Dionne G.F. Magnetic Oxides

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150 3 Magnetic Exchange in Oxides

28. J.B. Goodenough, Magnetism and the Chemical Bond, (Wiley Interscience, New York, 1963),

Chapter 3, Table XII; also J.B. Goodenough, Phys. Rev. 117, 1442 (1960)

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2225 (2005)

30. L. N´eel, Ann. Phys. (Paris) 17, 64 (1932)

31. A.H. Morrish, The Physical Principles of Magnetism, (John Wiley, New York, 1965), Chapter 8

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35. R.K. Nesbet, Phys. Rev. 122, 1497 (1961)

36. J.B. Goodenough and J.M. Longo, Crystallographic and Magnetic Properties of Perovskite and

Perovskite-Related Compounds, Landolt-Bornstein, Volume 4a (Springer-Verlag, New York,

1970) pp. 126–314

37. G.H Jonker and J.H. Van Santen, Physica XVI, 337 (1950)

38. J.B. Goodenough, A. Wold, N. Menyuk, and R.J. Arnott, Phys. Rev. 124, 373 (1961)

39. J.B. Goodenough and J.M. Longo, Crystallographic and Magnetic Properties of Perovskite and

Perovskite-Related Compounds, Landolt-Bornstein, Volume 4a (Springer-Verlag, New York,

1970), Fig. 39

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Chapter 4

Ferrimagnetism

In the previous chapters, the origins of spontaneous magnetism for parallel

(ferromagnetism) and antiparallel spin alignments (antiferromagnetism) have been

reviewed. In their pristine forms, the former occurs through direct exchange in

metals and alloys, and the latter in nonmetallic ionic compounds comprising oxy-

gen or other elements from the right-hand side of the Periodic table as the anion

lattice. Utilitarian applications of ferromagnets are self-evident to even the most

casual observer of physical phenomena, but the situation is much less so in the case

of antiferromagnetism. For the most part, antiferromagnetism has been a portal to

fundamental research in materials, particularly involving the diagnostic methods of

neutron and more recently, muon diffraction and scattering.

There are, however, select groups of transition-metal oxides that combine the

magnetic properties of ferromagnetic metals with the electrically insulating char-

acteristics of the antiferromagnetic compounds described in the previous section.

These magnetic insulators are termed ferrimagnets, and the phenomenon that char-

acterizes their magnetic properties is called ferrimagnetism. Ferrimagnetic oxides

have also served as rich sources of knowledge about the fundamental physics of ma-

terials, but unlike the antiferromagnetic oxides, they continue to add to their already

widespread uses in modern electronics technology. For these reasons, the proper-

ties of ferrites, as they are commonly designated, will be treated generously for the

remainder of this book.

4.1 Ferrimagnetic Order

In the previous chapter, the concept of multiple magnetic sublattices was intro-

duced to explain the phenomenology of antiferromagnetism. From this starting

point, we may deﬁne a ferrimagnet as an antiferromagnet with unbalanced mag-

netic sublattices due to either differing populations of similar spins or sublattices

with ions of different spin values altogether. For this to occur, it is apparent that

some kind of crystallographic selection must be involved to distinguish the sub-

lattices. In the common oxide systems where ferrimagnetism occurs, the sublattices

are deﬁned by cation sites of different oxygen coordinations: octahedral, tetrahedral,

G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 4,

 Springer Science+Business Media, LLC 2009

151

152 4 Ferrimagnetism

and dodecahedral, as described in Sect. 2.2.2. In a crystal lattice, these sites form

interspersed but ordered arrays that enable each to be occupied by entirely different

ionic species, or ions of the same atomic species but of different valence, or various

combinations of both. In the simplest case, a two-sublattice molecular ﬁeld theory

can be applied, but now requiring three coefﬁcients instead of the two for the an-

tiferromagnet – one each for the ions in the same sublattice (intrasublattice), and a

third (intersublattice) linking ions between the sublattices.

4.1.1 Generic Ferrimagnetic Systems

In Fig. 4.1, a one-dimensional sketch is offered to clarify the difference between

a ferromagnet, an antiferromagnet, and a ferrimagnet. In practical terms, a ferri-

magnet behaves magnetically as a ferromagnet and can be analyzed in terms of

the Brillouin–Weiss theory from Sect. 1.3.2. It must be recognized, however, that

all of the bonding linkages produce principally superexchange and therefore favor

antiparallel spin alignments. In an ideal ferrite with collinear moments, the ﬁnal

ordering of the magnetic moments in the sublattices is therefore the result of com-

peting exchange ﬁelds that are generally dominated by the antiparallel inﬂuence

from intersublattice coupling. The resultant magnetic moment then becomes the

arithmetic difference of the two opposing sublattice moments.

