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Санкт-Петербургский государственный электротехнический университет "ЛЭТИ"

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

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			9.2 DOMAIN WALL STRUCTURE	279
and so
df		2
A			¼ g(f):	(9:11)
	dx

Comparing this with Equation 9.5, we see that the exchange and anisotropy energies are equal everywhere in the wall. This means that where the anisotropy energy is the highest, which is where the magnetization points in a hard direction, the rate of change of magnetization angle @f/@x is the greatest.

From Equation 9.11,

df	¼			A
df		pg(f)
dx			p
	¼	p			pg(f)
dx				A			(9:12)
dx				A			(9:12)
						df	:
						df

x ¼ pA ð pg(f)

This is the general equation relating x and w in the domain wall. Note that it appears as x ¼ f(f), not f ¼ f(x).

The simplest case is a 1808 domain wall in a material with uniaxial anisotropy, where
g(f) ¼ Ku sin2f, p[g(f)] ¼ pKu sin f,				and		the	local magnetization		rotates through
1808, from 0 to p. Therefore			r
	x		r			df :			(9:13)
				A
		¼			ð
		¼		Ku	ð	sin f
This is a standard integral, listed in handbooks; the result is
x	¼	r		ln		tan		,	(9:14)
x		r		ln		tan	2	,	(9:14)
			Ku				2
			A				f

which gives the relationship between x and f through the domain wall. It is shown graphically in Fig. 9.3.

The thickness of the domain wall is formally inﬁnite. An effective wall thickness can be

deﬁned as the thickness of a wall with a constant value of df/dx equal to that at the center p

of the wall. For uniaxial anisotropy, the slope df/dx has its maximum value (A/Ku) at the center of the wall, so the effective wall thickness is

(9:15)

as shown by the dashed line in Fig. 9.3.

Turning now to the calculation of the wall energy, we recall Equation 9.5:

swall

sex

g(f)#dx:

(9:5)

280 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.3 Variation of magnetization direction through a 1808 domain wall. Dashed line shows deﬁnition of wall width.

Since the two terms are everywhere equal, we can rewrite this as

swall

¼ ð1

2 g(f) dx

(9:16)

and from Equation 9.12,

pg(f) :

g(f)

s180 ¼ 2pA

df ¼ 2pA ð

g(f) df:

(9:17)

pg(f)

For uniaxial anisotropy, p[g(f)] ¼ pKu sin f, giving

suniaxial180 ¼ 2

AKu ð

AKuj cos fj0p

sin f df ¼ 2

(9:18)

AKu:

For cubic anisotropy described by a single positive anisotropy constant K1 . 0, g(f) ¼ K1 sin2 f cos2 f if the domain wall lies in a f100g plane. Then

p	¼		¼	p
g(f)		pK1 sin f cos f			K1	sin 2f:	(9:19)
g(f)		pK1 sin f cos f		2		sin 2f:	(9:19)
				2

However, a complication arises when we attempt to calculate the domain wall thickness and energy of a 1808 wall in this case. There is an easy direction halfway through the wall, where f ¼ p/2, as indicated in Fig. 9.4. We might expect the 1808 wall to separate into two 908 walls, with a new domain between them. This does not in fact occur, because of

			9.2 DOMAIN WALL STRUCTURE								281

Fig. 9.4 Splitting of a 1808 wall into two 908 walls.

magnetostriction. If the original domains, separated by the 1808 wall, were magnetized

¯	f001g plane, they would be elongated along the
along [100] and [100] directions in a

[100] easy axis. The new domain would be magnetized and elongated along a [010] direction. The misﬁt would create local elastic strains, which would increase the energy associated with the wall. The magnitude of the energy increase will depend primarily on the magnetostriction constants and the elastic constants of the material, and to some extent on the surrounding domain conﬁguration. (B. A. Lilley Phil. Mag., 41 (1950) p. 792] has made this calculation under some simplifying assumptions, and has calculated domain wall energies and thicknesses for various additional cases, including cubic Ku , 0.

A simple approach is to calculate the wall energy and thickness for a 908 wall in a cubic material, and double these values for the 1808 wall. The calculated values will both be too

low, but not by a large factor.

Proceeding this way, Equation 9.13 becomes

ð pg(f) ¼

2pK1 sin 2f ¼

ð sin 2f

ln (tan f)

(9:20)

which clearly has the same form as Equations 9.13 and 9.14.

The domain wall energy is given by

scubic90 ¼ 2pA ð

p=2

g(f) df ¼ 2

AK1

p=2

sin 2f df

¼ 2 AK1:

(9:21)

The energy of the 1808 is just twice that of the 908 wall, so

cubic

s180

AK1:

(9:22)

Note that for any domain wall, the wall thickness d is proportional to (A/K) and the wall p

energy s is proportional to (AK).

