Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

Fig. 11.30 (a) Cross section of 1808 Bloch wall in thin film. (b) Elliptic cylinder.

where C ¼ 4p (cgs) or 1 (SI). The magnetostatic energy density of the wall is then

This must be multiplied by d to obtain the magnetostatic energy per unit area of wall in the yz plane:

C d2M2

gms,B ¼ þ ds : (11:48) 2 t

This energy is negligible when t/d is large, as in bulk specimens, but not when it is of the order of unity or less.

When the film thickness t is small, the magnetostatic energy of the wall can be reduced if the spins in the wall rotate, not about the wall normal x, but about the film normal z. The result is a Ne´el wall. Free poles are then formed, not on the film surface, but on the wall surface, and spins everywhere in the film, both within the domains and within the walls, are parallel to the film surface (Fig. 11.31).

If we again approximate the wall by an elliptic cylinder, its magnetostatic energy per unit area is

C tdM2

gms,N ¼ þ ds : (11:49) 2 t

402 FINE PARTICLES AND THIN FILMS

Fig. 11.31 Structure of Ne´el wall. (a) Section parallel to film surface. (b) Cross section of wall.

The ratio of the magnetostatic energies of the two kinds of wall is then

The magnetostatic energy of a Ne´el wall is less than that of a Bloch wall when the film thickness t becomes less than the wall thickness d. This relation is inexact, not only because of the approximations involved in its derivation, but also because d varies with film thickness. And to know with confidence which kind of wall is more stable at a given thickness, one must calculate the total wall energy g, which contains magnetostatic, exchange, and anisotropy terms. The result of such a calculation is shown in Fig. 11.32, which has been carried out with the values appropriate to 80 permalloy. We see that the total energy of a Ne´el wall, as well as the magnetostatic energy, is less that of a

˚

Bloch wall when the film is very thin, less than about 500 A. (Actually, for thicknesses of a few hundred angstroms, the total energy of either kind of wall is almost entirely magnetostatic.) The widths of Bloch and Ne´el walls also vary in different ways with film thickness: the thinner the film, the narrower the Bloch wall and the wider the Ne´el wall.

The second new kind of wall observed in thin films is the cross-tie wall. It consists of a special kind of Ne´el wall, crossed at regular intervals by Ne´el wall segments. As shown in Fig. 11.32, its energy is less than that of a Bloch wall or a Ne´el wall in a certain range of film thickness; the cross-tie wall therefore constitutes a transition form between the Bloch walls of thick films and the Ne´el walls of very thin films. Figure 11.33 shows the appearance of a cross-tie wall, and Fig. 11.34b its structure. In Fig. 11.34a a Ne´el wall is shown, separating two oppositely magnetized domains. It is not a normal Ne´el wall because it consists of segments of opposite polarity; these have formed in an attempt to mix the north and south poles on the wall surface more intimately and thus reduce magnetostatic energy. The regions within the wall where the polarity changes, marked with small circles, and where the magnetization is normal to the film surface, are called Bloch lines. However, this hypothetical wall would have very large energy, because the fields due to

404 FINE PARTICLES AND THIN FILMS

Fig. 11.34 Sections parallel to film surface of (a) hypothetical Ne´el wall with sections of opposite polarity, and (b) cross-tie wall.

the poles on the wall, sometimes called stray fields, are antiparallel to the domain magnetization in the regions opposite the Bloch lines marked A. As a result, spike walls form in these regions, as shown in Fig. 11.34b, and the stray fields close in a clockwise direction between the cross ties.