FERROMAGNETISM

> 0

ANTIFERROMAGNETISM

| > |N

< 0

FERRIMAGNETISM

| > |N

< 0

Fig. 4.1 One-dimensional exchange models of spontaneous spin alignment: (a) for a single mag-

netic lattice ferromagnetism: N

>0;(b) for two equal sublattices with N

as the dominant

coefﬁcient, antiferromagnetism: N

<0, N

< or >0,(c) for two unbalanced sublattices with N

dominant, ferrimagnetism: N

<0, N

, N

< or >0

4.1 Ferrimagnetic Order 153

Although there are exotic chemical compounds in which ferrimagnetic sublattice

spin arrangements exist, our discussion of ferrimagnetism will be limited to three

families of transition-metal oxides that have become the foundation of this branch

of magnetism. The ﬁrst of these systems to be recognized is the spinels designated

by the generic formula AŒB

 O

,whereA is the tetrahedral site with O

coor-

dination and B is the octahedral site with O

coordination. The brackets around

B serve to indicate the octahedral sites in the actual chemical formulae. Later the

magnetic garnets with the generic formula

Œa

.d

/ O

were synthesized and

have become equally important particularly in microwave and optical applications.

The bracket/site designations are

for dodecahedral with O

coordination, [d]

for tetrahedral, and (a) for octahedral. It is important to note that unlike the spinels

in which the octahedral sites dominate the tetrahedral sites by a ratio of 2:1, in the

garnets the tetrahedral sites dominate by a ratio of 3:2.

A third family of ferrimagnetic compounds is the magnetoplumbites, named

after the naturally occurring of PbFe

(lead ferrite). These compounds are

commonly referred to as hexagonal ferrites or hexaferrites because of their sixfold

symmetrical uniaxial crystallographic structures. In many respects, hexaferrites re-

semble spinels because they feature the same ratio of octahedral to tetrahedral sites,

but also include one trigonal bipyramid .O

/ site (see Fig. 2.7) that contains an iron

ion, and one large site to house the usually divalent Pb, Ba, or Sr. These compounds

are important for permanent magnet applications and will be described in more de-

tail along with the spinels and garnets in Sect. 4.3.

4.1.2 Molecular Field Theory of Ferrimagnetism

Recalling the exposition of the molecular ﬁeld model of antiferromagnetism from

Sect. 3.2.2, we can now apply this formalism in the manner of N´eel to the case of

a ferrimagnet [1]. For the individual sublattices of a two-site system, (3.35) with

¤ N

can be used to express the individual sublattice magnetizations as



H C N

C N



; (4.1)



H C N

C N



;

where

C 1/

; (4.2)



C 1



154 4 Ferrimagnetism

As in the discussion of antiferromagnetism, a Curie temperature relation can be

extracted from solutions of (4.1) in the paramagnetic region above T

andinthe

magnetically ordered region below T

. Some of the details of the mathematical ma-

nipulations will be left to the readers, who can also consult standard text books, e.g.,

Morrish [2]. The analytical results, however, can serve as helpful approximations as

well as providers of physical insight into the stability of the ferromagnetic state.

In the paramagnetic region, (4.1) can be solved simultaneously to produce in-

dividual relations for M

and M

as a function of H . A susceptibility can then be

deﬁned as  D



C M



=H , which then leads to a rather complicated expression

for 1= vs. T that takes the form of a hyperbolic function displayed graphically in

Fig. 4.2 for which the asymptote as T !1is given by



C C





; (4.3)

where





C C





C C

C 2C



: (4.4)

The asymptotic or paramagnetic Curie temperature is the intercept with the T axis,

given by



C C



: (4.5)

By the convention adopted so far in this text, all of the molecular ﬁeld coefﬁcients

are treated as negative quantities, so that 

and therefore 

will be negative. Equa-

tion (4.3) can be expressed as the Curie-Weiss law

 D

C C

T C 

: (4.6)

Fig. 4.2 Inverse susceptibility of a ferrimagnet above the Curie temperature, including the asymp-

tote of the hyperbola. Magnetization curve is added to illustrate the behavior at T<T

4.1 Ferrimagnetic Order 155

The above analytical method can be used to determine values of N

, N

,andN

various situations that are explained in Sect. 4.2.