To calculate numerical values of domain wall energy and thickness, we must have values for the exchange constant A and the anisotropy constant K. Values of K are reasonably well

282 DOMAINS AND THE MAGNETIZATION PROCESS

known, but A ¼ (nJS2)/a presents some difﬁculty. The value of A is directly proportional to the exchange constant J, which is not directly measurable. It is usually estimated from the Curie temperature as J 0.3kTc, as noted in Section 4.3. It can also be estimated from the

variation of the saturation magnetization with temperature at low temperatures. For iron, J 4 10214 erg, and so

nJS2

2 4 10 14 8 21 2

10 7 erg=cm

10 12

J=m:

(9:23)

2:87

The wall energy is then

180

scubic

4:5

2 erg=cm

2 10

J=m

(9:24)

The thickness of a 908 wall in iron is

dcubic90

4 10 6cm

40 nm

400 A

(9:25)

which is about 150 atom diameters. The average angle Df between adjacent spins is therefore 908/150 or 0.68, much smaller than the value of 308 shown in Fig. 9.2. The thickness

8 ˚

of a 180 wall will be somewhat more than twice 400 A, because of the tendency to separate into two 908 walls as in Fig. 9.4; the degree of separation will depend on the value of the magnetostriction constants as well as on the elastic constants of the material, just as the wall energy does.

Most ferromagnetic metals have domain wall energies of a few erg/cm2 and wall thicknesses of a few hundred atom diameters. This thickness means that domain walls will not interact strongly with point defects, such as vacancies or single interstitial atoms.

Two types of 908 wall are possible, and they have different structures. One kind is shown in Fig. 9.5a; the spins in the adjoining domains are parallel to the wall, which therefore has a structure identical with half of a 1808 wall. In the other kind, represented in Fig. 9.5b, the spins in the domains are at 458 to the wall. The structure of this kind is harder to visualize: the spins rotate through the wall in such a way that they make a constant angle of 458 both with the wall normal and the wall surface.

The energy of a domain wall also depends on the orientation of the wall in the crystal. This point is discussed brieﬂy in Section 9.4.

Fig. 9.5 (a) and (b) Two types of 908 wall.

9.2 DOMAIN WALL STRUCTURE

283

In materials with K1 , 0, such as nickel, the easy directions are of the form k111l. The domain wall that separates domains with magnetizations in opposite directions along a k111l axis is then a 1808 wall. Instead of 908 walls, however, there are 708 and 1098 walls.

Materials with very high values of anisotropy, such as rare-earth metals and their intermetallic compounds, may have domain walls only a few atoms thick, with correspondingly high energy. The continuum approximation used above may no longer be appropriate and a model based on individual atom moments must be used. In this case there may be an energy difference between a wall with an atom located at the midplane and a wall with the midplane between two atoms. This energy difference leads to an inherent obstacle to wall motion, or an intrinsic coercive ﬁeld, which will exist even in a perfect single crystal. In the cases where this effect has been calculated or measured, the coercive ﬁelds are not large enough to be of commercial interest: less than 100 Oe or 8 kA/m.

Various methods have been applied to the measurement of the wall energy s. For example, the equilibrium spacing of domain walls in a sample of known geometry depends directly on the wall energy, if there are no barriers to wall nucleation and motion. The resulting values of wall energy are not very accurate because of the simplifying assumptions that must be made, but generally agree well with values calculated by the method given above. Better methods of directly measuring s would be useful, not only because of interest in the wall energy itself, but also because such measurements would lead, through Equation 9.21 or 9.22, to better values of the fundamental constant J, the exchange integral.

The structure of domain walls in ferrimagnetic materials has apparently not been considered in detail. A drawing of such a wall would necessarily be more complicated than Fig. 9.2, because of the presence of chemically different atoms with oppositely directed spins. However, one can still think of the spin axis as slowly rotating through the wall from one domain to the other.

9.2.1Ne´el Walls

When the thickness of a sample becomes comparable to the thickness of the domain wall, the energy associated with the free poles that arise where a Bloch wall meets the surface (Fig. 9.6) becomes signiﬁcant. This can lead to a change in wall structure, with the

Fig. 9.6 Bloch wall. Note free poles appearing where the wall intersects the surfaces.

284 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.7 Ne´el wall in thin sample. Free poles appear along the surface of the wall, but not at the sample surface.

magnetization rotating in the plane of the sample rather than in the plane of the wall (Fig. 9.7). This creates free poles on the wall, since the normal component of magnetization in the wall is no longer constant, but nevertheless gives a lower overall energy. This kind of wall is called a Ne´el wall, and the term Bloch wall is now reserved for the normal wall structure in Fig. 9.6. There are also calculations and evidence to show that the spin variation in a Bloch wall is modiﬁed where the wall meets the surface of the sample. This topic is considered further in Section 11.11.

