Dionne G.F. Magnetic Oxides

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5.4 Magnetization Process and Hysteresis 251

“single-domain” state, multiple domains have not entered the discussion. Only in the

case of hard magnets where the volume of the individual crystallites is too small to

support domain walls can the assumption of a single domain particle be made in the

unsaturated state. In this section the role of domains and domain-wall energies in

the partially magnetized state will be considered as we examine the effects of mag-

netocrystalline anisotropy and magnetostriction on the magnetization process and

the shapes of hysteresis loops that occur because of domains.

5.4.1 Initial Permeability and Coercivity

In Fig. 5.30 the essential parameters of a generic 4M vs. H hysteresis loop are

indicated. From the origin, the magnetization process begins with reversible rotation

of individual magnetic domain vectors toward the axis the H ﬁeld. For a polycrystal

with randomly oriented grains of cubic lattice symmetry, the slope (susceptibility)



of the curve at H D 0 deﬁnes the average (isotropic) initial permeability dB=dH

according to [79]



<0/D 1 C 4



.4/ 3M

C 

sin





>0/D 1 C 4



.4/ M

C 

sin

; (5.54a)

Fig. 5.30 Tutorial diagram of the low-ﬁeld part of a generic hysteresis loop, with relevant param-

eters indicated

252 5 Anisotropy and Magnetoelastic Properties

where  is the angle between H and the easy axis. When sin

 is replaced by an

average value, e.g.,

sin



D 2=3 if we employ the uniaxial model, (5.54a) can be

reduced to an isotropic approximation



<0/ 

2M

C 



>0/ 

4M

C 

: (5.54b)

As H is increased further, domains with M vectors favorable to the H vector grow

at the expense of the others through the mechanism of domain wall movement. This

beginning of the process is depicted simply in Fig. 1.2b, where two 180

domains

are provided with magnetic ﬂux closure by a pair of orthogonal domains bounded by

diagonal walls at the long axis ends. In real cases, the domain pattern, e.g., Fig. 1.2c,

can be complex and is the basis for the ﬁeld of micromagnetism where it has been

the subject of extensive studies [80,81]. The resulting pattern is determined by the

tradeoff between the increase in energy of the walls that originates from exchange

forces and anisotropy, and the reduction in magnetostatic energy that occurs when

the external ﬂux is collapsed within the material. The geometrical shape is also an

important factor. In terms of the relevant magnetic parameters, a general relation for

the domain wall surface energy density is given by



ŒA

C 



1=2

; (5.55)

where A D JS

represents the exchange energy between adjacent spins in the

domain wall, and is therefore related to the Curie temperature. Two models of do-

main walls are generally considered: Bloch walls usually have the lower energy in

bulk materials; N´eel walls are the preferred approximation for thin ﬁlms. Our dis-

cussion will adopt the Bloch model of (5.55), with typical values for 

falling in the

range of 1–5 ergs=cm

. A corresponding expression for the domain wall thickness is





C 



1=2

; (5.56)

which yields values on the order of 10

5

cm .0:1 m/, or about 200 lattice

parameters.

The existence of a domain wall requires a particle dimension exceeding thick-

ness ı

. Otherwise, the magnetic body behaves as a “single-domain” particle with

initial permeability that is determined entirely by magnetic moment rotation from

an easy direction to that of the magnetizing ﬁeld. However, it is the energy of the

domain wall balanced against the demagnetizing energy of the magnetized particle

that ultimately determines whether a particle can exist as a single domain. For a

cubic particle, the critical dimension for single domain is estimated by [82]

2

M

ŒA

C 



1=2

: (5.57)

5.4 Magnetization Process and Hysteresis 253

Discussions of this subject can be found in most standard texts, including that by

Craik and Tebble [83].

The coercive ﬁeld (or force) H

of a hysteresis loop is generally deﬁned as

the reverse magnetic ﬁeld required to reduce the magnetization of a magne-

tized body to zero. For a single domain in which the only mechanism available

to alter the magnetic state is reversible rotation of the M vector, the value of

 C

rotat

C 

. In most situations of multidomains, demagnetization

in an applied ﬁeld occurs by movement of domain walls irreversibly overcoming

impediments such as grain boundaries and nonmagnetic inclusions e.g., air pores.

