Dionne G.F. Magnetic Oxides

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130 3 Magnetic Exchange in Oxides

of the molecular (or superexchange) ﬁeld in spontaneous magnetism that was intro-

duced in Sect. 1.3.1. Recalling (1.40), we express

eff

D H C H

D H C N

M; (3.26)

where N





from (1.68) for the ferromagnetic case. For the two sublat-

tices i and j ,





;





; (3.27)

where n

and n

are the respective densities of spins of the i and j sublattices and z

and z

are the respective numbers of exchange-coupled neighbors.

From (3.15) we also see that the molecular ﬁeld coefﬁcient can be expressed in

terms of the covalent stabilization energy

D





: (3.28)

Since the molecular ﬁeld in (3.26) from the j sublattice H

.j /

D N

acts on a

magnetic moment m

D g

in the i sublattice, the antiferromagnetic stabiliza-

tion energy for the i sublattice is expressed as

.i/

Dm

.j /

Dg

: (3.29)

Therefore, we can substitute for M

D n

and for N

from (3.28) to obtain

.j /

D z

: (3.30)

As a result, the Brillouin-Weiss function parameter a

with the applied ﬁeld H D 0

can be expressed as

.j /

: (3.31)

Note that there is no direct indication that m

.j /

is of magnetic origin. In generic

systems where orbital angular momentum and spin-orbit coupling effects are ab-

sent, i.e., for S -state ions such as Fe

or some crystal-ﬁeld quenched systems of

the d

electron groups, the Pauli principle and Hund’s rule are responsible for the

superexchange stabilization. The inﬂuence of orbital angular momentum will be ex-

amined in Sect. 5.1.

3.2 Antiferromagnetism 131

3.2.2 Molecular Field Theory of Antiferromagnetism

To appreciate the importance of the molecular ﬁeld concept in applying the theory

of superexchange to magnetic oxides, we must ﬁrst review N´eel’s extension of the

Brillouin-Weiss theory of ferromagnetism to the case of opposing sublattices that are

characteristic of antiferromagnetism and its more complex cousin ferrimagnetism

[30]. For these situations, the spin values represent the number of orbital states of

the two ions that are jointly participating in the superexchangecovalent stabilization.

Although many instances of antiferromagnetism involve multiple sublattices, we

limit this discussion to the simplest case of two sublattices comprising nearest-

neighbor sites i and j occupied by ions with alternating spin directions. This is

the case of a lattice in which the magnetic ions occupy the corners of a simple cubic

structure, typical of a cubic perovskite to be examined in Sect. 3.3.2.Therearealso

situations where the magnetic sublattices are formed between next-nearest neigh-

bors, and these will be discussed later. In the ideal situation at T D 0 K, the spin

directions are assumed to be exactly parallel or antiparallel. Recalling the notions

of the molecular ﬁeld introduced in Sect. 1.3., we express the magnitudes of the

resultant effective magnetic ﬁelds at the individual sites as

D H C N

C N

;

D H C N

C N

; (3.32)

where H is the applied magnetic ﬁeld, N



D N



,andN



D N



are the corre-

sponding intra and intersublattice molecular ﬁeld coefﬁcients for the two sublattices

of magnetizations M

and M

. Because the interaction between sublattices is anti-

ferromagnetic, the N

coefﬁcient is negative, while N

and N

could be positive or

negative depending on the nature of the particular superexchange discussed in the

previous section.

At thermal equilibrium, the individual sublattice magnetizations can be expressed

D n

/; (3.33)

where n

is the volume density of spins S

, B

is the Brillouin-Weiss function and

as deﬁned previously from (3.31)and(3.32). As suggested by the

introductory analysis of ferromagnetism in Sect. 1.3, a value of the threshold (N´eel)

temperature for spontaneous antiferromagnetic alignment will emerge from a solu-

tion of (3.33). To determine the behavior at or above the N´eel temperature, we use

the approximation for B

/ ! Œ.S

C 1/ =3S

a

near the limit where a

<< 1

from (1.37) to express the individual sublattice magnetizations as

C 1/

3kT



H C N

C N



;



C 1



3kT



H C N

C N



; (3.34)

132 3 Magnetic Exchange in Oxides

where the directions of H , M

,andM

can be treated as parallel in the paramagnetic

regime above T

. If we assume that each sublattice contains an equal division of

the total population n, which comprises the same ionic species with spin S and

spectroscopic factor g, S

D S

D S, g

D g

D g,andn

D n

D n=2,the

resultant magnetization becomes

M D M

C M

S.SC 1/

3kT



H C



C N





(3.35)

from which the susceptibility  D M=H can be written in terms of the paramagnetic

N´eel temperature 

 D

T C 

; (3.36)

where C D

S.SC1/

and 

D



C N



. Note that N

is assumed to

be negative and of greater magnitude than N

. Therefore, 

is positive and con-

sistent with (1.43). A graphical representation of the different situations of (3.36)is

presented in Fig.1.7. In situations where N

becomes equal to or greater than N

reordering of the sublattice spin alignments is expected to take place.