By proper manipulation of a mask during the deposition of an evaporated film, a film can be deposited whose its thickness varies continuously from one end to the other. The Bitter pattern of such a film is shown in Fig. 11.35. At the thin end on the left the walls are of the Ne´el type, and as the thickness increases, cross-tie and then Bloch walls appear. The film thicknesses at which transitions occur from one type to another are in good agreement with the curves of Fig. 11.32. (Ne´el walls are more easily visible than Bloch walls, which sometimes appear only as a series of dots. This is due to the different modes of flux closure

Fig. 11.35 Domain walls in a tapered film of 80 permalloy. Easy axis horizontal. Dark-field Bitter pattern. [S. Methfessel et al., J. Appl. Phys., 31 (1960) p. 302S.]

where each type of wall intersects the film surface. At the edge of a Ne´el wall the stray fields are parallel to the film surface, and these fields attract chains of colloid particles, transverse to the wall, which close the flux. The stray fields at the edge of a Bloch wall are normal to the film surface, have a larger spatial extent, and attract the colloid less strongly.)

11.12DOMAINS IN FILMS

One does not expect a polycrystalline specimen to be a single domain in zero field. It would normally break up into domains arranged in conformity with the easy-axis direction in each grain. A thin film of 80 permalloy, however, if saturated in the easy direction, can remain a single domain after the saturating field is removed, as shown in Fig. 11.36a. The reason is that the demagnetizing field in the plane of the film is nearly zero, because the film is so thin; in addition, the grain size is very small. If each grain became a domain, large numbers of free poles would form at grain boundaries, and the magnetostatic energy would be high.

If a small reverse field is applied, reverse domains are nucleated at the ends of the film, as in Fig. 11.36b. It is also common to see small reverse domains in the remanent state (H ¼ 0), nucleated by demagnetizing fields due to imperfections at the film edges; these fields need not be large, because this alloy is magnetically so soft that a reverse field of only 2 or 3 oersteds (160 or 240 A/m), applied along the easy axis, is enough to reverse the magnetization completely.

When the film is demagnetized by an alternating easy-axis field of decreasing amplitude, the resulting domain structure is shown in Fig. 11.36c. This structure is typical. It consists of elongated domains more or less parallel to the easy axis, bounded by gently curved,

406 FINE PARTICLES AND THIN FILMS

Fig. 11.36 Domains in an 81 permalloy film observed by the Kerr effect. The film is 8 mm in diameter,

˚ ¼

2000 A thick, and the easy axis is vertical. (a) Single-domain remanent state, H 0. (b) Domains nucleated at edges by a reverse easy-axis field of 0.2Hc. (c) After demagnetizing in a 60 Hz alternating field of decreasing amplitude. (d) After demagnetizing by a transverse field equal to 1.5 HK or more. [Courtesy of R. W. Olmen, Sperry Rand Univac Division, St. Paul, Minnesota.]

rather than straight, 1808 walls. The domain wall thickness is generally larger than the grain size. A film can also be demagnetized by saturating it in a hard direction, in the plane of the film and at right angles to the easy axis, and then removing the saturating field. The film then breaks up into the much narrower domains shown in Fig. 11.36d. The width of these domains varies from about 2 to 50 mm from one film to another.

This structure (Fig. 11.36d) is due to anisotropy dispersion. By this is meant a variation, from place to place in the film, of the direction of the easy axis and/or the magnitude of the anisotropy constant, because of inhomogeneities in the structure of the film. As a result the direction of the local magnetization Ms varies slightly from one point to another even within a domain. The nonparallelism of the Ms vectors adds exchange energy to the system; in addition, free poles are created within the domain because of the divergence of M (Section 2.6), causing stray fields and magnetostatic energy. In order to minimize this exchange and magnetostatic energy, the Ms direction varies in a wavelike manner, called

408 FINE PARTICLES AND THIN FILMS

magnetization ripple, rather than randomly, even though the film inhomogeneities which cause the effect are themselves randomly distributed. This ripple is illustrated in Fig. 11.37a, where the film is shown in the single-domain remanent state after saturation in an easy direction. When a field is now applied in a hard direction, as in Fig. 11.37b, the ripple still persists, and its relation to the local easy-axis directions is indicated. When the hard-axis field is removed, half of the Ms vectors rotate clockwise toward the local easy axis and half counterclockwise, forming the domain structure of the demagnetized state shown in Fig. 11.37c. The vectors indicated in Fig. 11.37b and c apply only to a narrow horizontal strip; the ripple persists in the vertical direction in each domain. The easy-axis dispersion is much exaggerated in these drawings; it is typically only 1 or 28.

