H. RIEDEL : Micromagnetic Theory of Large-Angle Ripples 449

phys. stat. sol. (a) 24, 449 (1974)

Subject classification: 18.2.1

Institut fur Physik a m Max- Planck- Institut fur Metallforschung, Stuttgart, and Institut fur Theoretische und Angewandte Physik der Universitat Stuttgart

Micromagnetic Theory of Large-Angle Ripples and Stripe Domains in Thin Ferromagnetic Films

BY H. RIEDEL~)

I n polycrystalline ferromagnetic films the magnetization exhibits wave-like fluctuations about its mean direction, which are known as magnetization ripples. The analytical method developed previously for a micromagnet’ic treatment of Nee1 walls is extended to magnetization ripples and to the stripe domain structures, into which the ripple is trans- formed as the ripple angle increases. The theory is applied to magnetization reversal in the easy direction and to hard-axis fallback as well as to the magnetization curves for these cases.

Die Magnetisierung polykristalliner dunner ferromagnetischer Schichten weist wellen- artige Schwankungen urn die mittlere Magnetisierringsrichtung auf (“ripples”). Die in einer friiheren Arbeit zur mikromagnetischen Behandlung der Nkelwand entwickelte analytische Methode wird erweitert auf die ripple-Struktur und die StreifendomLnen- struktur, die fur wachsende ripple-Winkel aus der ripple-Struktur hervorgeht. Als An- wendungsbeispiele werden die Ummagnetisierung in der leichten Richtung und die Ent- magnetisierung nach Sattigung in der schweren Richtung untersucht und die zugehorigen Magnetisierungskurven berechnet.

1. Introduction Evaporated ferromagnetic films frequently exhibit a uniaxial anisotropy

which, together with the external magnetic field, determines the mean direction of the film magnetization [ 11. By Lorentz-deflection electron microscopy, Puller and Hale [2], and Boersch et al. [3] discovered wave-like fluctuations of the magnetization about the mean magnetization direction in polycrystalline films. These so-called magnetization ripples have their origin mainly in the random orientation of the crystallites, whose crystal anisotropies are superimposed on the uniaxial film anisotropy. The magnetization is not able to follow the local anisotropies independently in each crystallite. Exchange- and stray-field forces couple the magnetization over characteristic coupling lengths. According to the linear ripple theory worked out by Rother [4], Hoffmann [5 ,6] , and others [7 to 91, the coupling lengths are

(1) 1 - 1: - ( iy2

parallel to the mean magnetization as a consequence of the exchange energy, and

1, = En 2 1/2 I, l/E (AK,h)-lI4 (2)

1) Present address: Max-Planck-Institut fur Eisenforschung, 4 Diisseldorf, FRG.

29’

450 H. RIEDEL

Fig. 1. Top view of a thin ferromagnetic film with the definition of the angles q, rpo, 6, y

perpendicular to the mean magnetization as a consequence of the transverse magnetic stray field Hs,z [6]. Here A denotes the exchange constant, K , the energy constant of the uniaxial film anisotropy, B the film thickness, I, the spontaneous magnetization, and h the reduced field defined by

(3) I H e x

2Ku h = 2 ~ cos (6 - yo) + cos 2%

(for the angles 6 and yo see Fig. 1). H e , is the external field applied in the film plane.

Another result of the linear theory is the mean-square magnetization disper- sion, or ripple amplitude

(piz)l/z - h-312 . (4) Equation (4) shows that the prerequisite of the linear theory, the smallness of the ripple angle, is violated if the reduced field h tends towards zero.

A first attempt towards a non-linear ripple theory was made by Hoffmann [17 to 191. From the solution of the linear theory, Hoffmann calculated the longitudinal stray field Hs,%, which is of higher than first order in rp so that it does not appear in the linear theory; he suggested that the magnetization be- conies locally unstable as soon as Hs,% cancels the other contributions to the effective field, which happens first at the zeros of rp(z). As a consequence of this local instability, Hoffmann predicted a discontinuous jump of the ripple struc- ture into a “blocked” state. This state is characterized by the inability of the magnetization to rearrange itself according to the energy minimum, as the small- angle (“free”) ripple does [16] when the external field changes.

