ON THE CALCULATION OF MAGNETOSTATIC ENERGIES ASSOCIATED TO DOMAIN STRUCTURES IN FERROMAGNETIC THIN FILMS

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ON THE CALCULATION OF MAGNETOSTATIC ENERGIES ASSOCIATED TO DOMAIN

STRUCTURES IN FERROMAGNETIC THIN FILMS

A. Corciovei, Gh. Adam

To cite this version:

A. Corciovei, Gh. Adam. ON THE CALCULATION OF MAGNETOSTATIC ENERGIES ASSO- CIATED TO DOMAIN STRUCTURES IN FERROMAGNETIC THIN FILMS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-408-C1-409. �10.1051/jphyscol:19711143�. �jpa-00213962�

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JOURNAL DE PHYSIQUE Colloque C 1, supplément au rfi 2-3, Tome 32, Février-Mars 1971, page C l - 408

ON THE CALCULATION OF MAGNETO STATIC ENERGIES ASSOCIATED TO DOMAIN STRUCTURES IN FERROMAGNETIC THIN FILMS

by A. CORCIOVEI and GH. ADAM

Institute for Atomic Physics, POB 35, Bucharest, Romania

Résumé. — En supposant une dépendance périodique de l'aimantation des coordonnées x et y, dans le plan de la couche mince, la dépendance par rapport à z restant arbitraire, on obtient une formule générale pour l'énergie magnéto- statique, en fonction de coefficients de Fourier de l'aimantation. Par un choix particulier de la dépendance par rapport à z, on dérive une formule pour l'énergie de désaimantation des structures de domaines bidimensionnelles. D'autres sim- plifications permettent la déduction d'une formule simple applicable à toutes les structures de domaines unidimensionnelles.

Abstract. — Assuming a periodical dependence of the magnetization vector along the x, y axes in the film plane, and no restrictions with respect to its dependence on the z normal coordinate, a general formula for the magnetostatic energy of a thin film is obtained in terms of the Fourier coefficients of the magnetization. By choosing a particular z-dependence, a formula for the demagnetizing energies of bidimensional domain structures is derived. Further simplifications permit the deriving of a simple formula applicable to all unidimensional domain structures.

The finding of an adequate expression for the magnetostatic energy associated to a ferromagnetic specimen is, perhaps, the most difficult problem to be solved, in order to use successfully the general procedure of the total free energy minimization, which gets informations about the magnetization direction inside the specimen. In the frame of domain theory, Kittel [1], and subsequently Goodenough [2], have developed a method for the calculation of the magne- tostatic energies associated to planar distributions of magnetic poles. The results obtained permitted the investigation of a wide class of domain structures.

Particularly fruitful results have been obtained in the investigation of unidimensional domain structures in thin ferromagnetic films with a normal, or an inclined easy axis [3], [7]. Two remarks are, howe- ver, to be made. Firstly, the assumption of a planar distribution of magnetic poles is equivalent to a priori restrictions on the dependence on the coordinate normal to the film plane of the magnetization direction. Secondly, the procedure followed [4], [7], is groundless long : for each new domain structure a tedious calculation of the corresponding magnetostatic potential as well as other manipulations are needed.

In this paper a general formula for the magnetostatic energy of a thin film is obtained in terms of the Fourier coefficients of the magnetization only, without imposing any artificial restrictions on the dependence of the magnetization direction on the normal coordinate. Therefrom, by some constraints upon this dependence, the original result of Good- enough [2] is refound. Further, in the case of very thin specimens the first approximation gives an expres- sion corresponding to a uniform magnetization. Finally, a simple formula applicable to all possible unidimen- sional domain configurations is found in terms of the Fourier coefficients of the magnetization (easily found once a given domain structure is assumed), and in this way the needlessness of the aforementioned calculations is proved.

Let a ferromagnetic thin specimen, rectangular in shape, of a volume V. A rectangular coordinate system is chosen with the origin in the centre of the film, so that this is limited by | x \ < lx, \ y | ^ ly,

| z | < d, (d <^ l^x, l^y) and the (x, v) plane, named hereafter the film plane, is parallel to the film surfaces.

Following Kittel [1], the density of the magnetostatic energy per unit surface of the film, associated with an arbitrary distribution of the magnetization (*) is

*"«... = - (8 lx ly) -x \ M(r) Hm(r) dv , (1) where Hm(r) denotes the intensity of the magnetostatic field created by the volume and surface poles of the magnetization. The magnetostatic field is to be determined from the magnetostatic Maxwell equations with appropriate boundary conditions (Harte [8]).