Although irreversible domain rotation will contribute to the coercive ﬁeld, pinning

of domain walls at inclusions or lamellar precipitates caused by grain boundaries

is the microstructural source of coercivity [84]. To accommodate the nonmagnetic

volume of inclusions or grain boundaries within the wall volume, the surface energy

of the wall must increase through expansion of the surface. The resulting increase in

wall motion impedance produces two additional contributions to the coercive ﬁeld

which can be combined directly as H

incl

C H

grain

. As a generalization, all three

effects could be viewed analogously to a resistor network, with and H

rotat

repre-

senting an alternative mechanism acting in parallel. Consequently, we approximate

rotat

incl

C H

grain

1

: (5.58)

Since the anisotropy and magnetostriction constants inﬂuence the coercive ﬁeld

through the wall surface energy density [85], H

incl

and H

grain n

can be related to

the magnetoelastic parameters according to

incl

C 

incl

2=3

and H

grain

C 

grain

: (5.59)

where p is the volume fraction of inclusions (porosity), and d

incl

and d

grain

are the

average pore and grain dimensions, respectively. Because the rotation coercivity is

generally much greater than that allowed by domain wall movement, a more useful

relation could be reduced from (5.55), (5.56), and (5.59)as







incl

2=3

grain



: (5.60)

The magnetization process that begins with magnetic vector rotation (often called

domain rotation) will undergo 180

-domain reversal at the coercive ﬁeld. Beyond

this point, rotation ensues until the magnetization reaches alignment with the ﬁeld

(at the anisotropy ﬁeld). In addition, demagnetizing effects of the porosity continue

to be overcome as the body approaches magnetic saturation. These topics will be

discussed next.

254 5 Anisotropy and Magnetoelastic Properties

5.4.2 Anisotropy Field and Remanence Ratio

Other loop parameters of importance are the anisotropy ﬁeld H

and remanent mag-

netization 4M

, which is deﬁned as the net moment per unit volume that remains

after the material is returned to the H D 0 state following saturation. To appreciate

theroleofK

and 

in the remanent state, we begin by examining the case of a

single crystal magnetized along a hard axis. When the orientation is not along an

easy axis, the anisotropy ﬁeld H

deﬁned as 2K

=M delays saturation by impos-

ing an effective demagnetizing effect. The inﬂuence of the anisotropy is illustrated

in Fig. 5.31, which indicates the magnetization characteristics for materials of cubic

and uniaxial crystallographic structure magnetized along a hard axis.

The relations for anisotropy ﬁelds with magnetostriction contributions may be

found in most of the standard texts cited previously. In a

110

plane with hard axis

along a h100i axis (as deﬁned in Fig. 5.15), we restate (5.35)

K

D



100

: (5.61)

In Fig. 5.32 the hard-axis magnetization curve for single-crystal YIG is shown as

measured directly from an oscilloscope. Note the near-linear curve between the

remanent point and the saturation point at H D H

K

. Figure 5.16 introduced previ-

ously shows the effects of a compressive stress  applied along the hard axis, which

reduces H

K

and raises 4M as the material undergoes a uniaxial strain [59].

The inﬂuence of magnetostriction on the remanence ratio of polycrystalline ma-

terials can be important in low anisotropy materials where the internal stress is

sufﬁcient to compete with K

,i.e.,

100

 1. Examples of this effect on

the remanence region are shown schematically in Fig. 5.33. In practical situations,

the issue concerning magnetostriction arises where H

is small, which often oc-

curs in low switching-energy applications [86]. In these cases, the design strategy

is to reduce the value of 

by chemical alteration of the material. Figure 5.34

Fig. 5.31 Schematic illustration of the magnetocrystalline anisotropy effects on the magnetization

process with zero coercive ﬁeld H

is assumed to be zero: (a) the cubic case (rotation in a f110

planeg) where the remanence ratio M

D 0:58 in a hard direction, and (b), the uniaxial case,

where M

D 0 in a hard direction

5.4 Magnetization Process and Hysteresis 255

Fig. 5.32 Oscilloscope photograph of hysteresis property of Y

single-crystal described

in Fig. 5.16. There is effectively no coercive ﬁeld and both R and H

agree precisely with theory

[59]. Figure reprinted from [59] with permission.