To obtain an expression for the actual N´eel temperature T

,(3.36) can be used

in the limit of H ! 0,



C N



;



C N



: (3.37)

For M

and M

each to be nonzero, the determinant of the coefﬁcients of M

and M

must be zero. Therefore, the solution for T D T

becomes

D



 N



: (3.38)

At this point, it is possible to compare the paramagnetic temperature with the N´eel

temperature by taking the ratio of (3.37)and(3.38):



D

C N

 N

: (3.39)

There are limitations to the applicability of (3.39). In most cases N

and N

are

both negative, and the ratio 

>0.WhereN

is negligible compared with

, T

 

; where these two coefﬁcients are comparable in size, instability in the

ordering will occur; if N

were to dominate, the static spin ordering would assume

a different pattern.

To complete the picture of the susceptibility as a function of temperature for

a single-crystal antiferromagnet, we must examine the condition of the spin sys-

tems below the N´eel temperature. Here the two sublattices tend to be antiparallel

3.2 Antiferromagnetism 133

because of the superexchange stabilization, but are still inﬂuenced by an applied H .

Because of the existence of crystalline anisotropy, the direction of the applied ﬁeld

relative to the preferred direction of the spins must be taken into account. The impor-

tance of magnetocrystalline anisotropy will be realized in our discussions of ferrite

in Chap. 4. For this exercise, the anisotropy is considered to be uniaxial and our dis-

cussions will be limited to the cases of H parallel and perpendicular to the easy axis.

Because of the complexity of the analytical procedure, only the results will be pre-

sented in this text. For details of the derivation, the reader is advised to consult texts

such as Morrish [31]. The relations for the parallel and perpendicular susceptibility

are stated as

.T/ D

kT 



C N



;



.T / D



C N



; (3.40)

where B

/ is the ﬁrst derivative of the Brillouin-Weiss function and



 N





; (3.41)

where M

D M

DM

at H D 0. If it is assumed that N

D 0,(3.40) can

be plotted as function of T=T

with S as a variable parameter, without assigning a

speciﬁc value to N

, as shown by Lidiard’s .T/calculated graphs in Fig. 3.13 [32].

In the 1= vs. T format where N

¤ 0, the curves appear as sketched in Fig. 3.14.

Fig. 3.13 Susceptibility of an antiferromagnet computed as a function of temperature in units

reduced by the Neel temperature T

. Image is based on computations of Lidiard [32]

134 3 Magnetic Exchange in Oxides

Fig. 3.14 Inverse

antiferromagnetic

susceptibility modeled from

the Curie law as a function of

temperature, indicating the

asymptotic Neel temperature



discussed in Sect. 1.4

For polycrystalline or powdered materials consisting of randomly oriented

particles, this discussion can be extended by means of a simple argument. In

general, the applied ﬁeld will form an angle  with the easy axis of a particular

crystallite. Therefore, each single crystal grain will contribute a parallel and a per-

pendicular component to the magnetization, and thereby to the overall susceptibility

which can be expressed as



cos  C

sin : (3.42)

Since 

H cos 

and 

H sin

,(3.42) becomes



D 

cos

 C 

sin

; (3.43)

and after averaging over all particle orientations, the susceptibility of a polycrys-

talline specimen is



D 

cos

 C 

sin

 D



: (3.44)

One additional estimate for the susceptibility should be added to this discussion.

The ratio of the polycrystalline susceptibility at T D 0 to its value at T D T

can

be obtained from (3.44) by setting 

D 0 at T D 0,and

D 

(a constant

value) at T D T

to yield



.0/



: (3.45)

This polycrystal susceptibility can be estimated from a simple interpolation sug-

gested by (3.44) between the 

and 

curves in Fig. 3.14.

Application of these theoretical results to speciﬁc materials systems has been

expounded in a number of standard texts and comprehensive reviews. For our

purposes, some generic examples will be given to illustrate the relation between

covalent bonding, superexchange, and the magnetic sublattice arrangement.