From Maxwell equations it follows that Hm(r) may be expressed as — grad U(t), where the magnetostatic potential U(r) satisfies a Poisson equation :

A£/(r) = 4 ?rdiv M(r) = 4 np(r). (2) The boundary conditions require the vanishing of the z component of Hm when z -* ± oo. On the z = + d planes the continuity for the tangential component of Hm and for the normal component of the induction B, are imposed. As concerns the continuity conditions along x and y axes, the usual cyclic conditions are imposed. Then M(r) may be expanded in Fourier series :

M(r) = £ M^z) eikr<, | z | < d (3a)

k

= 0 , \z\> d (3b) where rx denotes a vector in the film plane, and

k = n(njlx, ny/ly, 0), nx, ny = 0, + 1, + 2, ... (4) The Fourier coefficients of the volume poles density, p(r), (see eq. (2)), are

pk(z) = iz Mk(z) + ikMt(z) , (5) where prima indicates the derivation with respect

to z, and iz denotes the unit vector along the z-axis.

The magnetostatic potential may be also expanded in Fourier series [8]. Denoting by Uk(z) the Fourier coefficient in the region — d < z < d, eq. (2) provides us with the solution :

(]) The magnetization vector, M(r), is considered to be of constant magnitude everywhere in the film, and to vary only in direction.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711143

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ON THE CALCULATION OF MAGNETOSTATIC ENERGIES ASSOCIATED TO DOMAIN STRUCTURES C 1 - ⁴⁰⁹

2 n where k, ⁼k/k. If it is supposed that M(r) is directed Uk(z) = - - ^X

k along the normal of the film, then F, = 0, and the

d demagnetizing energy per unit surface of the film

X

( 1

^p&) e-klz-cl d[ - iz Mk([) e-klz-ci is given by eq. (8a). This is twice the expression used

- d

1

^-

^]

by Goodenough [2] in rick-rack patterns investigations.

The factor two appears because both surfaces of k = l k l . ₍₆₎ the film have been taken into account.

Now, substituting H, by - grad U in eq. (l), In the case of very small thicknesses, since the essen- F,.,. may be written as the sum of surface term Fs tial contributions to sums are given by the terms and a volume term F,. Using the Fourier develop- with small k, the exponential may be expanded in ments of M, U, and p, and performing the integrations power series and only the two first terms retained.

over X and y one obtains : Then we can take F, = 0, and by the theorem of closure we obtain

Thus the magnetostatic energy is obtained as a func- tion of the Fourier coefficients of the magnetization only. Once the magnetization distribution is known, eq. (5), (7) permit a straightforward calculation of the corresponding magnetostatic energy. As the knowledge of t h e true magnetization distribution is an unsolved problem in general, in the frame of pheno- menological theories some ^Gnatural w assumptions about the magnetization distribution inside the thin film are made. So, in micromagnetic ripple theory [g], and in the frame of domain theory, the independence of the magnetization direction on the coordinate normal to the film plane is assumed. Then Mk(z) is a constant, M, say, pk = ikMk, and Uk(z) is easily calculated from eq. (6), so that

X - 1 (i, M,) (i, M-,) (1 - e -k.2d) , (8a) k

- 2 n C ^-1 (ko Mk) (k, M-,) (1 - e-k.2d) , (8b)

k # O k

which is exactly the demagnetizing energy of a uni- formly magnetized very thin specimen.

Let us return now to eq. (8). If the periodicity along the y-direction is supressed (i. e. M(r) depends only on ^X),a unidimensional domain structure is obtained.

Let D be the domain width (half of the wavelength along the X-direction). Then k = (nn/D, 0, 0). If M(r) has a constant component along x-axis, then the F, term vanishes. Moreover, if the sample is in a demagnetized state, i, M, = 0, so that the demagnetizing energy per unit area of the film associated to any unidimensional structure is given by :

" 1

F,.,. = 4 D

C

^-(i, M,) (i, M-,) ^X

n = l n

x [l - e x -

1 .

⁽¹⁰⁾

Therefore the knowledge of the dependence of M, ⁼i z M on the coordinate normal to domains in the film plane permits a straightforward calculation of the Fourier coefficients, and then the demagnetizing energy is immediately found from eq. (10). Particularly, the demagnetizing energies associated to open domain structures (Kittel [3], MAlek and Kamberskf [4], Jacubovics [5]), or to stripe domain structures (Saito et al. [6], Sukiennicki [7]) have been determined [9]

in agreement with the original results reported by these authors.

References

[l] KITTEL (C.), Revs. Mod. Phys., 1949, 21, 541. [6] SA~TO (N.), FUJIWARA (H.) and SUGITA (Y.), J. Phys.

r2] GOODENOUGH (J. B.), Phys. Rev., 1956, 102, 356. Soc. Japan, 1964, 19, 1116.

[3] KITTEL (C.), Phys. Rev., 1946, 70, 965. [7] SUKIENNICKI (A.), Phys. Stat. Sol., 1968, 29, 417.

[4] MALEK (Z.), and KAMBERSKY (V.), Czech. J. Phys., [8] HARTE (K. J.), J. Appl. Phys., 1968, ^39,1503.

1958, 8, 416. [9] ADAM (G. H.), Studii Cerc. Fiz. (in press).

[5] JACUBOVICS (J. P.), Phil. Mag., 1966, 14, 881.