Dionne G.F. Magnetic Oxides

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140 3 Magnetic Exchange in Oxides

The magnitudes of the N´eel temperatures presented in Table 3.5, which are

correlated with the z

through the relation of (3.48), were analyzed by

considering the various covalent linkages that occur in the face-centered cubic sys-

tem [35]. Inspection of these results immediately reveals the seemingly paradoxical

conclusion that stability of the antiferromagnetic order varies inversely with the

spin values in a systematic progression for which an explanation is not readily

obvious. Insight can be gained by examining the nature of the chemical bonding

in this structure. Recalling the implications of (3.41) when competing molecu-

lar ﬁeld coefﬁcients are present, we ﬁrst recognize that the main e

–p¢–e

180

antiferromagnetic couplings in Fig. 3.15a are between next-nearest neighbors along

the cube edges. The nearest neighbor cation spins are along face diagonals with

no intermediary oxygen ions. The issue, therefore, is whether competing ferromag-

netic t

–t

direct exchange along the diagonals can offset the antiferromagnetism

expected from the superexchange combinations of e

–p¢–e

and t

–p–t

be-

tween next-nearest neighbors. Figure 3.15a indicates that such a mechanism could

be t

–t

bonds acting consecutively across face diagonals. For this discussion, we

refer to this direct exchange as



–t



, with an individual stabilization energy

designated as b

Before we attempt to sort out these effects, an important point must be estab-

lished. If the intersublattice coefﬁcient is greater than that of the intrasublattice

coefﬁcient, i.e., N

> .3=4/ N

, we recall from (3.49)thatT

DCN

=4,and

we see that only N

is involved. This arises from the sublattice conﬁguration of lay-

ered <111> planes shown in Fig. 3.16. This structure has 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

contributions.

However, the nearest neighbor couplings can still affect the value of N

.Ifwein-

clude the nearest-neighbor inﬂuence from two consecutive t

 t

couplings that

favor ferromagnetism, the computational models developed in Appendix 3A specif-

ically for this problem can be used to deduce the values of b

 20 meV and





 2 meV, and the energy of the ferromagnetic coupling b

5 meV.

Since this latter contribution represents the direct exchange J

term in (3.14), the

negative sign must be used in this convention.

A summary of the values employed in the calculations of Appendix 3A is given

in Table3.6. From the resulting totals for the calculated stabilization energies, T

values are determined and compared with the measured values listed in Tables 3.4–

3.6 and plotted in Fig. 3.19. These data will be useful in interpreting the properties

of certain perovskites to be described next.

3.3.2 ABO

and A

Perovskites

The early investigations of antiferromagnetic oxides logically focused on the one-

metal systems. In later years, however, efforts were directed toward the perovskite

family of compounds that also feature M–O–M 180

bonds along the cubic (or

3.3 Antiferromagnetic Oxides 141

Table 3.6 Magnetic exchange contributions to M

2

antiferromagnetic

compounds

2

MnO FeO CoO NiO CuO

Half-ﬁlled orbitals 2e

2g 

–

240 240 240 240 120



36 24 12 – –

180 120 60 ––

Net

96 144 192 240 120

90 136 182 270 156

calc:

130 210 309 462 348

exp:

122 198 293 520 453

The model employed here is based on the next-nearest neighbor values of

D 20meV, b



D 2meV, and 5meV for the second-order direct

exchange contribution from nearest neighbors b

, which is designated as neg-

ative because of its destabilizing inﬂuence on the antiferromagnetic ordering.

Note that z

D 12 for the nearest neighbors, which is double the z D 6 for the

next-nearest neighbors

Fig. 3.19 N´eel temperatures of simple transition-metal oxides. Data are listed in Tables 3.4–3.6

tetragonal) cell edges, but without the presence of direct exchange-coupled nearest

neighbors (see Fig. 3.15b). One of the reasons for this emphasis is the availability

of many of these compounds in stable ceramic or single-crystal form where their

magnetic and dielectric properties could be utilized in practical applications. The

discussion of this family will be conﬁned to a description of two basic crystal struc-

tures and unique behavior that will be examined in more detail in later sections. For

a comprehensive listing of this family and its various properties, the reader should

consult Goodenough and Longo [36]. Reference to more recent developments will

be mentioned in their proper context.