In the magnetically ordered region, an expression for the true Curie temperature

can be derived from (4.1) by allowing H D 0 and solving the determinant of the

coefﬁcients of M

and M

to yield



C C





 C



C 4C

: (4.7)

In spinel and garnet ferrites, it will be shown that C

 C

and since N

usually the dominant coefﬁcient, 4C



 C



, allowing (4.7)to

be simpliﬁed to





C C



: (4.8)

A ﬁrst-order approximation that is often used with generally disappointing results is



. Because N

and N

are negative, N

estimates arrived at by this

approximation can be substantially larger than the true values.

The most important application of the N´eel molecular ﬁeld theory of ferrimag-

netism is the computation of the spontaneous magnetization characteristic of a given

chemical composition as a function of temperature (thermomagnetization). In the

case the resultant magnetization of the opposing sublattices is given by

M D

 M

: (4.9)

The procedure once again involves the Brillouin–Weiss function, which is applied

to each sublattice according to

.T / D M

.0/B

.T / ; (4.10)

.T / D M

.0/B

.T / ;

where

.T / D

.i/



C N



; (4.11)

.T / D

.j /



C N



;

and M

.0/ D n

and M

.0/ D n

. Solution of (4.11) cannot be

done in closed form, but can be accomplished by self-consistent iteration proce-

dures involving multiple sublattices simultaneously. Computer programs for this

156 4 Ferrimagnetism

Fig. 4.3 Thermomagnetic characteristics of a ferromagnetic metal and a two-sublattice ferrite.

Note greater magnetization expected from the itinerant magnetism metal, for which no oxygen

occupies lattice sites and there are no opposing sublattices

purpose are listed in MIT Lincoln Laboratory Technical Reports ([3] (garnet), [4]

(RE garnet), [5] (spinel)).

A sketch of a typical thermomagnetization computation

isgiveninFig.4.3. The usefulness of the molecular ﬁeld model has proven to be

enormous over the past four decades, particularly the reﬁned versions of it that have

made possible the explanation and prediction of thermomagnetismbehavior of com-

pounds in which the sublattice moments are diluted with diamagnetic substitutions

for the purpose of tailoring magnetic properties to speciﬁc applications.

In most of these situations, dilution of the magnetic sublattices has been accom-

panied by departures from ideal magnetic spin alignments, commonly referred to as

“spin canting.” These reductions in the effective magnetic moments occur beyond

the normal disruptions of the magnetic ordering arising from the thermal random-

ization accounted for in the application of the Brillouin–Weiss function. Before

the more general theory of thermomagnetization is discussed, some background on

canting effects must be reviewed.

These published documents can be readily obtained from the U.S. National Technical Information

Service (NTIS).

4.1 Ferrimagnetic Order 157

4.1.3 Magnetic Frustration and Spin Canting

In ferrimagnetism, the molecular ﬁelds comprise contributions from magnetically

opposing sublattices. As a consequence, there exists the possibility that breakdown

of the long-range magnetic order by local cancellation (or even reversal through

overcompensation) of the magnetic moments can occur through variations in spa-

tial ordering of these individual moments from inhomogeneous site distributions.

Another cause for cancellation is magnetic dilution, i.e., the replacement of the

magnetic ions by diamagnetic S D 0 substitutes (which could include actual lat-

tice vacancies). For a site in the i sublattice that is missing even one of its nearest

neighbor spins, there is a ﬁnite probability that the ion does not participate in the

exchange stabilization. Such an occurrence is called magnetic frustration, which

effectively renders the ion paramagnetic at the site in question. The resulting spin

canting is a departure from collinearity of the spin directions independent of the

thermal lattice vibrational disruptions that accompany rising temperatures. The gen-

eral subject of spin canting has been examined from various approaches and will be

reviewed in the approximate chronological order in which they were reported.