 1969 by the IEEE

Fig. 5.33 Hysteresis loop models indicating the inﬂuence of K

and 

presents measured results of Mn

substitution mostly in octahedral sites of NiZn

spinel ferrite. Similar effects were also reported for YIG-based compounds [87,88].

As described in Sect. 5.3.4, the positive Jahn–Teller extensions of the local ligand

group surrounding the Mn

ion has the effect of compensating the normal negative

256 5 Anisotropy and Magnetoelastic Properties

Fig. 5.34 Comparison of low-ﬁeld hysteresis loops of Ni

0:65

0:35

and Ni

0:65

0:35

1:8

0:2

spinel ferrites, showing the cancellation of internal stress effects by local Jahn–Teller

effects of Mn





in the octahedral sublattice. Figure reprinted from [86] with permission.



1986 by the American Institute of Physics

magnetostriction of the Fe

lattices. A comprehensive discussion of stress effects

on hysteresis loops can be found in Bozorth [52].

For the uniaxial case, the relation is similar, H

D2K

=M and the mag-

netostriction effect is usually inconsequential because of the extraordinarily large

values of K

.HeretheM vs. H relation along a hard axis is expected to be linear

for a single crystal, reaching from the origin to the saturation value M

at H

,as

sketched in Fig. 5.31b.

In polycrystals with grains of random crystallographic orientation, H

K

of each

grain varies according to the particular alignment with H . The averaging that occurs

in the magnetization produces the rounded corners and edges of the loop. Calculated

loops that can approximate these effects will be discussed, but ﬁrst it is important to

examine the mechanism of magnetization when H exceeds the coercive ﬁeld, i.e.,

in the approach to saturation.

5.4.3 Approach to Saturation

Once the domain walls have moved to complete the reversal of the M direction, the

balance of the magnetization with increasing magnetic ﬁeld beyond H

is gained by

5.4 Magnetization Process and Hysteresis 257

Fig. 5.35 Sequence of diagrams showing how a nonmagnetic inclusion can reduce remanence by

nucleating 90

spike domains and also impede the movement of a domain wall by reducing the

wall area [82]

M vector rotation up to the maximum H

K

and by overcoming the demagnetizing

ﬁelds centered about pores and other nonmagnetic entities. These latter effects gen-

erally involve reverse spike or closure domains on either side of the pores, with the

progress of N´eel 90

closure domains during switching depicted in Fig. 5.35.The

relative magnetization ratio R in the approach to saturation is traditionally written

as a semiempirical law

R D

D 1 



 (5.62)

where a  .4M

eff

,isthemagnetic hardness constant that is proportional to

the effective porosity p

eff

, b is related to K

,andc, d are the coefﬁcients of

higher-order terms that represent magnetostriction contributions from internal stress

that show their effects at low magnetic ﬁelds [89]. If the ﬁeld-dependent terms are

small and cross terms are ignored, (5.62) can be an approximation to an exponential

function

R  exp



a=H  b=H

 c=H

 d=H





: (5.63)

Careful analysis of R data from the high-ﬁeld portion of the hysteresis loop yielded

accurate values of p

eff

and K

in magnetic garnets and spinels [71]. This model

breaks down as H ! 0 from the return from saturation, denying the determination

of a value for the remanence ratio R

or the possibility of completing a hystere-

sis loop. If the coercive ﬁeld is treated as a negative bias in the outward leg and

as a positive ﬁeld in the return part, the ﬁrst term of (5.63) can be expressed as

R  exp Œa= .H ˙ H

/, thereby allowing the formation of a hysteresis loop and

permitting the establishment of a basic remanence ratio R

D exp .–a=H

/.Ina

more complete form, a nucleation ﬁeld H

for reverse domains is added as a posi-

tive bias in the .H ˙ H

C H

/ factor [80,84].

258 5 Anisotropy and Magnetoelastic Properties

5.4.4 Demagnetization and Permanent Magnets

For the loops discussed so far in this text, the assumption is made that shape demag-

netization effects discussed in Chap. 1 are not present, i.e., the magnetic circuit is

fully closed as in the case of a toroid with uniform susceptibility and cross-section.