The basic perovskite (ABO

) unit cell is sketched in Fig. 3.20,whereAdes-

ignates a large cation site with 12-fold dodecahedral oxygen coordination, often

occupied by a trivalent member of the lanthanide (rare-earth) series, and B is an

octahedral site that harbors ions of the 3d

transition series. N´eel temperatures are

142 3 Magnetic Exchange in Oxides

Fig. 3.20 Diagrams of A (octahedral) and B (dodecahedral) cation sites in generic perovskites.

generally low because e

–p¢ bonds in the 3C ions are not operative in most cases,

and many properties of interest occur well below room temperature.

Variations of the A

compounds, such as A

also occur (the

original compound from which the term perovskite was coined is diamagnetic

), as well as solid solutions of multiple cations in each type of site

that maintain the required 6C total cation charge. The most frequently encountered

example of a B

compound is the ferroelectric Ba

(BTO) with its rela-

tives Sr

(STO), Ca

,andPb

in which the Ti

ion is

and therefore diamagnetic.

The A

form, often referred to as the K

MnF

or K

NiF

structure sketched

as part of Fig. 8.1, is tetragonal and therefore has a different crystal ﬁeld symmetry at

the octahedral (B) site. The elongation of the c axis results in an orbital state energy

level structure of the type sketched in Fig. 2.29 for a tetragonal c=a > 1 distortion

of the d

case. One implication of this situation is the splitting of the E

doublet,

which allows the d

state to stabilize relative to its d

y

partner. In cases

where the e

levels are only partially ﬁlled, i.e., with one electron or one hole and

the exchange occurs by means of d

–p¢ bonds as illustrated in Fig. 3.21.Wherethe

B ion has a d

(e.g., Mn

)ord

(e.g., Cu

) conﬁguration, the Jahn-Teller effect

that would be expected to remove the E

degeneracy from an undistorted cubic or

rhombohedral site is preempted by the overriding D

lattice distortion.

In A

and synthetic variations of ABO

compounds, mixed-valence Cu ions

in these tetragonally distorted sites can supply charge carriers for superconductivity.

This subject will be examined in Chap. 8. The magnetic behavior of mixed-valence

Mn ion combinations, however, lends credence to the notions of superexchange and

charge transfer exchange explained in the foregoing sections.

Ions of the transition elements with partially ﬁlled e

orbitals can be formed from Mn, Fe, Co,

Ni, and Cu in either the 2C or 3C states (in the case of Mn, the 4C state as well). However, only

, which has special magnetoelastic properties, and Fe

occur readily in the ABO

structure,

which often departs from cubic symmetry.

3.3 Antiferromagnetic Oxides 143

Fig. 3.21 Tutorial model of

spin transfer in a

tetragonally-split e

doublet

with an active lower z

state

–y

M⬘

3.3.3 The Mixed-Valence Manganite Anomaly

Jahn-Teller distortions from cubic (O

) to tetragonal (D

) or orthorhombic (D

)

are prevalent in oxygen coordinated octahedral sites containing Mn





ions.

In mixed-valence ABO

compounds, however, unexpected magnetic and crystallo-

graphic effects are observed. The typical manganite composition La

has

been the subject of detailed investigations into its crystallography and magnetic be-

havior, particularly when Sr

or Ca

ions are blended with La

in the A sites

to create a mixture of Mn

and Mn

ions in the B sublattice, according to the

formula



1x



1x

[37].

