- Email: [email protected]
Simulation of transverse and longitudinal magnetic ripple structures induced by surface anisotropy
ELSEVIER
Journal of Magnetismand MagneticMaterials 163 (1996) 285-291
journal of magnetism and magnetic ~ I ~ materials
Simulation of transverse and longitudinal magnetic ripple structures induced by surface anisotropy Lu Hua 1, J.E.L. Bishop, J.W. Tucker * Department of Physics, The University of Sheffield, Sheffield $3 7RH, UK Received 15 March 1996
Abstract Micromagnetic ripple structures on the surfaces of thick specimens of ultra-soft magnetic material having strong surface anisotropy K s favouring out-of-surface magnetization have been calculated. These ripples have wavelengths of the order of 0.1 ixm and extend to a depth ~ f A / M s, where A is the exchange constant and Ms is the saturation magnetization. The wave-vectors of the ripple s~uctures are either transverse or parallel to the bulk magnetization. Both structures have lower energy than the one-dimensional structure discussed by O'Handley and Woods, and they exhibit stronger normal magnetization. The transverse structure requires a surface anisotropy K S> 0.80K0, where K o = (2xrA)l/2Ms is that required for the one-dimensional structure. The threshold for longitudinal ripples is 0.84K 0. It is suggested that the transverse structure probably constitutes the ground state. The magnitudes of K s and A should be obtainable from measurements of the ripple wavelength and amplitude, and Ms.
Keywords: Surface anisotropy;Computer simulation;Micromagnetism;Ripple structures
1. I n t r o d u c t i o n With the progressive miniaturisation of magnetic devices, the influence of surface anisotropy on micromagnetic structure is becoming of increasing importance. This influence is most significant when the sign and strength of the surface anisotropy are such as to oppose, and locally overcome, an otherwise dominant bulk or shape anisotropy. A surface anisot-
* Corresponding author. Fax: +44-144-272-8079; email: j [email protected]. 1 Present address: School of Computing and Mathematical Sciences, Universityof Greenwich,Woolwich,London SE18 6PF, UK.
ropy of appropriate sign and strength may give rise to magnetization normal to the surface of a soft magnetic material despite the substantial increase in demagnetization energy that results. O'Handley and Woods ( O ' H W ) [1] have provided a one-dimensional treatment of this situation. Their model consists of a magnetic half-space, z > 0, with zero bulk anisotropy and positive surface a n i s o t r o p y e n e r g y K s sine(0(0)). Here O(z) is the angle between the magnetization vector M and the z-axis, which is normal to the surface. O(z) was obtained by minimizing the sum of the exchange, dipolar, Zeeman and surface anisotropy energies of the system, expressed as functionals of the one-dimensional function 0(z). In the limit of zero applied field, it was found that unless K s exceeds a threshold, K o, O(z)
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0304-8853(96)00345-9
L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
286
= 7r/2 throughout and M is uniformly aligned with an axis (chosen to be the x-axis) parallel to the surface. However, for K~ > K 0, M was found to deviate towards the normal in a thin layer at the surface. This deviation, O(z), is described by the following expression:
O(z)=arctan[sinh(ao+boz)],
K s > K 0.
sion of the layer into strips of alternatively positive and negative M z. These strips - which we term r i p p l e s - are somewhat analogous to the familiar Kittel open-flux domain structure of stripe domains of alternating magnetization normal to the surfaces of a film or sheet sample that has strong uniaxial bulk anisotropy favouring normal magnetization [2]. In this paper we consider two types of magnetic ripple with quite different symmetries (Fig. 1). In the first, which we term the longitudinal ripple structure, the magnetization near the surface varies periodically in the x-direction, i.e. the ripple wavevector is parallel to the bulk magnetization. Thus the magnetization vector M remains confined to the x - z plane, as in the O ' H W model, but it is now a function of two variables: M = M ( x , z ) . These longitudinal ripples were first proposed by us at the International Conference on Magnetism (Warsaw) [3] and we showed them to have lower energy than the one-dimensional O ' H W structure. In addition, they have a lower surface anisotropy threshold than K o.
(1)
Here a 0 and b 0 are material constants defined by the equations b0~ = 2 ~ M s ~ / A and tanh(a 0) = Ko/Ks; A is the exchange constant, M s is the saturation magnetization, and the threshold K 0 = Ab o. Although this O ' H W model provides a useful benchmark for the occurrence of out-of-surface magnetization, the fact that it restricts any variation of M to only one dimension - depth - means that it cannot provide an adequate representation of the energetic optimum magnetic structure. This is because the dipolar energy associated with the normal magnetization component M z = M cos(0) in the surface layer can be substantially reduced by a subdivi-
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(b) Fig. 1. Examples of cross sections ~rough ~ e magnetic ripple structures just benea~ the surface (top of fi~res) of ~ ulna-soft magnetic hal~space with s~Nce anisotropy easy axis norton to the surface. (a) Tr~sverse ripple s ~ c t u r e ~ r K s = 2.0 e r g / c m 2 w i ~ imposed wavelength 128 nm. The bulk magnetization is normal to the p l ~ e of ~ e fig~e ~ d is not shown. (b) ~ngitudinal ripple s~ucmre of Ref. [3] ~ r K~ - 2.25 e r g / c m 2 with imposed wavelen~h 76 nm. The ~gnetization vector is confined to the p l ~ e of ~ e f i ~ e .
