Dionne G.F. Magnetic Oxides

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190 4 Ferrimagnetism

lattice symmetry. Anisotropy of J

following the [100, 111], and [110] axes were

also reported [60].

Another important ﬁnding from these experiments that is consistent with the sub-

lattice canting discussion of Sect. 4.2 was that the molecular ﬁeld coefﬁcients N

and N

are reduced by a frustration factor .1  ˛k

/,where˛ D 0:185 for Sc

ions substituted into the a sublattice. The value of  D 51:5

at low H was found to

be insensitive to k

, thereby conﬁrming the above contention that the exchange ﬁeld

from the iron sublattices does not contribute to the J

reduction and that the effect

is caused by crystal ﬁeld quenching of the orbital component of the total angular

momentum. It is also noteworthy that the value of  decreased with applied H and

declined more sharply at ﬁelds above 10 T, beginning with the larger values of k

(representing the smallest H

), as expected.

When one considers the breadth of the practical applications of these versatile

materials, it is not surprising that the studies of the magnetic garnets have become

exhaustive, as documented in the book by Winkler [61]. In subsequent chapters,

these materials will be examined in terms of their magnetoelastic, microwave, and

magneto-optical properties for which they have found use in their technology of in-

formation storage, radar, and communications systems. Before the focus is shifted

to these magnetic related phenomena, the basic structural and magnetization prop-

erties of the magnetically hard hexagonal ferrites will be included.

4.3.5 Hexagonal Ferrites

The spinel and garnet ferrites discussed in the previous sections have proven to

be rich in physics because their cubic crystal structure has made both measurement

and theoretical analysis easy compared with another technologically important class

of ferrimagnetic oxides that features a crystallographic axis of sixfold symmetry

(designated as the c axis and usually selected as the z direction in a Cartesian

system). These hexagonal ferrites, “hexaferrites” as they are commonly called,

comprise the same Fe

sites as the cubic ferrites: octahedral and tetrahedral. In ad-

dition, they also contain a low density of Fe

trigonal bipyramid sites (see Fig. 2.7)

that enhance the uniaxial or planar anisotropy that is strongly reﬂected in their mag-

netic properties.

Although this family has not been a vehicle of choice for the exploration of basic

physical knowledge, and in some respects remains uncharted territory with regard

to the origins of its properties, at least an introductory treatment is mandated by this

book because of their widespread application. In this section, we shall outline the

salient features of the family and how it contrasts with the cubic systems.

The hexaferrite family consists of several structures comprising a stack of alter-

nating building blocks shown in Fig. 4.22 for a type-M lattice structure, beginning

with a spinel cell, labeled S, but aligned with the c axis along a <111> body di-

agonal. As a consequence, the ferromagnetic spin arrangements of the spinel are

in place automatically, but with only one easy (or hard) magnetic direction instead

of four in a cubic structure. Moreover, the symmetry is already threefold in this

4.3 Ferrimagnetic Oxides 191

Fig. 4.22 Cross section of a

unit cell of the

magnetoplumbite M-type

hexagonal ferrite structure

BaFe

.TheH blocks are

hexagonal and S are spinel.

As presented, H



represents a

partial rotation about the c

axis and S



indicates a

rotation orthogonal to the c

axis. The spheres are as

labeled on the image. Note

that the Fe

ions with

vertical hatching occupy a

ligand coordination that

forms a trigonal bipyramid.

These sites, which are due to

the absence of an O

2

ion,

occur as only one among the

12 iron sites, but are likely

responsible for at least part of

the sixfold axial symmetry of

the magnetic properties.