In the generic



02C



MnO

system, all three oxidation states of Mn can

occur, offering a variety of 180

superexchange interactions. For the cubic octahe-

dral coordination, crystal-ﬁeld effects dictate that the t

orbital states are of lower

energy and are half-ﬁlled to satisfy the Hund’s rule spin polarization requirement

for each of Mn





,Mn





,andMn





. For all combinations of ex-

change pairs, the t

electrons favor antiferromagnetism via -bonding as in the

case of Mn

–O

2

–Mn

.ForMn

–O

2

–Mn

, the stabilization is determined

by the stronger ¢-bonding e

states. For Mn

–O

2

–Mn

, a single electron in the

states (Mn

case) can be stabilized by a static Jahn-Teller (J-T) orthorhombic

distortion that splits energy levels as shown in Fig. 2.29. Where the distortions are

cooperative, a tetragonal/orthorhombic phase will appear with axis ratio c=a > 1

and the half-ﬁlled d

orbital is stabilized relative to the empty d

x2y2

state. In the

perovskites, however, Mn

–O

2

–Mn



–d



couplings do not always follow

these rules.

In (La,Ca)MnO

, three types of exchange can occur: d

–d

, d

–d

,and

–d

, as listed in Table3.7. Based on the qualitative estimates of exchange

energy summarized in Table 3.6 (including footnote), the d

–d

couplings should

be strongly antiferromagnetic with 3b



C b

. 26 meV/ stabilization

energy, the d

–d

(and d

–d

) couplings should be strongly ferromagnetic with

. 20 meV/ stabilization and the d

–d

couplings should be antiferromag-

netic with 3b



. 6 meV/ stabilization. For a random distribution of the Ca

ions in the lattice, we would expect a corresponding randomization of the Mn

144 3 Magnetic Exchange in Oxides

Table 3.7 Magnetic exchange of d

electrons in manganites

y

Ion pair d

, d

a b Axes c Axis Net

$ d

"# 

"# ¢

str

"# ¢

str

$ d

"# 

"" ¢

str

– ""

str

––"# ¢

str

$ d

"# 

"# ¢

str

– "#

str

$ d

"# 

"" ¢

mod

"" ¢

mod

$ d

"# 

"" ¢

str

– ""

mod=str

$ d

"# 

––"#

Based on rules developed by J.B. Goodenough [28]

J-T splitting of e

orbitals producing c=a > 1 distortion

Proposed quasi-static J-T version of d

$ d

that causes ferromagnetism

through charge transfer into empty orbital states. It occurs in a rhombohedral

structural phase that exists when the static cooperative distortions that cause

orthorhombic phases are absent

Conditions similar to those of footnote a, but featuring charge transfer

superexchange that promotes ferromagnetism

and Mn

dispersal that would lead to antiferromagnetic ordering for almost ev-

ery value of x, except possibly near x D 0:5,wherethed

–d

couplings could

dominate with ideal charge ordering.

When the fraction of Mn

is in the range from 0.1 to 0.5, however, mag-

netization measurements indicate that ferromagnetism prevails [37] with Curie

temperatures varying according to the data of Fig. 3.22. For ferromagnetism to oc-

cur at x values where the most abundant exchange combination is antiferromagnetic

–d

, the ready explanation is that J-T effects of the d

ions could suppress the

antiferromagnetic stabilization over this concentration range. Such an effect was

proposed by Goodenough et al. to account for the observed ferromagnetic result

[38]. The features of the phenomenon that are peculiar to this system will be de-

scribed in connection with magnetoresistance properties in Sect. 8.3 as resulting

from a vibronic breathing mode of the octahedral ligands that occurs when the sym-

metry is cubic or rhombohedral (trigonal) with a degenerate E

term. This latter

condition was supported by experiment as indicated by the correspondence between

crystallographic and magnetic states in Fig. 8.10 [39].

In Appendix 3B, a mathematical model utilizing empirical exponential functions

is constructed to represent the peculiar variations in superexchange between d

and

ions as a function of the d

ion fraction x. Apparently, the transition from an-

tiferromagnetism to ferromagnetism near x D 0:1 and back to antiferromagnetism

above x D 0:5, is inﬂuenced by the concentration of Mn

ions on crystallographic

symmetry, which in turn allows for an E

term degeneracy and the vibronic J-T

effect that occurs only in this limited x range. The results of the model applied to

the calculation of Curie temperatures ﬁtted to the data of Figs. 3.22 and 8.10 are

plotted in Fig. 3.24. For this example, a random dispersal of valence states is as-