L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
In the second, which we term the transverse ripple structure, and discuss for the first time here, the magnetization near the surface varies periodically transverse to the bulk magnetization direction. (As it is convenient to retain the x-axis parallel to the ripple wave-vector for both types of ripple, the y-axis is chosen parallel to the bulk magnetization for these transverse ripples only.) While for both types of ripple, M = M ( x , z ) is a function of two variables only, so that OM/Oy = 0, the orientation of M for transverse ripples is not confined to a plane as in the case of the longitudinal ripples. The periodic behaviour of the component of M transverse to the bulk magnetization axis y is depicted in Fig. la.
2. Calculations Our model is similar to that of O ' H W in that the half-space z > 0 is filled with an ultra-soft material having a saturation magnetization M s = 800 e m u / c m 3, an exchange constant A = 10 .6 e r g / c m (both values typical of permalloys) and a positive surface anisotropy K s sin2(0(0)). The value of K s was varied over the range 1.5 _< K s _< 3.5 e r g / c m 2. This range includes the threshold values K L and K T that were found to be required for the occurrence of longitudinal and transverse ripple structures, respectively. It proved computationally convenient to remove the degeneracy associated with the orientation of the magnetization in the bulk limit z ~ ~ by introducing a very weak uniaxial bulk anisotropy g U = 1 0 3 e r g / c m 3) with its easy axis along x (for longitudinal ripples) or along y (for transverse ripples); this bulk anisotropy had a negligible influence (roughly equivalent to a 0.012% increase in M s) on either the form or the energy of the ripple structures obtained. In each case, the structure M ( x , z ) was obtained by minimizing (numerically) its energy U~urf per unit area of the surface of the magnetic half-space. The (arbitrary) zero of Usurf was chosen to be that of the corresponding state of uniform magnetization along the weak bulk easy direction. (Consequently any small contribution to gsurf that arises from the weak bulk anisotropy K U is necessarily positive.) For numerical computation, a rectangular region of width L in the x-direction, and depth D in the z-direction,
287
was selected. (With OM/Oy uniformly zero, the problem is two-dimensional, with a nominal length in the y-direction of 1 cm.) The computational region was discretised into N x × N z cells (30 < N x < 176, N z = 50) each 2 nm square in cross section. The depth D ( = 100 rim) was always sufficient for M ( x , D) to depart negligibly from the bulk anisotropy axis. (The longitudinal ripple calculations reported in Ref. [3] were carried out with D = 60 nm; in order to compare like with like, all longitudinal results reported here have been recalculated to the same (enhanced) precision used for the transverse ripple.) The boundary condition at the surface z = 0 required to account for the surface anisotropy has been discussed by Brown [4]. It was implemented in discrete form by introducing a set of N x additional computational elements located in the plane z = 0. Elements of this type were introduced by LaBonte in his treatment of free surfaces [5], The energetic optimum ripple wavelength A was not known at the outset; periodic boundary conditions were imposed at x = 0 and x = L, fixing the period "~irnp0sed = L (or a submultiple of L) and the calculations were repeated for various (fixed) values of L in order to determine the value of wavelength, A, that minimises the value obtained for Usurf: In the continuum limit Usurf is defined as follows:
Usu,:f=L-l£Ldx[-Ksrn2z(x,O)+fodz
{
1
× Kum~-~HD'M
where m x, my, m z, are the components of the unit vector m = M / M s, m I is the component of m transverse to the bulk easy axis, and H D is the demagnetising field that arises from the divergence of the magnetization M and the surface charges introduced by the extended ripples. Equilibrium ripple structures were obtained b y numerically integrating the Landan-Lifshitz-Gilbert equation:
dm/dt=TomXIteff-AmX(mXHeff),
(3)
in which 3'0 is the gyromagnetic ratio and A a
L. Hua et al. /Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
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damping constant. Herf is an effective field defined as - 8U~urf/gM, such that M × Herf gives the torque on M. Because our calculations were restricted to stable equilibrium states, the gyromagnetic term in Eq. (3) could be ignored [6]. In these circumstances the integration becomes essentially a numerical procedure leading to an equilibrium state accessible by monotonically decreasing the energy. The choice of a value for A is then equivalent to selecting a time step for integration. Examples of the magnetic ripple patterns obtained in this way are presented in Fig. 1. Though not of optimum wavelength, both have significantly lower energy than the O ' H W structure for the same material constants.
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Surface Anisotropy K s (erg/cm2 )
Surface A n i s o t r o p y Ks ( e r g / c m 2) Fig. 2. Illustrating the dependence on surface anisotropy, Ks, of Usu~, the energy per unit area of the sample surface, of the three magnetic surface structures discussed in the text. (a) Transverse ripple structure; (b) longitudinal tipple s~ucture of Ref. [3]; (c) structure with depth variation only as proposed in Ref. [1]. The lines through the points for (a) and (b) are least squares polynomial fits as guides to the eye only.
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Fig. 3. Dependence on surface anisotropy K~, of the energetically optimum wavelength of the ripple structures. (a) Transverse ripple structure; (b) longitudinal ripple structure. The lines are least squares fits as guides to the eye.