3.2 Antiferromagnetism 135

3.2.3 Antiferromagnetic Spin Conﬁgurations

In nonmetallic compounds, antiferromagnetism reveals itself in a vast number of

chemical structures that include, besides the oxides, sulﬁdes, selenides, tellurides,

and of course, halides of both simple and more complex combinations of cations in

various crystal structures both natural and synthetic. To explore this ocean of pos-

sibilities in any depth would defeat the purpose of this text, but it is, nonetheless,

important that the reader gain an appreciation of the background and the more basic

facets of this most common type of magnetic interaction. As explained in Sect. 3.1,

there are two overriding factors that determine the nature of the spin alignment

in most situations: (1) direct overlap e

 p¢ bonding is expected to dominate the

exchange between octahedral sites and (2) the sign of the exchange constant J would

be negative in situations where the cations have similar valence charges and elec-

tronic conﬁgurations.

For particular crystal lattices, the sites that make up the opposing sublattice, des-

ignated as j , are not always the crystallographic nearest neighbors. The determining

factor in each case is the disposition of the anion (oxygen) relative to the cation

neighbor, because it is the anion that provides the chemical bond which establishes

the stabilization energy. There are three common magnetic structures that need to

be distinguished: simple cubic (perovskites), body-centered cubic (rutile), and face-

centered cubic (one-metal oxides). The ﬁrst two are conventional in the sense that

the j sublattice consists of nearest neighbor cations and the N´eel temperature ex-

pressedby(3.38),

D

 N

/ for rutile and perovskite



for

: (3.46)

The general relations between J ,

and T

for this antiferromagnet are

J D

Dq

3kT

2zS.SC 1/

; (3.47)

from which we can express

S C 1

.q D 2; simple cube/;

S C 1

.q D 4; face-centered cube/; (3.48)

S C 1

.q D 1/ :

136 3 Magnetic Exchange in Oxides

-t

PEROVSKITE

FACE-CENTERED CUBE

-2p

-e

Fig. 3.15 Comparison of bonding linkages across cube faces for (a) a face-centered cube and (b)

a perovskite. The difference is contribution of direct t

t

 bonds in (a)

Note that q D 2 for antiferromagnetism and originates from the deﬁnition of (3.38);

it is also double that of the ferromagnetic case (q D 1). A more interesting situation

is the face-centered cube shown in Fig. 3.15a, for which q D 4. In this case, the op-

posing magnetic sublattice comprises next-nearest neighbors for reasons that can be

seen from the diagram of the face. Along the unit cell edges 180

M–O–M linkages

between cations occupying octahedral corner sites provide the strongest superex-

change from e

–p¢ bonds, and these next-nearest-neighbor interactions dominate

over the diagonal interactions between the nearest-neighbor corner to face-center

sites. If the intersublattice coefﬁcient is greater than that of the intrasublattice coef-

ﬁcient in this structure, i.e., jN

j > .3=4/ jN

j, and from [33]

D

.one-metal oxide/; (3.49)

and we see that only N

can be involved and the factor of 2 in the denominator of

(3.38) is increased to 4. This occurs because of the sublattice conﬁguration of lay-

ered <111> planes shown in Fig. 3.16 for this structure must have the 12 nearest

neighbors occurring six in-plane of the same spin alignment and six in the adjacent

plane of opposite spin alignment, thereby producing a cancellation of the N

contri-

butions. However, coupling to nearest neighbors can still inﬂuence the value of N

through direct linkages in tandem. The relations corresponding to (3.38) apply here

also, but with q D 4.

Experimental evidence of the spin conﬁgurations was obtained by the powerful

tool of neutron diffraction, which can sort out planes of common spin orientations

in a manner similar to that of X-ray diffraction with ordinary crystal lattices. An

example of these data above and below the N´eel temperature for MnO is shown in

Fig. 3.17 [34]. Note the radical difference in the patterns for the two temperature

regimes.

3.2 Antiferromagnetism 137

Fig. 3.16 Antiferromagnetic structure of a diatomic metal oxide of cubic symmetry

Fig. 3.17 Neutron-diffraction pattern from MnO, which is antiferromagnetic below the Neel tem-

perature at 120 K. The upper trace was from the antiferromagnetic state; the lower one is for

paramagnetism at room temperature. Image is adapted from Lax and Button presentation ([13],

Fig. 3.22) of data by Shull et al. [34]

Figure3.18 shows another common antiferromagnetic structure rutile, which is

a tetragonal with magnetic cations centering octahedrally coordinated ligands with

axes directed along twofold lattice symmetry axes. The local antiferromagnetic 180

cation–anion–cationarrangements produce the particular spin ordering shown in the

ﬁgure. In this case the N´eel temperature T

D.C =2/



 N



follows (3.46).