Image is adapted from Braun

[66]

c-axis

Ba Fe (oct)

Fe (tet) Fe (trig)

direction. To obtain the sixfold symmetry, the S blocks are systematically rotated

by 180

relative to one another with the rotated designation given by S



.Inthe

stack are also blocks or layers that contain large cations, Ba

(radius 1.36

A), Sr

(1.16

A), Pb

(1.18

A) or combinations thereof. These blocks can also contain the

ﬁvefold oxygen-coordinated trigonal bipyramid sites which occur because only one

ion is available for two tetrahedral sites due to the presence of the large cation

in these layers. Additional blocks in the ﬁgure feature hexagonal symmetry and are

labeled accordingly as H (or H



The most common hexaferrite barium (M -type) is a solid solution BaO 6Fe

written in molecular form BaFe

and derived from the mineral magneto-

plumbite PbFe

7:5

3:5

0:5

. For technical applications Sr is regularly

used with Ba because it provides slightly better hysteresis loop properties. Crys-

tallographically, this compound is described as HSH



with 10 atomic layers per

unit cell with a c-axis length of 23.2

A. Magnetically, the 12 Fe

ions per for-

mula unit consist of the single trigonal-site ion

in the H block directed up and the 2

As pointed out by Smit and Wijn [62], the source of very large magnetocrystalline anisotropy in

Ba ferrite (type M) is a not explained by a conventional spin dipole-dipole alignment mechanism.

Because all of the magnetic ions are S -state

L D 0

, there should neither be ﬁrst-order

spin-orbit or Jahn-Teller stabilizations. A curious suggestion regarding the single highly distorted

ligand coordination is that the trigonal crystal ﬁeld might stabilize the d

electrons into a low-spin

192 4 Ferrimagnetism

Table 4.10 Hexagonal ferrite structural and magnetic data

Composition Type Block order 4M

300 K

(kG) H

300 K

(kOe)

(Ba,Sr)Fe

MRSR



4.8 17–19

BaMe

D MS



4.0– 4.8 2.2–19

BaMe

D MS



BaMeFe

1.5–3.8 9–28





D MY



RSR





3:8





11





Z RSTSR



3.1– 3.9 13 (Co

)

(Ba,Ca)Fe

F– 1.2(Ca

)2.2(Ca

)

H.P.J. Wijn [62]

Curie temperatures are typically in the vicinity of 700 K, and are accordingly lower when the

room temperature magnetization is higher, e.g., where Me D Zn

can be lowered by substituting Sc

or In

for Fe

in S-block octahedral sites which

contribute most of the anisotropy energy. Conversely, Al

substitutions will raise H

, but mainly

by reducing the magnetization

octahedral-site ions down; in the spinel S block, 7 octahedral ions are up and 2 tetra-

hedral ones are down. For each Fe

ion contributing 5m

, the theoretical magnetic

moment is 20m

, which has been conﬁrmed by measurements at low temperatures.

Although other forms of hexagonal ferrites exist and occasionally ﬁnd use, e.g.,

W, X, Y, and Z , [63] according to the listing presented in Table 4.10, only the M

structure has proven to be of important technical value. The reasons for this lie prin-

cipally in its uniaxial magnetic direction, as opposed to others that feature a planar

magnetic orientation. Because of the comparatively high magnetic moment relative

to ferrites in general Ba and Sr ferrite have become permanent magnet materials. As

in the case of the spinel and garnet ferrites, dilution of the magnetic moment can be

accomplished by substitutions of diamagnetic ions that favor octahedral sites. Ex-

perimental work by DeBitetto has shown that Al

ions substituting for Fe

will

lower the magnetization [64]. Similar work by Roschmann et al. [65] and Wilber

et al. [66] with In

and Sc

ions will not only reduce the magnetization but also

lower the magnetic anisotropy that is responsible for the magnetic hardness property

of the material. This topic will be examined in Sect. 5.4 on hysteresis properties.

For a more complete treatment of the hexagonal ferrite family, the reader is

directed to seminal works by Braun [67], Went et al. [68], Jonker et al. [69], sub-

sequent texts by Smit and Wijn [62],LaxandButton[70], and the comprehensive

review by Wijn [63].

In our discussions up to this point, the origin of the magnetization has been the

primary emphasis. Equally important is the process of magnetization, i.e., the ini-

tial permeability of a material and the factors that inﬂuence it. Prominent among

these effects is the magnetoelasticity that enhances the anisotropy of the magnetic

moment vector relative to the symmetry axes of the crystal in which it occurs. From

these effects come many of the utilitarian properties of magnetic oxides, including

magnetostriction, magnetic domains, and hysteresis.

S D 1=2

conﬁguration, rendering some of the Fe

cations highly magnetoelastic with a degen-

erate spin-orbit stabilized ground state.