The ﬁrst examination of noncollinearity of spins in magnetically ordered systems

was reported by Yafet and Kittel (Y-K) [6] as an attempt to explain the magnetic be-

havior of spinel ferrites with A sites diluted by zinc. In the example of Fig. 4.4

Fig. 4.4 Magnetic moment of lithium zinc spinel ferrite at T ! 0 K, showing the canting effects

from Zn

dilution of the minority tetrahedral A sublattice. Note departure from the linear N´eel

model and peak in data at z  0:4. Data are from Gorter [7]. Figure reprinted from G.F. Dionne,

J. Appl. Phys. 45, 3621 (1974) with permission.

 1974 by the American Institute of Physics

158 4 Ferrimagnetism

for a lithium ferrite host [7], i.e., Fe

1z



0:5z=2

1:5Cz=2



, the saturation

moment expressed in Bohr magnetons per formula unit n

is shown to fall below

the collinear N´eel model and reach a peak as the zinc content z ! 0:4. The ba-

sic notion of the Y-K model was that the i–i and j –j interactions under the right

conditions could form antiferromagnetic spin alignments within the their own sub-

lattices, thereby breaking up the main i –j antiferromagnetic ordering to produce

four sublattices and causing radical changes in the net magnetization. Although the

Y-K model failed to ﬁt the data of the ferrite anomalies, it called attention to the ex-

istence of canted spins that was later conﬁrmed in the spinel CuCr

by neutron

diffraction [8]. Another important result of this concept was the reasoning by de

Gennes that the canting of a sublattice would be principally the result of dilution of

the opposing sublattice [9].

There have been several attempts to devise a theory of canting that could place the

concept on a more quantitative basis. The seminal work was carried out by Gilleo

in an analysis of magnetic moment departures from the N´eel model for the mag-

netic garnets. In his most successful model, he assumed that Fe

ions linked to

no more than one nearest neighbor nonmagnetic ions would not contribute to the

spontaneous magnetization [10]. This condition can be more stringent than that of

frustration, which can occur by means of a cancellation of exchange ﬁelds and there-

fore requires less dilution, but the results of the model nonetheless provided some

degree of satisfaction. In this model, the net magnetic moment per molecule in the

garnet system is

/ D n

/  n

/; (4.12)

where n

D 15 .1  k

/Œ1 E

/ and n

D 10 .1  k

/Œ1 E

/.

(Note that each Fe

ion carries a magnetic moment of 5m

.) The parameters

/ D 6k

5k

and E

/ D 4k

3k

are the respective sublattice

canting probabilities from opposite sublattice dilution as determined from random

probability theory. From these relations both the magnetic moment at T D 0 K

and the Curie temperature can be estimated. In this attempt to ﬁt measurement

data, only the intersublattice interactions were taken into account. Nonetheless,

the results as applied to the magnetic garnet compositions with a-site dilution

by Sc

in the forms of

ŒFe

2x

.Fe

/ O

or by Zr

with charge com-

pensating c-site Ca

in the form of

3x

ŒFe

2x

.Fe

/ O

have given

reasonable qualitative agreement with experiment, as shown in Fig. 4.5.Ford -site

dilution by Ge

or Si

with charge compensating c-site Ca

in the form of

3x

ŒFe

.Fe

3x

/ O

3x

ŒFe

.Fe

3x

/ O

, the model

has also given similar agreement with experiment, as shown in Fig. 4.6.

By the same procedure, Gilleo also deduced a model for the ferrimagnetic spinel

system AB

, with

/ D n

/  n

/; (4.13)

4.1 Ferrimagnetic Order 159

Fig. 4.5 Initial canting effects on magnetic moment of a-sublattice diluted yttrium-iron garnet at

T  0 K. Data of fY

3x

2x

.

and fY

3x

2x

.

are from

Geller [15]. Figure reprinted from G.F. Dionne, J. Appl. Phys. 41, 4874 (1970) with permission.

 1970 by the American Institute of Physics

where n

D n

.1  k

/Œ1 E

/, n

D n

.1  k

/Œ1 E

/,

and E

/ D 12k

 11k

, E

/ D 6k

 5k

. In this case the

respective undiluted Bohr magnetons per molecule are left as variables n

and

because of the greater likelihood of varying ionic spin values, i.e., other than

S D 5=2, in the spinel system.

In fashioning a physical description of spin canting in the magnetic garnets,

Geller realized the importance of the intrasublattice exchange ﬁelds and proceeded

to picture the evolution of the antiferromagnetic ground state from its initial ferri-

magnetism by examining the inﬂuence of the intrasublattice a–a and d –d exchange

ﬁelds on the stronger intersublattice a–d interaction [11]. The reasoning proceeds as

follows: When dilution of the a sublattice, for example, is made, the initial effect is a

net increase of n

which would proceed initially as a linear function of k

;however,