(~)s)max = (90 - 0(0)max )° of the ripples at the sur-
face are shown in Figs. 3 and 4, respectively. In Figs. 2 - 4 the points labelled (a) and (b) refer to transverse and longitudinal ripples respectively; the lines through the points are empirical least squares polynomial fits. The lines denoted (c) in Figs. 2 and 4 represent simple analytic expressions pertaining to the ripple-free structure of O ' H W in which M is permitted to vary with depth z only. It was not
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3.~ Results and discussion Fig. 2 shows the dependence of the energy per unit area of surface, on the surface anisotropy constant K S for the two types of ripple structure treated here, and also for the ripple-free smacture determined analytically by O ' H W in Ref. [1]. The dependence on K s of the energetically optimum ripple wavelength A and of the angular amplitude Usurf,
0 1.5
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Surface Anisotropy K s (erg/cm2) Fig. 4. Amplitude of the magnetization tilt angle ( ~ ) s ) m a x at the surface as a function of the surface anisotropy K s. (a) Transverse ripple; (b) longitudinal tipple; (c) magnetization varying with depth only [1]. The lines through the points for (a) and (b) are fitted as guides to the eye.
L. Hua et al. / Journal of Magnetism and Magnetic Materials 1.63 (1996) 285-291
possible to vary the wavelength L imposed on the ripple calculations other than in finite steps without altering other parameters such as D, thereby possibly introducing systematic errors. The values indicated by the points in the figures relate to the energetic optimum wavelength A and were obtained from the data for neighbouring wavelengths by polynomial interpolation. Because the ripple energies U~u~f vary quadratically with ripple wavelength at their minima, they can be estimated much more precisely in this way than can the optimum wavelengths A themselves - compare the scatter of the data in Figs. 2 and 3. It is evident from the data in Fig. 2 that the transverse ripples have lower energy than the longitudinal ripples which in turn have lower energy than the O ' H W structure. (The O ' H W structure may be regarded as constituting the long wavelength limit for both ripple structures.) The surface energy of the O ' H W structure may readily be shown to be given by the following simple formula: UsO'HW
=
- (Ks - K0)
2
/Ks.
(4)
From Eq. (5) it follows that for values of K S not far above the threshold Ko, ~/(-rrO'HW~ Vsur f ] increases almost linearly with K S. It is clear from Fig. 2 that the behaviour of ~/(-Usurf) for the ripple structures is similar, indeed rather more nearly linear. (The onset of the non-uniform magnetic state at K s = K 0 is analogous to a second-order phase transition; gsurf has exponent 2 with respect to the parameter k = ( K s Ko)/Ko.) Extrapolating the ripple data in Fig. 2 to zero Usurf yields K0T = 1.61_+0.02 e r g / c m 2 and K0L = 1.68 _+ 0.02 e r g / c m 2 for the values of the surface anisotropy thresholds for transverse and longitudinal ripples respectively; the corresponding threshold value for the O ' H W structure in this material is K 0 = 2.01 e r g / c m ; . The reason why the transverse ripples have lower energy than the longitudinal ripples is readily understood qualitatively from a comparison of their structures as shown in Fig. 1. Firstly, in the transverse case rotation of the magnetization at the boundary between ripples of opposite sign assists closure of the dipolar flux associated with the normal magnetization M z, giving rise to the vortex structures evident in Fig. la. No comparably effective closure -
289
process is available in the transition regions (i.e. where M z ~ 0) of the longitudinal ripple. Secondly, with OMy/Oy = 0, the volume pole density of the transverse ripples is free from any contribution from the principal magnetization component My. The corresponding contribution OMz/OZ is non-zero for the longitudinal ripples which, in this respect, are somewhat analogous to Ngel domain walls - see Fig. lb. In this context one should remark that while the transverse ripples are symmetric about their planes of maximum M z, the longitudinal ripples are not; this is because the magnetization in successive transition regions is alternately parallel and anti-parallel to the x-component of the magnetic field H o at the surface. The dependence of the ripple wavelength on K S is shown in Fig. 3 for both types of ripple. Because the energies of the ripple structures are insensitive to small changes in wavelength near the optimum value, we could not determine A(KS) to an accuracy better than about _+3 nm. At their thresholds, the transverse and longitudinal ripples have wavelengths A0 r ~ 123 + 3 nm and A0 L ~ 103 __ 3 nm, respectively. AT is consistently greater than AL presumably because of the better flux closure at the transverse ripple boundaries discussed in the preceding paragraph. With increasing K s the width of both types of ripple increases while the difference between their wavelengths decreases. Fig. 4 shows the dependence of the ripple amplitude ( ~ s ) m a x = ( 9 0 - 0(0)max)° on K s. For the ripple-free O ' H W structure, the corresponding amplitude ~b°'uw is given by the simple relation cos(~b°'Hw) = sin(0(0)) = K o / K S [1]. Consequently sin(~b°'Hw) = [1 - (Ko/Ks)2] 1/2, and ~b°'Hw rises abruptly at the threshold K 0. The behaviour of the amplitudes of the two ripple structures with increasing K s closely mimics that of v-s~hO'HW.In each case the abrupt rise assists estimation of the corresponding threshold values of the surface anisotropyl From the data in Fig. 4 one obtains K0T = 1 . 6 2 _ 0.02 e r g / c m 2 and K0L = 1.69 _ 0.02 e r g / c m 2, in very close agreement with the values K0T = 1.61 _ 0.02 e r g / c m 2 and K0L = 1.68 _+ 0.02 e r g / c m 2 derived from the data for U~urf in Fig. 2. We have shown that the transverse ripple structure has lower energy than either the longitudinal ripple structure or the one-dimensional O ' H W struc-