In fact, many experiments (for an early review see Feldtkeller [lo]) seem to confirm the idea of blocking:

1. When an external field, saturating the film in a hard direction, is diminish- ed, the ripple angle increases and finally the magnetization ripple turns into a stripe-domain structure with NBel walls between the domains (hard-axis fallback). The energy minimum would correspond to an infinite domain width (compare Tig. 3, Section 4), in contrast to the observed finite domain widths (5 to 50 pin) [11 to 131.

2. Also in the case of magnetization reversal in the easy direction, a stripe- domain structure originating from the ripple structure has been observed [ 141, the domain walls being orientated perpendicular to the reversing field.

3. Rotational hysteresis [el, 221 increases drastically when the stray field Hs,% becomes of the order of magnitude of the other contributions to the total effective field.

z, From a mathematical point of view, the term “local instability” makes no sense in connection with variational problems including spatial derivatives [20] (in our case coming from the exchange energy which Hoffmann erroneously disregarded in his stability con- siderations).

~ -

Micromagnetic Theory of Large-Angle Ripples 45 1

It is, however, not possible to attribute the observed blocking conclusively to a local instabilityz), as Hoffmann did. Blocking might also be caused by coercive forces preventing the small-angle NQel walls from rearranging them- selves according to the energy minimum.

Harte [9] expanded the micromagnetic torque equation up to third powers of the ripple angle y . He arrived at a non-linear integral equation for an effective field, which seems to be inaccessible to a rigorous treatment. Harte’s simple “ansatz” with three parameters, to be determined in a self-consistent manner, does not include the characteristics of NQel walls, namely the narrow cores and extended tails [23] .

In this paper, a direct solution has been attempted of the micromagnetic equation for large-angle ripple and stripe domains by the extension of an ana- lytical method developed previously for NQel walls [23] .

2. The Ripple Equation In sufficiently thin films the magnetization vector lies in the film plane and

does not depend on the coordinate perpendicular to the film. Since for large- angle ripple ( h + 0) the ratio &/ZX ((1) and ( 2 ) ) tends towards infinity, we con- fine ourselves to the one-dimensional problem, i.e., the ripple angle q depends on one variable x only. The energy of the magnetization is then given by

m

+ K, sin2 ( y - y)} dx . ( 5 ) Here K , is the energy constant of the crystal anisotropy, which is assumed to be hexagonal. Cubic crystal anisotropy may be accounted for a t the end by putting K , + 0.39 Kpbic) (Harte [9], Appendix B). Variation with respect to the mean magnetization direction qo g‘ ives us

1,HeX sin (6 - po) - K , sin 2y0 = 0 . Variation with respect to the ripple angle q yields the ripple equation

2-41” + Ig(I?ex cos (6 - yo) + Hs,z) sin q + K , cos 2y0 sin 2q =

= K , sin 2q4, (cos y - cos 2 y ) - K, cos 2 y sin 2q1 + K, sin 2y cos 2y

The stray field H s , ~ consists of a local and a non-local contribution [24] w

H s , z ( ~ ) = - 4nI, COB V ( X ) + 21, J F ( z - x’) cos F ( X ’ ) dx’ -m

with the integral kernel (D film thickness)

On the right-hand side of (7) we neglect the third term, as all previous workers have done, with the justification that its mean value vanishes (Brown [S], equations (62) to (69) in Harte’s work [9]). The first term on the right of (7), which does not appear in the linear theory, generally introduces great mathe- matical difficulties. We therefore confine the present theory to the important

452 H. RIEDEL

cases where this term vanishes:

vo = 0, mean magnetization in the easy direction sin 2v0 = 0 -+ 37

yo = - , mean magnetization in the hard direction. 2 In all other cases the ripple becomes non-symmetric with respect to

After dividing (7) by sin q~ the second term on the right-hand side of (7) , describing the statistical effective field of the crystal anisotropy is replaced by its mean value

= 0.

(11) K , sin 2~ cos 29

sin y - = 2KuG,

where the constant Cl will be obtained by averaging (11). This procedure re- produces the results of the linear ripple theory in the limit of small ripple angles, yielding G = 2h. For larger ripple angles the statistical effective field becomes negligible anyway in comparison to the other contributions to (7).