In many practical situations, however, the magnetic circuit is not complete and air

gaps that produce demagnetizing ﬁelds ranging from insigniﬁcant to the full 4M

can inﬂuence the resultant magnetization curve. For this reason, “permanent” mag-

netism requires that the bias ﬁeld provided by coercivity be maximized to offset

the sloping of the loop sides caused by demagnetizing effects. The skewing of

the loop can be seen in terms of reduced effective susceptibility by a simple one-

dimensional analysis that begins by deﬁning an effective internal ﬁeld reduced by a

demagnetizing ﬁeld:

eff

D H  H

; (5.64)

where H

D N

.4M

/ D N

.4M

/R.IfM is expressed in terms of suscepti-

bility  and H

eff

M D ŒH  N

.4M / (5.65)



1 C N

.4M / 

;

which leads to the deﬁnition of an effective susceptibility



eff



1 C N

4

: (5.66)

If  !1or a single crystal magnetized along an easy direction, 

eff

! 1=4N

and the corresponding magnetization curve is plotted in Fig. 5.36.SinceN

D 1 in

the case of the ultimate “gap”, e.g., a thin disc magnetized normal to its ﬂat surfaces,

the ﬁeld required to saturate the magnetization is exactly equal to 4M

If a hysteresis loop model following the reasoning of the previous section is con-

structed for a circuit that contains a demagnetizing section, the demagnetizing ﬁeld

must be included so that

R  exp





H ˙ H

C H

 H





; (5.67)

where H

D N

.4M /, with 4M expressed as .4M

/R for computational

iteration purposes [90]. The issue of shape demagnetization arises mainly in the ap-

plication of permanent magnets, where H

 4M

is the condition to achieve the

maximum R value under conditions of N

 1. Figure 5.37 illustrates this extreme

condition.

A second concern that enters into the design of a hard magnet, whether oxide or

metallic, is the demagnetization “energy product” deﬁned as .BH/

max

. As indicated

5.4 Magnetization Process and Hysteresis 259

Fig. 5.36 Simple diagram of the shape demagnetizing ﬁeld along the hard axis of a uniaxial

specimen (typical of planar geometry with H normal to the plane)

Fig. 5.37 Schematic illustrating the skewing of hysteresis loops caused by demagnetizing gaps in

the magnetic circuit

by the rectangular-loop diagram in Fig. 5.38,theﬁeldH

max

for the maximum BH

product is found from the relation B D H C4M

that applies in the negative ﬁeld

quadrant, so that

D N

.4M / ; (5.68)

resulting in H

max

D4M

=2 and yields

.BH/

max

.4M

D .B

=2/

: (5.69)

From a loop design standpoint, we recognize that to achieve the theoretical limit

deﬁned by (5.68), H

 4M

=2. Moreover, it follows that any excess of H

over

260 5 Anisotropy and Magnetoelastic Properties

Fig. 5.38 Schematic diagram illustrating the relation between the coercive ﬁeld H

and the

maximum BH energy product, which dictates that H

4M

=2 does not increase the energy product. However, H

 4M

allows the

magnet to retain its magnetization up to H DH

, which is an important feature

of any square-knee permanent magnet.

Hexagonal ferrites are vehicles for permanent magnetism. For a comprehen-

sive review of these oxide families, the reader is referred to the chapter by Wijn

[91]. Among the various crystallographic structures that comprise the class, uni-

axial M-type (magnetoplumbite, with easy c-axis and K

<0) offers the highest

combination of magnetization and anisotropy ﬁeld. Typically 4M

 4;500 Gand

 17;000 Oe in undiluted barium ferrite .BaFe

/. Careful processing

that begins with submicron-size single-domain particles and involves cold-pressing

in a magnetic ﬁeld can produce good quality ceramic materials with H

values that

reach 4,000 Oe to satisfy the H

 4M

condition [92]. Easy-plane anisotropy

occurs in type-Y structures of formula Ba

, where M is a divalent ion

such as Co

or Zn

. These compounds provide lower anisotropy ﬁelds and are

of interest in cases where dilution of the M-type hexaferrites cannot achieve the

desired properties.