Appendix 4A 193

Table 4.11 Room-temperature magnetization of rare-earth orthoferrites

Y La CePr NdSmEuGdTbDyHoEr TmYbLu

0 0 3/2 4 9/2 5/2 0 7/2 6 15/2 8 15/2 6 7/2 0

8.4 6.6 – 5.7 4.9 6.7 6.6 7.5 10.9 10.2 7.3 6.5 11.2 11.4 9.5

Data are from Bobeck et al. [73]

All ions are 3C valence state

values are in Gauss units

4.3.6 Orthoferrites

Perovskites represent another oxide structure compositionally similar to the gar-

nets that can be made magnetically active. An advantage over the garnets is that

they can be grown as ﬁlms on a range of substrates. With 180

cation–anion–cation

bonds, perovskites are also preferred lattices for superexchange investigations.Mag-

netic perovskites, with the generic formula ABO

(see Fig. 3.21,whereAisan

thanide (rare-earth) series, and B houses a transition metal ion, usually Fe

have found applications in liquid-phase epitaxial ﬁlm form as cylindrical-domain

(bubble) memories. In later studies, magneto-optical properties of CeFeO

[71]and

Ba(Ti,Fe)O

[72] have been reported.

For rare-earth orthoferrites (RE)FeO

[73], the basic ferrimagnetism that arises

from a canted (0:5

) antiferromagnetic spin ordering of the Fe

–O

2

–Fe

in-

teractions in octahedral sites, a net magnetization on the order of 100 emu/cm

(about a factor of 10 less than the corresponding garnet) is created. A summary

of magnetization values for this series is given in Table 4.11. More importantly, the

N´eel (Curie) temperatures exceed 600 K, which is more than sufﬁcient to provide

collinear alignment of orbital and spin angular momentum in the A sublattice for

room-temperature operation. If the frustration of the 4f spins of RE ions leads

to such alignment with the resultant magnetization of the canted Fe

lattices, as

sketched in Fig. 4.23, a Faraday rotation effect could occur in proportion to the net

exchange ﬁelds. The subject of polarization rotation in magnetic oxides is

discussed further in Chap. 7.

Appendix 4A Molecular Field Analysis of LiZnTi Ferrite

For the lithium zinc titanium spinel system Li

0:5z=2Ct=2

1:5z=23z=2

there are two cation site distribution regimes to consider, z  t and z  t [22]. In

terms of generic dilution fractions, a set of general relations for the coefﬁcients can

be expressed as

twelve-coordinated oxygen site occupied commonly by magnetic ions of the lan-

194 4 Ferrimagnetism

Fig. 4.23 Two-dimensional schematic diagram comparing the likely spin ordering of

garnet and

orthoferrite (expressed as four molecules). Note that the

magnetically opposing tetrahedral (d) and octahedral (a) sites of the Fe sublattices in the garnet

are in the ratio of 3:2, while the net moment of the orthoferrite arises from the canting of spins

equally divided between the two antiferromagnetic lattices that comprise only octahedral sites.

Figure reprinted from M. Bolduc, A.R. Taussig, A. Rajamani, G.F. Dionne, and C.A. Ross, IEEE

Trans. Magn. 42, 3093 (2006) with permission;

 2006 IEEE

150



1 





;

60



1  k





;

D N

C273



1 



C k







;

(4.43)

where the superscripts t, l,andz refer to titanium, lithium and zinc dilutions.

For t  z, the chemical formula is

1t=2z=2

t=2z=2

ŒLi

0:5

1:5t



and the coefﬁcient relations are

150



1 



60



1 

t 



C273



1 

t.t C z/ 



(4.44)

Appendix 4B 195

For z  t,Fe

1z



0:5z=2Ct=2

1:5Cz=23t=2



, and the molecular ﬁeld

coefﬁcient relations are

150



1  t 



;

60



1 



;

D N

C273



1 

tz 



(4.45)