sumed and the previously determined stabilizations energies of b

D 20 meV

and 3b



D 6 meV are used to compute values effective exchange constants

3.3 Antiferromagnetic Oxides 145

Fig. 3.22 Classic plot of

Curie temperature vs. x for

(La

1x

/Mn

1x

Data are from Jonker and Van

Santen [40]. See also

Fig. 8.10

D 2:2 meV, J

(and J

/ D 3:3 meV, and J

D1:2 meV. The resultant an-

tiferromagnetic stabilization energy for end-member CaMnO

(at x D 1) becomes

36 meV, which equates to T

D 116 K as computed from the appropriate relation

in (3.49), in reasonably good agreement with the measured value of 130 K. For the

opposite end-member LaMnO

(at x D 0), a net stabilization of 156 meV would

produce an expected T

D 453 K, far from an experimental value of only 60 K.

At x D 0:5, the combined d

–d

and d

–d

couplings with random charge or-

dering would have a total ferromagnetic stabilization energy of 240 meV capable

of producing a T

D 364 K, if acting independently. Clearly the data and model

constructed to ﬁt them indicate that the d

–d

couplings do not behave in the con-

ventional manner particularly over the lower half of the Mn

concentration range.

At a point in the regime x  0:1, the crystallographic phase changes to rhombo-

hedral (trigonal symmetry with no J-T splitting of the e

levels) and remains such

until x  0:5 is reached. With half the Mn ions in the 4C state, the vibronic J-T

condition breaks down, and the remaining Mn

–O

2

–Mn

couplings revert to

antiferromagnetism, combining with the growing population of antiferromagnetic

–O

2

–Mn

couplings to produce mainly antiferromagnetism in the range

from 0:5  x  1:0. At higher temperatures, changes in the cation charge distri-

bution could enable the rhombohedral phase to extend beyond x D 0:5, to possibly

account for a second antiferromagnetic/ferromagnetic transition ﬁrst reported by

Jonker and Van Santen [40]for



0:3

0:7



MnO

146 3 Magnetic Exchange in Oxides

Appendix 3A Analysis of M

2

Exchange Interactions

If we consider that the nearest neighbor inﬂuence on N

arises from two consecutive

–t

direct couplings that favor ferromagnetism according to the direct exchange

term of (3.14), the following model can be constructed: We begin by assuming

that the values of z

deduced from measured T

values listed in Table 3.4

are the arithmetic sum of contributions from b

and b



superexchange and

direct exchange. Simultaneous linear equations are then formulated from the data

according to

MnOW 2zb

C 3zb



C 3z

D 90 meV;

FeOW 2zb

C 2zb



C 2z

D 136 meV;

CoOW 2zb

C zb



C z

D 182 meV; (3.50)

NiOW 2zb

C 0 C 0 D 270 meV;

CuOW zb

C 0 C 0 D 156 meV;

where z and z

are the respective numbers of neighbors in each case. Solution of

these equations produces nearly uniform values that are averaged to zb



120 meV and zb



C z

D48 meV.

Further dissection of these effects can lead to estimates of the relative values of



and b

bonding components. Estimates of these superexchange contri-

butions can be obtained from measurements of the N´eel temperatures of Mn





in the cubic perovskite CaMnO

and Ni





in face-centered cubic NiO because

there is no possibility of ferromagnetic effects from some form of t

–t

bonding.

The d

conﬁguration of VO shown in Fig. 3.23 involves superexchange of the three

half-ﬁlled t

states, with no contributions from the e

shell, i.e., t

in conven-

tional nomenclature. On the contrary, the d

conﬁguration shown in Fig. 3.23bhasa

ﬁlled t

shell, but provides full superexchange from its two half-ﬁlled e

states, i.e.,

. As a consequence, these two situations are good examples for comparing the

superexchange effects originating from individual  and -bonding orbitals in this

Fig. 3.23 Model comparing the Aufbau cation spin occupation for V





and Ni





that determines the superexchange energy and the ultimate values of T

Appendix 3B 147

lattice geometry. In both cases, the cations occupy octahedral sites and are linked

covalently through 180

bonds. From (3.48),





D 2

S C 1

for CaMnO

;