9.3DOMAIN WALL OBSERVATION

Domains are normally so small that one must use some kind of microscope to see them. Exactly what one sees depends on the technique involved. Generally the observation techniques fall into two groups:

1.Those which disclose domain walls (Bitter method, scanning probe microscope, Lorentz microscopy in the Fresnel mode). The individual domains, whatever their direction of magnetization, look more or less the same, but the domain walls are delineated.

2.Those which disclose domains (optical methods involving the Kerr or Faraday effects; polarized electron analysis; differential phase contrast). Here domains magnetized in different directions appear as areas of different color or brightness, and the domain wall separating them appears merely as a line of demarcation where one hue changes to the other.

9.3.1Bitter Method

The ﬁrst successful observation of domains made use of the Bitter or powder method. This involves the application of a liquid suspension of extremely ﬁne (colloidal) particles of ferrimagnetic magnetite (Fe3O4) to the polished surface of the specimen. Imagine a 1808 wall intersecting the surface, as in Fig. 9.8a, where the spins in the wall are represented simply by the one in the center, normal to the surface. A north pole is therefore formed as shown, and this is the origin of an H ﬁeld gradient above the surface as indicated. The ﬁne particles of magnetite are attracted to this region of nonuniform ﬁeld, depositing as a band along the edge of the domain wall, normal to the plane of the drawing. If the surface is then examined with a metallurgical (reﬂecting) microscope under the usual

9.3 DOMAIN WALL OBSERVATION

285

Fig. 9.8 Bitter method or powder method for observation of domain walls.

conditions of “bright-ﬁeld” illumination, as in Fig. 9.8b, the domain wall will show up as a dark line on a light background; the domains on either side of the wall reﬂect the vertically incident light back into the microscope and so appear light, while the particles on top of the wall scatter the light to the side and appear dark. Better contrast is observed under “dark-ﬁeld” illumination (Fig. 9.8c), where the incident light strikes the specimen at an angle. The domain wall appears as a light line on a dark background, because only light reﬂected from the particles along the wall is scattered into the microscope objective. In either case, the arrangement of lines (domain walls) seen under the microscope is commonly called a powder pattern or Bitter pattern.

Suitable liquid suspensions of colloidal magnetite are available commercially under the name Ferroﬂuids. A drop of the Ferroﬂuid suspension is placed on the surface to be examined, and covered with a thin microscope cover glass to spread out the suspension into a uniform ﬁlm.

Careful specimen preparation is extremely important. The surface of a metallic specimen is ﬁrst mechanically polished and then electrolytically polished to remove the strained layer produced by the mechanical polishing. Electropolishing is accomplished by making the specimen the anode in a suitable electrolytic cell and passing a fairly heavy current, which removes a signiﬁcant number of surface atom layers while keeping the surface smooth. This second step is essential. Early workers did not realize this and obtained “maze patterns,” which are determined entirely by the strains left in the surface after mechanical polishing and reveal nothing of the true domain structure. An example of the domain structure observed on a crystal of Fe þ 3.8 wt% Si is shown before and after electropolishing in Fig. 9.9. The interpretation of this rather complex structure will be given later. Ferrites, being

286 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.9 Powder patterns on a single crystal of Fe þ 3.8% Si, using dark-ﬁeld illumination. (a) after mechanical polishing, (b) after electropolishing. The area viewed is about 0.5 0.45 mm.

nonconductors, cannot be electropolished, but strain-free surfaces can be prepared by careful mechanical polishing with diamond powder, followed by heating to about 12008C.

The essential features of most domain structures can be seen clearly at magniﬁcations of a few hundred diameters. However, simple interpretable domain structures are normally observed only when the sample surface contains an easy direction of magnetization; if anisotropy energy forces the magnetization direction to have a component normal to the surface, a complex closure domain pattern forms, which obscures the simpler basic domain structure of the sample. This qualiﬁcation is itself subject to qualiﬁcations: a sample with vanishingly small anisotropy, such as an amorphous alloy, may show a simple domain structure, and a sample with a very large anisotropy with its easy axis normal to the surface, and a sufﬁciently small magnetization, may have a stable simple domain structure with the magnetization normal to the surface everywhere except in the domain walls. This occurs in the bubble domain structure, treated brieﬂy in Chapter 14.

The Bitter method can detect slowly moving, as well as stationary, domain walls. When a wall moves in response to an applied ﬁeld or stress, the line of colloid particles follows the intersection of the wall with the surface, as long as the wall is moving slowly. Observation of the way walls move, causing one domain to grow at the expense of another, can be very fruitful. For this purpose it is desirable to have some means, such as Helmholtz coils or a small electromagnet, to apply a ﬁeld to the specimen while it is under the microscope.

