar decay constants of Eqs. F 2 2 sinh 38 and 39 , that correspond to the analogous problem on a Q 0 x x 0 xD/2 sinh 0D/2 semi-infinite medium. The four constants Ax, A0, Sx, and S0 2 x x Q2 Q sinh xD/2 cosh 0D/2 are obtained by applying the boundary conditions of null 2 normal derivatives of the magnetization at the upper and 0 0 Q2 Q cosh xD/2 sinh 0D/2 . A9 lower surfaces, and both H tang Q or equivalently Q) and The change in energy per unit surface area due to this single B normal Fourier component Q follows from use of Eq. 23 : Q continuous at the upper and lower surfaces of the film. These constants become EQ 2Q 2 2 1 u 2 d 2 iM iM A 2 M0 0 x x 0 F Q Q Q A 0 u d 0 u d x 2F d0 Q Q , A0 dx Q Q , Q 2FQ 2 u d u d Q Q cos Q Q sinh xD/2 sinh 0D/2 iM iM 1 S 0 u d 0 u d u d u d u d x 2B a0 Q Q , S0 ax Q Q Q 2 Q 2 2 Q Q cos Q Q Q 2BQ BQ A7 with cosh xD/2 cosh 0D/2 . A10 1 J.C. Slonczewski, Phys. Rev. Lett. 67, 3172 1991 . 6 R. Kawakami, E.J. Escorcia-Aparicio, and J. Qiu, Phys. Rev. 2 S.O. Demokritov, J. Phys. D 31, 925 1998 . Lett. 77, 2570 1996 . 3 J. Chen and J.L. Erskine, Phys. Rev. Lett. 68, 1212 1992 . 7 H.J. Choi, Z.Q. Qiu, J. Pearson, J. S. Jiang, D. Li, and S.D. Bader, 4 A. Berger, U. Linke, and H.P. Oepen, Phys. Rev. Lett. 68, 839 Phys. Rev. B 57, R12 713 1998 . 1992 . 8 R.K. Kawakami, M.O. Bowen, H.J. Choi, E.J. Escorcia-Aparicio, 5 W. Weber, C.H. Back, A. Bischof, Ch. Wursch, and R. Allens- and Z.Q. Qiu, Phys. Rev. B 58, R5924 1988 . pach, Phys. Rev. Lett. 76, 1940 1996 .