290
L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
ture, but does it constitute the ground state of this simple ideal system, or does some other as yet unknown structure have still lower energy? The transverse ripple structure is a two-dimensional model with M = M ( x , z ) and removing the restriction ~ M / O y = 0 might permit us to obtain an M ( x , y , z ) with a lower U~u~f. However, to perform a three dimensional simulation of comparable precision would be extremely demanding computationally, and there are reasons for suspecting that the transverse ripple structure is the ground state. As mentioned earlier, the problem of an ultra-soft half-space with a strong surface anisotropy favouring normal magnetization is quite closely analogous to the well known problem of a thin sheet with a strong uniaxial bulk anisotropy supporting out-of-plane magnetization. This situation has been very thoroughly studied both theoretically and experimentally for many decades and it has been established convincingly that for sufficiently thin sheet the Kittel structure of parallel bar domains that span the sheet thickness minimises (for optimum domain width) the sum of the dipolar and domain wall energies when the wall energy per unit area is treated as a constant (see Ref. [7] and the work cited therein). The more complex rick-rack and flower-like surface domain patterns seen on these materials require sheet thicknesses very much greater than the exchange length and domain wall width [8,9]. The analogy with thin films is appropriate to the present problem because of the limited penetration ( ~ v/-A/Ms) of the normal magnetization into the bulk of the material. Furthermore, we have shown here that the boundaries between the ripples have lower energy per unit length in the transverse than in the longitudinal configuration: this provides an additional bias in favour of a (transverse) ripple structure with straight parallel ripple boundaries, a bias not present in the Kittel domain analogue. In this paper, for concreteness, we have adopted specific values A = 10 .6 e r g / c m and M s = 800 e m u / c m 3) for the two magnetic constants A and M S that characterise the magnetic half-space. These two constants determine the natural units of length and energy density (per unit area) for the problem at hand. They a r e respectively l o = 1 / b 0 = [ A / ( 2 7 r M 2 ) ] 1/2 and K o = Ab o. The existence and nature of the ripple structures depend on the value of the surface anisotropy (kre d = K J K o) in these re-
duced units. Experimental observations of the amplitude (~bs)m,x and wavelength A of these ripple structures on a fairly thick sample of ultra-soft material should, if the values of both A and M s are known, provide two independent estimates of the value of ,' the surface anisotropy K s by reference to the data provided here in Figs. 3 and 4, after appropriate scaling to the reduced units. Agreement between the two estimates would serve as a confirmation of the validity of our model. With many materials, however, the value of the exchange constant A is not known with much precision. In that case, the observed value of the amplitude (qSs)m,x will still enable an estimate of the reduced surface anisotropy kre d to be obtained from Fig. 4. Given this value of krea, a predicted value of the reduced ripple period /~red = h / l o will in turn be obtainable with the aid of Fig. 3 (with axes rescaled). A value of A for the sample can then be obtained from the value of the reduced length l 0 appropriate to the sample given by the ratio hobs/Ared that the observed ripple period bears to the reduced value.
4. Conclusions We have obtained the micromagnetic structures of two distinct types of magnetic ripple that are expected to occur on the surfaces of ultra-soft magnetic materials that possess a sufficiently strong surface anisotropy that locally favours magnetization normal to the surface. These ripples have wavelengths of the order of 0.1 Ixm and penetrate to a rather lesser depth in the material. In this paper only the limiting case of material much thicker than these dimensions is treated. Both ripple structures are shown to have lower energy than the one-dimensional structures predicted by O'Handley and Woods [1] and exhibit a greater component of normal magnetization at the surface. The O ' H W structure may be regarded as representing the long wavelength limit of these ripple structures. In one structure the ripple wave-vector is transverse to the bulk magnetization direction, in the other it is parallel to it. The former, transverse, structure has a lower energy and reasons are advanced that it probably constitutes the ground state. The transverse structure requires a surface anisotropy K s > 0.80K o, where K 0 = (21rA)1/2Ms is the corre-
L. Hua et aL / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
sponding threshold for the one-dimensional O'HW structure. The threshold for the longitudinal ripple structure i s 0.84K 0. Measurements of the wavelength of these ripples and of the amplitude of the normal surface magnetization they exhibit should provide a method for obtaining the magnitudes of the exchange constant of the material and also its surface anisotropy.
Acknowledgements This work was supported by grants GR/H26758 and GR/J89859 from the Engineering and Physical Sciences Research Council of the UK.
291
References [1] R.C. O'Handley and J.P. Woods, Phys. Rev. B 42 (1990) 6568. [2] C. Kittel, Rev. Mod. Phys. 21 (1949) 541. [3] Lu Hua, J.E.L. Bishop and J.W. Tucker, J. Magn. Magn. Mater. 140-144 (1995) 655. [4] W.F. Brown, Jr., Micromagnetics (Wiley Interscience, New York, 1963) chap. 4. [5] A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450. [6] M.E. Schabes and H.N. Bertram, J. Appl. Phys. 64 (1988) 1347. [7] B. Kaplan and G.A. Gehring, J. Magn. Magn. Mater. 128 (1993) 111. [8] R. Szymczak, Elec. Tech. 1 (1968) 5. [9] A. Hubert, Phys. Status Solidi 24 (1967) 669.