138 3 Magnetic Exchange in Oxides

Fig. 3.18 Basic cell of rutile

indicating the octahedrally

coordinated ligands

surrounding the internal

cation with axes along a

lattice axes of twofold

symmetry

Table 3.4 Parameter data from selected antiferromagnetic compounds

Crystal

structure

mole

Compound T





(cgs units)

MnO fcc 122 610 5.0 0.69 4.40

FeO fcc 198 570 2.9 0.77 6.24

CoO fcc 293 280 0.96  3.0

NiO fcc 520 – – 0.67 –

CuO monoclin. 453 ––––

MnS fcc 165 528 3.2 0.82 4.30

MnF

rutile 72 113 1.6 0.75 4.08

FeF

rutile 79 117 1.5 0.72 3.9

CoF

rutile 38 53 1.4 – 3.3

NiF

rutile 73 116 1.6 – 1.5

MnO

rutile 84 – – 0.93 –

References may be found in the review article by Nagamiya et al. [19]

Mn is in the 4C state, which means that only the t

orbitals are involved in the superexchange

Because of the relationship between spin alignments and bonding, simple metal

oxides (M

2

) can form antiferromagnetic compounds of the face-centered cube

(MO) or rutile (MO

) types. Perovskites with magnetic structures that approximate

simple cubic usually feature M

cations that have inactive e

–p¢ bonds. When

these 2C ions are from the upper half of the series, the N´eel temperatures are sig-

niﬁcant because of the availability of strong e

–p¢ bonds. A comparison of the

properties of these ions in the two crystal structures is informative. As listed in

Table 3.4, T

and 

values [19] are greater for the face-centered cubic structure.

Part of the difference can be attributed to the 180

M–O–M bond angles, which

provide the largest overlap integral and greatest stabilization energy. For rutile, the

angle is less than 180

, which is more typical of ferrimagnetic oxides to be examined

in Chap. 4

3.3 Antiferromagnetic Oxides 139

3.3 Antiferromagnetic Oxides

Chemical compounds with antiferromagnetic spin alignments are the most common

of materials that exhibit magnetic properties. Even magnetically undiluted mate-

rials that are paramagnetic at room temperature usually reveal a N´eel transition

if the temperature is lowered far enough. Although the present discussion is re-

stricted to oxides, halides and other compounds that incorporate ions of the 3d

series will be included wherever they can illustrate important features. The crys-

tallographic systems that have been studied extensively both as vehicles for basic

science investigations and for practical applications are the chemically simple one-

metal compounds already introduced in the previous section and the more complex

perovskites.

3.3.1 One-Metal Oxides

In the previous section, magnetocaloric properties of some of the divalent 3d

metal

oxides were used as examples of the thermal effects that occur at the antiferro-

magnetic order–disorder transition. With the exception of CuO, which features a

noncubic structure inﬂuenced by the Jahn-Teller distortions of the normally octahe-

dral sites, each of them is of the face-centered cubic structure. The relation between

the electron conﬁgurations and the N´eel temperatures are summarized in Table 3.5.

Exchange stabilization energies z

are deduced from the T

values with

the aid of (3.48).

The ions from the lower half of the series are typically trivalent, which pre-

cludes their occurrence in the M

2

face-centered cubic oxides. As a result,

lower symmetry molecular structures are formed without 180 ˚ bonds and strong

antiferromagnetism does not appear. Since the ions from n D 1, 2, or 3 conﬁgu-

rations have only t

electrons, bonding is achieved by means of t

orbitals with

oxygen that can be stronger than the t

–p bonds of typical formations when 180

angles are available. In this case, the  overlaps are a combination of ¢ and .As

in all chemical compounds, the directionality of the t

orbital lobes in relation to

their 2p lobe bonding partners of the oxygen ligands is the determining inﬂuence in

establishing the particular stereochemistry of the molecular structure.

Table 3.5 Superexchange data of the 3d

ions in one-metal oxides

Ion Conﬁg. ST

meV

E

hop

meV

O t

5/2 122 1:2 90 100

O t

2 198 2:8 136 –

O t

3/2 293 6:8 182 300

O t

1 520 22:4 270 600

1/2 453 52:0 156 600

CuO

is of monoclinic structure probably because of the Jahn-Teller nature of the Cu

ioninan

octahederal site