With these simplifications the ripple equation, written in terms of the vari- able

reads E ( Z ) = 1 - cosy(2) (12)

E" (1 - E ) E'2 A h--G ~ - _ _ _ _ ~ . _ E + F E - - = o . E ( 2 - E ) &2(2 - E ) 2 P

___- Lengths are given in units of tA/ ( .K!p); p = (2nIE/Ku) - cos 2y0; the non- local stray field is given in the abbreviated notation

W

--m

We search for periodical solutions E ( X ) of the integro-differential equation (13) describing the ripple structure which, for large ripple angles, turns into periodi- cally arranged NBel walls.

3. Solution of the Ripple Equation In analogy to a previously developed analytical method for NBel walls [ 2 3 ]

we separate the integro-differential equation (13) into a differential equation for the cores of the small-angle NBel walls, and into an integral equation for the tails by putting the non-local stray field, as a zeroth approximation, equal to a constant A

FE = &,I . (15) The second-order differential equation (13) with (15) may be solved by multi-

plying equation (13) with E ' , integrating once, by separation of the variables E

and 5 and tabulated integration ( [25] formula 252.000). The resulting periodic function E,(x) describes the cores of periodically arranged NBel walls

Micromagnetic Theory of Large-Angle Ripples 453

The modulus k of the Jacobian elliptic function sn is related to the constant of integration e by

with the abbreviation h - G

P E m 3 En1 - -.

In (16) 2x0 = [ ( E W - @) (1 - e)]-1/2

is the width of the NBel wall cores positioned at the zeros of E,(x). From the differential equation (( 13) with (15)) follows a = 1. Treating a as an unspecified variational parameter, however, may improve the approximate analytical solu- tion of (13). The ripple-wavelength I (i.e., the distance between next nearest neighbouring NBel walls) is given by

(K(k) = complete elliptic integral of the first kind). In the limiting case e --f 0, k --f 0, il = 4n(x,/a), equation (16) represents the trigonometric functions of the linear ripple-theory; for e -+ ~ ~ 1 2 , E + 1, Iz + cx) the resulting hyperbolic func- tion describes an isolated NBel wall core. In the intermediate range of k-values, (16) describes the transformation of the ripple structure into stripe domains separated by NBel walls.

Inserting into (13) 4%) = E c ( 4 + E t ( 4 2 (21)

where cc(x) is the prescribed function (16), one obtains a complicated integro- differential equation for the function et(s) which describes the N6el-wall tails. In this equation, as the only approximation in the present solution of (13)) we neglect the exchange terms with the justification that in the tails the magneti- zation direction changes very smoothly [23]. Then, for E J X ) a linear integral equation

A

- st + FEt - = - F^E, (22)

remains. It can be solved by a Fourier series “ansatz”, since the integral kernel F depends on the difference x - x‘ only. With

4nnx A

m

E c ( 4 = 2 EC,% cos ~

n=o we get ,. M 4nnx

FE, = E ~ , ~ F, cos - n=O il

454 H. RIEDEL

with the tabulated [26] Fourier transform of F (9)

2~12, 1 1 - e-4nnW F K , p 4nnD 7 k -

Thus the solution of (22) is

Together with E,(x) (16), equation (25) represents the solution of (13). The parameters E ~ Z , G, e , a and the Fourier coefficients E ~ , % will be determined in Section 4.

4. Evaluation and Discussion E~~ follows from the self-consistency condition

i?E\z=O = En1

= 0 Y

or equivalently (see (22)) from the boundary condition

which must be solved numerically. The Fourier coefficients cc,% defined by

(27b) h n x dx (n = 1, 2, 3 . . .)