Journal of Magnetismand MagneticMaterials 163 (1996) 285-291
journal of magnetism and magnetic ~ I ~ materials
Simulation of transverse and longitudinal magnetic ripple structures induced by surface anisotropy Lu Hua 1, J.E.L. Bishop, J.W. Tucker * Department of Physics, The University of Sheffield, Sheffield $3 7RH, UK Received 15 March 1996
Abstract Micromagnetic ripple structures on the surfaces of thick specimens of ultra-soft magnetic material having strong surface anisotropy K s favouring out-of-surface magnetization have been calculated. These ripples have wavelengths of the order of 0.1 ixm and extend to a depth ~ f A / M s, where A is the exchange constant and Ms is the saturation magnetization. The wave-vectors of the ripple s~uctures are either transverse or parallel to the bulk magnetization. Both structures have lower energy than the one-dimensional structure discussed by O'Handley and Woods, and they exhibit stronger normal magnetization. The transverse structure requires a surface anisotropy K S> 0.80K0, where K o = (2xrA)l/2Ms is that required for the one-dimensional structure. The threshold for longitudinal ripples is 0.84K 0. It is suggested that the transverse structure probably constitutes the ground state. The magnitudes of K s and A should be obtainable from measurements of the ripple wavelength and amplitude, and Ms.
Keywords: Surface anisotropy;Computer simulation;Micromagnetism;Ripple structures
1. I n t r o d u c t i o n With the progressive miniaturisation of magnetic devices, the influence of surface anisotropy on micromagnetic structure is becoming of increasing importance. This influence is most significant when the sign and strength of the surface anisotropy are such as to oppose, and locally overcome, an otherwise dominant bulk or shape anisotropy. A surface anisot-
* Corresponding author. Fax: +44-144-272-8079; email: j [email protected]. 1 Present address: School of Computing and Mathematical Sciences, Universityof Greenwich,Woolwich,London SE18 6PF, UK.
ropy of appropriate sign and strength may give rise to magnetization normal to the surface of a soft magnetic material despite the substantial increase in demagnetization energy that results. O'Handley and Woods ( O ' H W ) [1] have provided a one-dimensional treatment of this situation. Their model consists of a magnetic half-space, z > 0, with zero bulk anisotropy and positive surface a n i s o t r o p y e n e r g y K s sine(0(0)). Here O(z) is the angle between the magnetization vector M and the z-axis, which is normal to the surface. O(z) was obtained by minimizing the sum of the exchange, dipolar, Zeeman and surface anisotropy energies of the system, expressed as functionals of the one-dimensional function 0(z). In the limit of zero applied field, it was found that unless K s exceeds a threshold, K o, O(z)
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0304-8853(96)00345-9
L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
286
= 7r/2 throughout and M is uniformly aligned with an axis (chosen to be the x-axis) parallel to the surface. However, for K~ > K 0, M was found to deviate towards the normal in a thin layer at the surface. This deviation, O(z), is described by the following expression:
O(z)=arctan[sinh(ao+boz)],
K s > K 0.
sion of the layer into strips of alternatively positive and negative M z. These strips - which we term r i p p l e s - are somewhat analogous to the familiar Kittel open-flux domain structure of stripe domains of alternating magnetization normal to the surfaces of a film or sheet sample that has strong uniaxial bulk anisotropy favouring normal magnetization [2]. In this paper we consider two types of magnetic ripple with quite different symmetries (Fig. 1). In the first, which we term the longitudinal ripple structure, the magnetization near the surface varies periodically in the x-direction, i.e. the ripple wavevector is parallel to the bulk magnetization. Thus the magnetization vector M remains confined to the x - z plane, as in the O ' H W model, but it is now a function of two variables: M = M ( x , z ) . These longitudinal ripples were first proposed by us at the International Conference on Magnetism (Warsaw) [3] and we showed them to have lower energy than the one-dimensional O ' H W structure. In addition, they have a lower surface anisotropy threshold than K o.
(1)
Here a 0 and b 0 are material constants defined by the equations b0~ = 2 ~ M s ~ / A and tanh(a 0) = Ko/Ks; A is the exchange constant, M s is the saturation magnetization, and the threshold K 0 = Ab o. Although this O ' H W model provides a useful benchmark for the occurrence of out-of-surface magnetization, the fact that it restricts any variation of M to only one dimension - depth - means that it cannot provide an adequate representation of the energetic optimum magnetic structure. This is because the dipolar energy associated with the normal magnetization component M z = M cos(0) in the surface layer can be substantially reduced by a subdivi-
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(b) Fig. 1. Examples of cross sections ~rough ~ e magnetic ripple structures just benea~ the surface (top of fi~res) of ~ ulna-soft magnetic hal~space with s~Nce anisotropy easy axis norton to the surface. (a) Tr~sverse ripple s ~ c t u r e ~ r K s = 2.0 e r g / c m 2 w i ~ imposed wavelength 128 nm. The bulk magnetization is normal to the p l ~ e of ~ e fig~e ~ d is not shown. (b) ~ngitudinal ripple s~ucmre of Ref. [3] ~ r K~ - 2.25 e r g / c m 2 with imposed wavelen~h 76 nm. The ~gnetization vector is confined to the p l ~ e of ~ e f i ~ e .