When greater detail is desired, static Bitter patterns may be examined at the much higher magniﬁcation of the electron microscope by a replica technique. A water-soluble plastic is added to the magnetite suspension, which is then spread on the specimen and allowed to dry. The powder particles go to the domain walls, as usual, and are trapped in these positions as the suspension dries. The result is a thin ﬁlm, containing the powder particles, which can be peeled away from the specimen surface and examined in a transmission electron microscope.

An important step in interpreting an observed domain structure is to determine the directions in which the various domains are magnetized. This can be done by observing the behavior of the colloid particles at accidental or deliberate scratches, or other irregularities, on the specimen surface. Figure 9.10 shows the principle involved. When Ms in a particular domain is parallel to the surface and at right angles to a scratch, as in Fig. 9.10a, the ﬂux lines tend to bow out into the air at the scratch, and this nonuniformity of ﬁeld attracts the

9.3 DOMAIN WALL OBSERVATION

287

Fig. 9.10 Effect of surface scratches on colloid collection.

powder particles. This effect does not occur when Ms is parallel to the scratch, as in Fig. 9.10b, and few powder particles are attracted. Thus scratches crossing a domain structure like that of Fig. 9.10c will appear dark when they are at right angles to Ms and light when they are parallel. It remains then to determine the sense of the Ms vector in each domain; this is done by applying a ﬁeld parallel to the Ms-axis of a certain domain, and noting whether that domain grows in volume or shrinks. If it grows, then Ms must be parallel, rather than antiparallel, to the applied ﬁeld.

The tendency of powder particles to collect at surface ﬂaws which are transverse to the magnetization is exploited, on a coarser scale, in the Magnaﬂux method of detecting cracks in steel objects. The object to be inspected is magnetized by a strong ﬁeld and immersed in a suspension of magnetic particles. When withdrawn, previously invisible cracks are made visible by the powder particles attracted to them.

The Bitter method has several limitations: (1) If the anisotropy constant K of the material becomes less than about 103 erg/cm3 or 102 J/m3, the domain walls become so broad that powder particles are only weakly attracted to them; (2) the method can be applied only over a rather restricted temperature range; (3) moving domain walls can be followed only at low velocities; and (4) the colloid suspension, especially if it water-based, dries up fairly quickly, limiting the time period during which domain observations can be made.

9.3.2Transmission Electron Microscopy

This instrument can disclose domain walls in specimens thin enough to transmit electrons,

generally about 1000 A (100 nm) or less. Since a moving electric charge is acted on by a force when it is in a magnetic ﬁeld, the electrons passing through the specimen will be deviated by an amount and in a direction determined by the magnitude and direction of

288 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.11 Domain photographs of an iron foil by transmission electron microscopy, 5600 . [J. T. Michalak and R. C. Glenn, J. Appl. Phys., 32 (1961) p. 1261.]

the local Ms vector. In a domain wall this vector has different orientations at different positions within the wall; the result is that the wall shows up as a line, either dark or light, on the image of the specimen. The microscope must be slightly underor overfocused in order to make the wall visible. This technique is often called Lorentz microscopy, because the force F on the electron is known as the Lorentz force. This force, on an electron of charge 2e, is given by 2e/c(v B), where v is the electron velocity, B the induction, and c the velocity of light (cgs units). The greater part of B at any point in the specimen is made up of the local value of Ms, the balance being due to the vector sum of any applied or demagnetizing H ﬁelds present.

Two kinds of specimen are of interest:

1.Those which are thin already, which will normally have been made by evaporation or sputtering or electrodeposition, and are called “thin ﬁlms.” Their magnetic properties are of great interest because of the number of current and potential applications in computer technology. Description of these materials is postponed to Chapter 11.

2.Bulk specimens which have been thinned down by grinding and etching. These are usually called “foils.” Figure 9.11 shows an example of domains in a foil of highpurity iron. The domain walls are the light and dark lines. Normally, crystal imperfections such as dislocations and stacking faults are also visible.

Lorentz microscopy has the advantage of high resolution, which allows the examination of the ﬁne detail of domain structure. It also permits the direct observation of interactions between domain walls and crystal imperfections and grain boundaries. In the case of foil samples, there is always a question of the extent to which the domain structure of the foil sample represents that of the bulk material.

The Lorentz microscopy described here is known the Fresnel mode. Other modes are possible, including differential phase contrast and holographic, which have certain advantages [See J. Chapman and M. Scheinfein, J. Magn. Magn. Mater., 200 (1999) p. 729.]

9.3.3Optical Effects

Two magneto-optic effects can distinguish one domain from another, either as a difference in color or in the degree of light and dark.

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