Magnetization ripple is evidenced in transmission electron micrographs by fine striations which are everywhere normal to the local direction of Ms, as in Fig. 11.38. The easy axis is roughly vertical in this photograph, and three 1808 walls (two light and one dark) run from top to bottom. Cross ties are also visible.

PROBLEMS

11.1Show that the ratio of the energy barriers DEms that must be overcome in reversing the moments of a two-sphere chain from u ¼ 0 to 1808 by fanning and by incoherent

rotation is

DEfanning ¼ m2 a3 ¼ 1 :

DEcoherent 3m2=a3 3

11.2For a prolate spheroid of axial ratio c/a ¼ 5, plot hci vs D/D0 for coherent rotation

and for curling, and find the critical value of D/D0 that separates the two modes. Assume k ¼ 1.2.

11.3Consider single crystal of iron, in the form of a perfect sphere 1.0 cm in diameter,

initially magnetized to saturation in an easy direction. Find the values of Hd and of 2K/Ms

a.Will the sphere spontaneously demagnetize by forming multiple domains?

b.Let the magnetized sphere magically grow along the axis of magnetization, becoming a prolate ellipsoid, keeping the same diameter. Is there a length at which the sample will stayed magnetized, i.e., not break up into domains? If so, find this length.

11.4Calculate the critical diameter Dp of iron particles for superparamagnetic behavior at room temperature, if the particles are (a) spheres or (b) prolate spheroids with an axial ratio of 1.5. Take the diameter of the spheroids as the length of the minor axis.

11.5Find the blocking temperature TB for iron particles 440 nm in diameter for the shapes

(a) and (b) of the previous problem.

11.6Derive Equation 11.34.

11.7Calculate the values of D/Dp corresponding to the following values of reduced magnetization Mr/Mi: 0.50, 0.90, and 0.99.

410 MAGNETIZATION DYNAMICS

Fig. 12.1 Eddy currents in a rod.

When the current iw is increasing, the direction of e is such as to set up an eddy current iec in the circular path shown. The direction of e and iec is known from Lenz’s law, which states that the direction of the induced emf is such as to oppose the cause producing it. Thus, e and iec are antiparallel to iw, when iw is increasing. Correspondingly, the field Hec due to the eddy current is antiparallel to the field Ha due to iw. When the current iw is in the same direction as shown but decreasing, as when the switch is opened, then iec and Hec reverse directions because they now try to maintain Ha at its former value. The magnitude of the emf acting around a circular path of radius r is given by

where A ¼ pr2 is the cross-sectional area of the rod within the path considered (cm2 or m2), B is the induction (gauss or tesla), f the flux (maxwells or webers), and t the time (sec). Several points should be noted:

1.This emf will be induced in any material, magnetic or not.

2.For a given dHa/dt, the induced emf will be larger, the larger the permeability m, because e depends on dB/dt and B ¼ mH. Thus the eddy-current effect is much stronger for magnetic materials, with m values of several hundred or thousand, than for nonmagnetics with m 1.

3.For a given dB/dt and e, the eddy currents will be larger, the lower the electrical resistivity r of the material. In ferrites, which are practically insulators, the eddycurrent effect is virtually absent.

In Fig. 12.1 only one ring of eddy current is shown. Actually, circular eddy currents are flowing all over the cross section of the rod in a series of concentric rings. Inside each ring the eddy current produces a field Hec in the 2z direction and outside it in the þz direction. It follows that the eddy-current field is strongest at the center of the rod, where the contributions of all the current rings add, and that it becomes weaker toward the surface. The variation of Hec across the bar diameter at one particular instant is sketched in Fig. 12.2a. Because the true field H acting in the material is equal to the vector sum