L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
In the second, which we term the transverse ripple structure, and discuss for the first time here, the magnetization near the surface varies periodically transverse to the bulk magnetization direction. (As it is convenient to retain the x-axis parallel to the ripple wave-vector for both types of ripple, the y-axis is chosen parallel to the bulk magnetization for these transverse ripples only.) While for both types of ripple, M = M ( x , z ) is a function of two variables only, so that OM/Oy = 0, the orientation of M for transverse ripples is not confined to a plane as in the case of the longitudinal ripples. The periodic behaviour of the component of M transverse to the bulk magnetization axis y is depicted in Fig. la.
2. Calculations Our model is similar to that of O ' H W in that the half-space z > 0 is filled with an ultra-soft material having a saturation magnetization M s = 800 e m u / c m 3, an exchange constant A = 10 .6 e r g / c m (both values typical of permalloys) and a positive surface anisotropy K s sin2(0(0)). The value of K s was varied over the range 1.5 _< K s _< 3.5 e r g / c m 2. This range includes the threshold values K L and K T that were found to be required for the occurrence of longitudinal and transverse ripple structures, respectively. It proved computationally convenient to remove the degeneracy associated with the orientation of the magnetization in the bulk limit z ~ ~ by introducing a very weak uniaxial bulk anisotropy g U = 1 0 3 e r g / c m 3) with its easy axis along x (for longitudinal ripples) or along y (for transverse ripples); this bulk anisotropy had a negligible influence (roughly equivalent to a 0.012% increase in M s) on either the form or the energy of the ripple structures obtained. In each case, the structure M ( x , z ) was obtained by minimizing (numerically) its energy U~urf per unit area of the surface of the magnetic half-space. The (arbitrary) zero of Usurf was chosen to be that of the corresponding state of uniform magnetization along the weak bulk easy direction. (Consequently any small contribution to gsurf that arises from the weak bulk anisotropy K U is necessarily positive.) For numerical computation, a rectangular region of width L in the x-direction, and depth D in the z-direction,
287
was selected. (With OM/Oy uniformly zero, the problem is two-dimensional, with a nominal length in the y-direction of 1 cm.) The computational region was discretised into N x × N z cells (30 < N x < 176, N z = 50) each 2 nm square in cross section. The depth D ( = 100 rim) was always sufficient for M ( x , D) to depart negligibly from the bulk anisotropy axis. (The longitudinal ripple calculations reported in Ref. [3] were carried out with D = 60 nm; in order to compare like with like, all longitudinal results reported here have been recalculated to the same (enhanced) precision used for the transverse ripple.) The boundary condition at the surface z = 0 required to account for the surface anisotropy has been discussed by Brown [4]. It was implemented in discrete form by introducing a set of N x additional computational elements located in the plane z = 0. Elements of this type were introduced by LaBonte in his treatment of free surfaces [5], The energetic optimum ripple wavelength A was not known at the outset; periodic boundary conditions were imposed at x = 0 and x = L, fixing the period "~irnp0sed = L (or a submultiple of L) and the calculations were repeated for various (fixed) values of L in order to determine the value of wavelength, A, that minimises the value obtained for Usurf: In the continuum limit Usurf is defined as follows:
Usu,:f=L-l£Ldx[-Ksrn2z(x,O)+fodz
{
1
× Kum~-~HD'M
where m x, my, m z, are the components of the unit vector m = M / M s, m I is the component of m transverse to the bulk easy axis, and H D is the demagnetising field that arises from the divergence of the magnetization M and the surface charges introduced by the extended ripples. Equilibrium ripple structures were obtained b y numerically integrating the Landan-Lifshitz-Gilbert equation:
dm/dt=TomXIteff-AmX(mXHeff),
(3)
in which 3'0 is the gyromagnetic ratio and A a
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damping constant. Herf is an effective field defined as - 8U~urf/gM, such that M × Herf gives the torque on M. Because our calculations were restricted to stable equilibrium states, the gyromagnetic term in Eq. (3) could be ignored [6]. In these circumstances the integration becomes essentially a numerical procedure leading to an equilibrium state accessible by monotonically decreasing the energy. The choice of a value for A is then equivalent to selecting a time step for integration. Examples of the magnetic ripple patterns obtained in this way are presented in Fig. 1. Though not of optimum wavelength, both have significantly lower energy than the O ' H W structure for the same material constants.
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Surface Anisotropy K s (erg/cm2 )
Surface A n i s o t r o p y Ks ( e r g / c m 2) Fig. 2. Illustrating the dependence on surface anisotropy, Ks, of Usu~, the energy per unit area of the sample surface, of the three magnetic surface structures discussed in the text. (a) Transverse ripple structure; (b) longitudinal tipple s~ucture of Ref. [3]; (c) structure with depth variation only as proposed in Ref. [1]. The lines through the points for (a) and (b) are least squares polynomial fits as guides to the eye only.
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Fig. 3. Dependence on surface anisotropy K~, of the energetically optimum wavelength of the ripple structures. (a) Transverse ripple structure; (b) longitudinal ripple structure. The lines are least squares fits as guides to the eye.