0

may be obtained by applying the addition formula for Jacobian elliptic func- tions [25]

(28) sn u cn (iv) dn (iv)

1 - k2 sn2 (iv) sn2 u * sn (u + iw) + sn (u - iv) =

By putting

(29) e - k2 sn2 (iv) = -___ 1 - e

the function E, (16) may be written in the form

sn u [sn (u + iv) + sn (u - iw)] . (30)

Inversion of (29) g' ives us w = F (arcsin ]/Ern - e, k ' )

with k'2 = 1 - k2, F being the incomplete elliptic integral of the first kind. Inserting (30) into (27), expanding the sn-functions into Fourier series [25],

Micromagnetic Theory of Large-Angle Ripples 455

and applying addition formulae and orthogonality realtions of trigonometric functions, we finally arrive at

(32 a )

12 = 1,2, 3 . . . (32b)

(K' = K(k')). Alternatively, cc,0 may be expressed in terms of Heuman's func- tion A, [27] by direct integration of (27a) [25 ]

For E ~ , ~ no closed representation in terms of tabulated special functions could be found. Because the sums (32) converge poorly for k -+ 1 (large angle ripple), this important limiting case is considered separately: if the core width of the NQel walls is small compared with the distance between the walls, one may replace the exact funtion E,(x) by distinctly separated wall cores

(34) 03 Ew(2 - Ew)

EC(X) =Em - 2 m = -03

with the tabulated [26] Fourier coefficients

4 AU

~ , , ~ = ~ ~ - - a r c c o s ( 1 - c W ) ,

I. (35b)

4nnx, sinh __ arccos (1 - E ~ )

h n x 0 837 [ l a ?La

sinh [r n] E,,n = - -

The parameters p , a, and G must be determined in a different manner for the two cases:

1. The ripple structure is able to attain the new energy minimum freely, when the external field changes. This case applies for small-angle ripple (<lo), and will be treated in Section 4.1.

2. If the ripple structure with increasing ripple angle is transformed into a stripe-domain structure, the wavelength is blocked, e.g., by coercive forces acting on the NQel wall cores between the stripe domains (Section 4.2).

456 H. RIEDEL

4.1 Free ripple

By minimizing the energy per unit volume (in units of Kup)

the parameter a and the constant of integration e, which determine the wave- length, are derived. If the exchange term of (36 ) is expanded to the first power of ctr as in [23 ] , the energy may be evaluated:

+ pp -- E i l + 2(a2 - 1) ~ ' E n l E r n - cos 2y0 cos 2y0 (37)

In the case of free ripples, the parameter G is determined by averaging (11) over areas, whose magnetization is coupled by the exchange energy and the transverse stray field Hs, z:

KZ sin 2y2 = 8K:GSZ. (38) For small cry~tallites~) (i.e., small compared with Zz), the average of the crystal- lite orientation is

where (T is the geometric mean value of sin 27y, being (T = 2/1/15for an isotropic distribution of the easy crystalline axes in space, and 0 = 11fi if the easy axes are confined to the film plane [17].

is the number of crystallites contributing to the average, with the crystallite volume VC and the transverse coupling length I , given by ( 2 ) .

With the mean value of E(X)

s, For other cases, which have also been observed [29], see [S] .

t loo:

1-5

Fig. 2. Maximum ripple amplitude for which the linear ripple theory may be applied vs. film thickness (in units of v A / ( K , p ) & 50 A

in permalloy), I ; / K , = 640, h = 1 --L

3 Y -

Fig. 2 shows the ripple angle for which the longitudinal stray field resulting from the linear theory,

(43) 2K, 1 -

Hs,.(O) = - -p ,vYF, - E;) Y

I€! cancels the single domainfield (2KU/IB) h. This ripple angle may be considered as a limiting angle for the validity of the linear theory. For thin films (4nD/A < < l), it is given by

1

l o - l 1 1 1 1 1 1 1 ~ I I I I I

An instability as claimed by Hoffmann [17 to 191 does not occur. The ripple wavelength, the ripple amplitude, and the relative Ndel wall core

width W = 4nx,/ail (Fig. 3 to 5) have been computed numerically from (26), (32), (37), and (42). For small ripple angles the present theory reproduces the

h - h -> Fig. 3. Ripple wavelength vs. reduced field ac- cording to the non-linear (solid line) and linear (dashed line) theory. D = 10 (L500 A in per- malloy), I:/& = 640, S = (K, /K,) (7$/4/fi) x

xaVCD-3/4p--7/s = 0.007

Fig. 4. Ripple amplitude vs. reduc- ed field according t o the linear (dash- ed line) and nonlinear theory (solid line). For film parameters see Fig. 3

Fig. 5. Relative wall core width W = 4nz0/ail vs. reduced field. The linear theory yields W = 1. For 0.1

film parameters see Fig. 3 Oio-4 103 10’2 10-1 i h-

458 H. RIEDEL

results of the linear theory. In contrast to the linear theory, the non-linear theory yields a finite ripple amplitude for h + 0 (Fig. 4). The wavelength of free ripples increases infinitely for h -+ 0, but more slowly than predicted by the linear theory (Fig. 3). In accordance with Suzuki’s observations [15], the wall core contraction (given by W , Fig. 5) takes place already a t ripple angles as small as 1”.