Appendix 4B High-Magnetization Limits

The search for ferrites with magnetizations signiﬁcantly greater than 5,000 G is

motivated by the desire for higher permeability media in general and for device

applications at millimeter wavelengths, which is discussed in Chap. 6. The basic

challenge, apart from sublattice spin canting that is unavoidable, is to retain as

much Fe as possible. Ideally, the maximum magnetization would be obtained if ev-

ery magnetic ion were Fe

according to the formula Fe

1x





where Q represents a combination of A-site ions (including vacancies) with an

effective valence of n D 3  1=x for the range 0:33  x  1. In practical

cases, this has never been achieved mainly because the required dilutants have

not been found. Figure 4.24 shows computed thermomagnetization curves in the

range of x<0:5[74]. Note that only the x D 0:4 curve is realizable if stoi-

chiometry is to be preserved; a curve for x D 0:33 (not shown) would represent

the case of one-third of the A-sublattice lost to vacancies, similar to maghemite



20:33



0:33



, but with all of the vacancies in the B sublattice (see

computed curve in Fig. 4.10).

Generally, the A sublattice accepts divalent ions more readily, and the closest one

can come to reaching the high magnetization goals is to use Zn

. Solid solutions of

magnetite Fe





and zinc ferrite Zn





will form the se-

ries Zn

1x



1x

1Cx



has produced room-temperature mag-

netization values well over 7,000G [75]. Unfortunately, these compositions are not

useful at microwave frequencies because the large Fe

content renders the material

dielectrically lossy. A possible solution to this impediment could be the monova-

lent cation system with generic formula A

1x



0:5x

1:5Cx



where A and B represent any combination of monovalent ions occupying the

tetrahedral and octahedral sites, respectively. The computed results of a thermomag-

netization analysis shown in Fig. 4.25 predicted that a room-temperature 4M 

6;500 G could be obtained with x in the 0.3–0.4 range [74]. This would require that

at least 60% of the monovalent cations, Li

,Na

,orCu

be forced into the

A sites.

196 4 Ferrimagnetism

Fig. 4.24 Thermomagnetism curves for ﬁctional inverse spinel



1x







.The

range of values x D 0, 0.1, 0.2, 0.3, and 0.4 cannot be realized in practice, but the exercise serves to

illustrate the challenge of the search for higher magnetization ferrites, particularly for microwave

applications where Fe

ions cannot be used. Figure reprinted from [74] with permission.

1987

by the American Institute of Physics

Fig. 4.25 Thermomagnetism curves of monovalent cation diluted spinel Fe

1x



0:5x

1:5Cx



for x D 0, 0.1, 0.2, 0.3, and 0.4. Figure reprinted from [74] with per-

mission.

 1987 by the American Institute of Physics

Appendix 4C Brillouin Functions in Exchange Energy Format

If the Brillouin functions are expressed in terms of exchange constants instead of

molecular ﬁelds coefﬁcients in the relations, the spin parameters are used in (4.30)

according to

.T / D S

.0/ B

References 197

.T / D S

.0/ B

/; (4.46)

.T / D S

.0/ B

where S

.T /, S

.T /,andS

.T / are average spin values of ions in the respective

sublattices that are randomized by lattice thermal energy. Employing the expressions

for the sublattice exchange energy from (4.32) (without the canting and dilution

factors), we deﬁne

.T / D

.d /

D

Œz

C z

;

.T / D

.a/

D

Œz

C z

; (4.47)

.T / D

.c/

D

Œz

C z

:

To convert back to magnetic moments, the relations listed previously in Sect. 4.3.2

can be applied. However, for the case of rare-earth ions from the upper half of the

series in the c sublattice two adjustments must be made: ﬁrst, the g

factor must

be that deﬁned by (4.33) to account for the fact that the contribution to the magnetic

moment M

comes from the relation

D 3g

C S

/; (4.48)

and second, J

must replace S

in the determination of the magnetic moment M

In addition, the molecular ﬁeld coefﬁcients N

and N

must be multiplied by the

factor S

to offset the change in M

and keep the N

and N

products

constant. A redeﬁnition of N

in terms of J

from that of (4.28)to





(4.49)

is necessary to take into account that the L

contribution to the moment does not

take part in the exchange stabilization. This spin-only format can be used as an

alternative to the conventional molecular ﬁeld approach that is described in the text,

but would require vigilance to make certain that J

is used in M

where necessary

and that the N

coefﬁcients are adjusted according to (4.49).

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