(~)s)max = (90 - 0(0)max )° of the ripples at the sur-
face are shown in Figs. 3 and 4, respectively. In Figs. 2 - 4 the points labelled (a) and (b) refer to transverse and longitudinal ripples respectively; the lines through the points are empirical least squares polynomial fits. The lines denoted (c) in Figs. 2 and 4 represent simple analytic expressions pertaining to the ripple-free structure of O ' H W in which M is permitted to vary with depth z only. It was not
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3.~ Results and discussion Fig. 2 shows the dependence of the energy per unit area of surface, on the surface anisotropy constant K S for the two types of ripple structure treated here, and also for the ripple-free smacture determined analytically by O ' H W in Ref. [1]. The dependence on K s of the energetically optimum ripple wavelength A and of the angular amplitude Usurf,
0 1.5
2.0
2.5
3.0
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Surface Anisotropy K s (erg/cm2) Fig. 4. Amplitude of the magnetization tilt angle ( ~ ) s ) m a x at the surface as a function of the surface anisotropy K s. (a) Transverse ripple; (b) longitudinal tipple; (c) magnetization varying with depth only [1]. The lines through the points for (a) and (b) are fitted as guides to the eye.
L. Hua et al. / Journal of Magnetism and Magnetic Materials 1.63 (1996) 285-291
possible to vary the wavelength L imposed on the ripple calculations other than in finite steps without altering other parameters such as D, thereby possibly introducing systematic errors. The values indicated by the points in the figures relate to the energetic optimum wavelength A and were obtained from the data for neighbouring wavelengths by polynomial interpolation. Because the ripple energies U~u~f vary quadratically with ripple wavelength at their minima, they can be estimated much more precisely in this way than can the optimum wavelengths A themselves - compare the scatter of the data in Figs. 2 and 3. It is evident from the data in Fig. 2 that the transverse ripples have lower energy than the longitudinal ripples which in turn have lower energy than the O ' H W structure. (The O ' H W structure may be regarded as constituting the long wavelength limit for both ripple structures.) The surface energy of the O ' H W structure may readily be shown to be given by the following simple formula: UsO'HW
=
- (Ks - K0)
2
/Ks.
(4)
From Eq. (5) it follows that for values of K S not far above the threshold Ko, ~/(-rrO'HW~ Vsur f ] increases almost linearly with K S. It is clear from Fig. 2 that the behaviour of ~/(-Usurf) for the ripple structures is similar, indeed rather more nearly linear. (The onset of the non-uniform magnetic state at K s = K 0 is analogous to a second-order phase transition; gsurf has exponent 2 with respect to the parameter k = ( K s Ko)/Ko.) Extrapolating the ripple data in Fig. 2 to zero Usurf yields K0T = 1.61_+0.02 e r g / c m 2 and K0L = 1.68 _+ 0.02 e r g / c m 2 for the values of the surface anisotropy thresholds for transverse and longitudinal ripples respectively; the corresponding threshold value for the O ' H W structure in this material is K 0 = 2.01 e r g / c m ; . The reason why the transverse ripples have lower energy than the longitudinal ripples is readily understood qualitatively from a comparison of their structures as shown in Fig. 1. Firstly, in the transverse case rotation of the magnetization at the boundary between ripples of opposite sign assists closure of the dipolar flux associated with the normal magnetization M z, giving rise to the vortex structures evident in Fig. la. No comparably effective closure -
289
process is available in the transition regions (i.e. where M z ~ 0) of the longitudinal ripple. Secondly, with OMy/Oy = 0, the volume pole density of the transverse ripples is free from any contribution from the principal magnetization component My. The corresponding contribution OMz/OZ is non-zero for the longitudinal ripples which, in this respect, are somewhat analogous to Ngel domain walls - see Fig. lb. In this context one should remark that while the transverse ripples are symmetric about their planes of maximum M z, the longitudinal ripples are not; this is because the magnetization in successive transition regions is alternately parallel and anti-parallel to the x-component of the magnetic field H o at the surface. The dependence of the ripple wavelength on K S is shown in Fig. 3 for both types of ripple. Because the energies of the ripple structures are insensitive to small changes in wavelength near the optimum value, we could not determine A(KS) to an accuracy better than about _+3 nm. At their thresholds, the transverse and longitudinal ripples have wavelengths A0 r ~ 123 + 3 nm and A0 L ~ 103 __ 3 nm, respectively. AT is consistently greater than AL presumably because of the better flux closure at the transverse ripple boundaries discussed in the preceding paragraph. With increasing K s the width of both types of ripple increases while the difference between their wavelengths decreases. Fig. 4 shows the dependence of the ripple amplitude ( ~ s ) m a x = ( 9 0 - 0(0)max)° on K s. For the ripple-free O ' H W structure, the corresponding amplitude ~b°'uw is given by the simple relation cos(~b°'Hw) = sin(0(0)) = K o / K S [1]. Consequently sin(~b°'Hw) = [1 - (Ko/Ks)2] 1/2, and ~b°'Hw rises abruptly at the threshold K 0. The behaviour of the amplitudes of the two ripple structures with increasing K s closely mimics that of v-s~hO'HW.In each case the abrupt rise assists estimation of the corresponding threshold values of the surface anisotropyl From the data in Fig. 4 one obtains K0T = 1 . 6 2 _ 0.02 e r g / c m 2 and K0L = 1.69 _ 0.02 e r g / c m 2, in very close agreement with the values K0T = 1.61 _ 0.02 e r g / c m 2 and K0L = 1.68 _+ 0.02 e r g / c m 2 derived from the data for U~urf in Fig. 2. We have shown that the transverse ripple structure has lower energy than either the longitudinal ripple structure or the one-dimensional O ' H W struc-