4.2 Blocked state

For large-angle ripples the wavelength is no longer determined by the energy minimum, but may be kept constant. In this case, the magnetization cannot respond to the statistical torques originating in the local crystal anisotropies, so we may put G = 0.

In the following, one must keep in mind that the tails of isolated NBel walls have a length of several pD (A 200pm in 500 A-thick Permalloy films) [28]. This exceeds typical stripe domain widths, which may result from hard-axis fallback (5 to 50 pm). The crucial effect is therefore the overlapping of the tails of the periodically arranged NBel walls.

Fig. 6 shows magnetization curves as determined by numerical solution of (26) and (32). The magnetization component along the external field is given by

M = 1,(1 - E ) . (45) As demonstrated in Fig. 6 the existence of “inverted” films4) (coercive field H , > 2K,/Z,) is a natural consequence of the overlapping NQel wall tails in- cluded in the present theory. For large negative external fields the magneti- zation curves given in Fig. 6 are not realistic, because Bloch lines will nucleate

Fig. 6. Magnetization curves in the hard direction for a) 1. = 4000 and b) I , = 20000, and in the easy direction for c) il = 4000 and d) 1 = 10000. For IsJHeYJ/2K1, > hmax the

boundary condition (26) has no solutions. For film parameters see Fig. 3

4, A number of authors, e.g. [30], have attributed the inversion of films to the existence of regions with a negative anisotropy constant, K,.

Micromagnetic Theory of Large- Angle Ripples 259

t ‘7 4

Fig. 7 il-

Fig. 8

Pig. 7. Remanent magnetization Mr of the hard-axis magnetization curves as a function of the wavelength A (= 2 x domain width)

Fig. 8 Energy per unit volume (in units of K ~ L , left scale) and energy per domain wall (in units of 1/AK,p, right scale) of a stripe domain structure with domain width A/2. The zero-point of the energy is defined by the state when the magnetization is aligned parallel to the easy axis. For ,I + 00 the energy of an isolated wall is reproduced (arrow). cos 2p, =

-. .

= -1 (hard-axis fallback), h = -1 ( H e x = 0) , D = 10

and resolve the energetically unfavourable stripe-domain structure [ 131 so that the magnetization is finally aligned parallel to the external field.

The reinanent magnetization (Fig. 7) and the energy (Fig. 8) decrease ap- proximately according to a logarithmic law for large, but not too large, wave- lengths reflecting the logarithmic decrease previously derived for the magneti- zation direction cosine in NBel-wall tails [23].

In all cases considered, the variational parameter a (16) deviates only little from unity (1 < n < l.l), which confirms the validity of the present method for solving the micromagnetic equation.

6 . Conclusions

1. If the ripple angle exceeds a few degrees, the linear ripple theory is not valid.

2. The increasing longitudinal stray field, which does not appear in the linear theory, continuously transforms the wave-like ripple structure into a stripe- domain structure with small-angle Nkel walls between the domains. In contrast to Hoffmann’s non-linear ripple theory [17 to 191, the present theory predicts no instability during the formation of stripe domains.

3. Macroscopic magnetic properties of thin films, e.g., magnetization curves are greatly influenced by the periodically arranged NBel walls formed out of the ripple structure. The present analytical method allows for the overlapping of NBel walls in an almost exact manner.

Acknowledgements

The author acknowledges valuable suggestions by Prof. Dr. A. Seeger, and Prof. Dr. H. Kronmuller’s interest in this work.

460 H. RIEDEL : Micromagnetic Theory of Large-Angle Ripples

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(Received April 24, 1974)