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L. Hua et al. / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
ture, but does it constitute the ground state of this simple ideal system, or does some other as yet unknown structure have still lower energy? The transverse ripple structure is a two-dimensional model with M = M ( x , z ) and removing the restriction ~ M / O y = 0 might permit us to obtain an M ( x , y , z ) with a lower U~u~f. However, to perform a three dimensional simulation of comparable precision would be extremely demanding computationally, and there are reasons for suspecting that the transverse ripple structure is the ground state. As mentioned earlier, the problem of an ultra-soft half-space with a strong surface anisotropy favouring normal magnetization is quite closely analogous to the well known problem of a thin sheet with a strong uniaxial bulk anisotropy supporting out-of-plane magnetization. This situation has been very thoroughly studied both theoretically and experimentally for many decades and it has been established convincingly that for sufficiently thin sheet the Kittel structure of parallel bar domains that span the sheet thickness minimises (for optimum domain width) the sum of the dipolar and domain wall energies when the wall energy per unit area is treated as a constant (see Ref. [7] and the work cited therein). The more complex rick-rack and flower-like surface domain patterns seen on these materials require sheet thicknesses very much greater than the exchange length and domain wall width [8,9]. The analogy with thin films is appropriate to the present problem because of the limited penetration ( ~ v/-A/Ms) of the normal magnetization into the bulk of the material. Furthermore, we have shown here that the boundaries between the ripples have lower energy per unit length in the transverse than in the longitudinal configuration: this provides an additional bias in favour of a (transverse) ripple structure with straight parallel ripple boundaries, a bias not present in the Kittel domain analogue. In this paper, for concreteness, we have adopted specific values A = 10 .6 e r g / c m and M s = 800 e m u / c m 3) for the two magnetic constants A and M S that characterise the magnetic half-space. These two constants determine the natural units of length and energy density (per unit area) for the problem at hand. They a r e respectively l o = 1 / b 0 = [ A / ( 2 7 r M 2 ) ] 1/2 and K o = Ab o. The existence and nature of the ripple structures depend on the value of the surface anisotropy (kre d = K J K o) in these re-
duced units. Experimental observations of the amplitude (~bs)m,x and wavelength A of these ripple structures on a fairly thick sample of ultra-soft material should, if the values of both A and M s are known, provide two independent estimates of the value of ,' the surface anisotropy K s by reference to the data provided here in Figs. 3 and 4, after appropriate scaling to the reduced units. Agreement between the two estimates would serve as a confirmation of the validity of our model. With many materials, however, the value of the exchange constant A is not known with much precision. In that case, the observed value of the amplitude (qSs)m,x will still enable an estimate of the reduced surface anisotropy kre d to be obtained from Fig. 4. Given this value of krea, a predicted value of the reduced ripple period /~red = h / l o will in turn be obtainable with the aid of Fig. 3 (with axes rescaled). A value of A for the sample can then be obtained from the value of the reduced length l 0 appropriate to the sample given by the ratio hobs/Ared that the observed ripple period bears to the reduced value.
4. Conclusions We have obtained the micromagnetic structures of two distinct types of magnetic ripple that are expected to occur on the surfaces of ultra-soft magnetic materials that possess a sufficiently strong surface anisotropy that locally favours magnetization normal to the surface. These ripples have wavelengths of the order of 0.1 Ixm and penetrate to a rather lesser depth in the material. In this paper only the limiting case of material much thicker than these dimensions is treated. Both ripple structures are shown to have lower energy than the one-dimensional structures predicted by O'Handley and Woods [1] and exhibit a greater component of normal magnetization at the surface. The O ' H W structure may be regarded as representing the long wavelength limit of these ripple structures. In one structure the ripple wave-vector is transverse to the bulk magnetization direction, in the other it is parallel to it. The former, transverse, structure has a lower energy and reasons are advanced that it probably constitutes the ground state. The transverse structure requires a surface anisotropy K s > 0.80K o, where K 0 = (21rA)1/2Ms is the corre-
L. Hua et aL / Journal of Magnetism and Magnetic Materials 163 (1996) 285-291
sponding threshold for the one-dimensional O'HW structure. The threshold for the longitudinal ripple structure i s 0.84K 0. Measurements of the wavelength of these ripples and of the amplitude of the normal surface magnetization they exhibit should provide a method for obtaining the magnitudes of the exchange constant of the material and also its surface anisotropy.
Acknowledgements This work was supported by grants GR/H26758 and GR/J89859 from the Engineering and Physical Sciences Research Council of the UK.
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References [1] R.C. O'Handley and J.P. Woods, Phys. Rev. B 42 (1990) 6568. [2] C. Kittel, Rev. Mod. Phys. 21 (1949) 541. [3] Lu Hua, J.E.L. Bishop and J.W. Tucker, J. Magn. Magn. Mater. 140-144 (1995) 655. [4] W.F. Brown, Jr., Micromagnetics (Wiley Interscience, New York, 1963) chap. 4. [5] A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450. [6] M.E. Schabes and H.N. Bertram, J. Appl. Phys. 64 (1988) 1347. [7] B. Kaplan and G.A. Gehring, J. Magn. Magn. Mater. 128 (1993) 111. [8] R. Szymczak, Elec. Tech. 1 (1968) 5. [9] A. Hubert, Phys. Status Solidi 24 (1967) 669.