D 4

S C 1

for NiO (3.51)

and the relative magnitudes for the average b



and b

ratio, weighted ac-

cordingly, can be expressed by





.Mn/

.Ni/



C 1



C 1



; (3.52)

where S

D 3=2 and S

D 1. Measurements have indicated that T

.Mn/ 

130 K[37]andT

.Ni /  520 K[19]. If U



 U

(an average value for the

oxides is about 8 eV, with variations due to differing ionization potentials, crystal-

ﬁeld splittings and overlap integrals), we arrive at b



W b

D 0:1.Ifthe

calculation is extended further, with the aid of (3.26) for nearest neighbors number

z D 6 and the Boltzmann constant k D 8:62510

5

eV=K, we estimate magnitudes

of b

 20 meV from the total of 120 meV listed in Table 3.4, and we can now

deduce zb



 12 meV, or the individual b



 2 meV.

From this result for b



, we can estimate the total stabilization energy for

the ferromagnetic contribution as 60 meV, i.e., 48 C12 meV. Since there are twelve

nearest neighbors .z

D 12/, the magnitude of the individual b

 5 meV.

Appendix 3B Curie Temperature Model for (La,Ca) MnO

If the spatial distribution of Mn

ions is assumed to be random, the gain in stabi-

lization energy per covalent coupling E

D z

due to magnetic ordering

may be approximated from the exchange Hamiltonian H

D2

 S

.By

setting E

DH

, we can work with positive energies and deﬁne further

D E

C E

D 2z



C p



cos ;

(3.53)

where z is the number of nearest neighbors, J

, J

.D J

/,andJ

are the re-

spective exchange constants of the individual magnetic pair interactions, and p

and

are the probabilities of occurrence of S

and S

spins in any given site, and  is

the angle between spins, and is assumed to be uniform throughout the spin system

with a value of 0 or  depending on whether the ordering is parallel or antiparallel.

148 3 Magnetic Exchange in Oxides

Since p

D 1  x and p

D x for a random distribution, (3.53) may be expressed

as a function of x:

D 2z

.1  x/

C 2x .1  x/ J

C x

cos 

D 2zJS

cos ; (3.54)

where S D .1  x/S

C xS

is a weighted average spin value, and J is the corre-

sponding average J

coefﬁcient.

The variation in E

with x may be inferred directly from the Curie/N´eel

temperature data of Fig. 3.23, which register a peak T

> 300 Katx  0:3.

A mathematical model is helpful in sorting out the contributions of the individual

terms in (3.54). To this end, the following empirical relation for J

was constructed

to simulate the vibronic J-T effects of the Mn

–O

2

–Mn

interactions as a func-

tion of x:

D J

1  exp Œ.x  x

/=d



1  exp Œ.x

 x/ =d



; (3.55)

where x

, d

, x

,andd

are parameters that may be structurally and temperature de-

pendent. To relate E

to the Curie temperature the relation kT

D E

.S C 1/ =3S

is employed.

Figure3.24 shows plots of the calculations based on (3.53) through (3.55) over

the full range of x. To obtain a good ﬁt with experiment the parameters were chosen

Fig. 3.24 Calculated plot of E

and kT

vs. x, including estimated contributions of E

, E

C E

. Parameter values used with exponential functions of (3.55)arex

D 0:1, d

D 0:4,

D 0:7,andd

D 0:14

References 149

to produce a kT

 25 meV .T

D 300 K/ with a peak at x D 0:3.Fromthis

exercise, J

D 2:2 and J

D 3:3 meV. The value of J

D1:2 meV is based on

the reported N´eel temperature T

 130 KforLa

[38]. (J

values

above x D 0:5 are shown as a dashed line that is an artifact of the mathematical

function of (3.55) and are not intended to represent any quantitative physical reality

in this regime.) It is, therefore, concluded that the relative importance of the terms

in (3.53) is heavily weighted toward the charge transfer E

C E

components in

the vicinity of x D 0:3. The source of ferromagnetism in this system lies both in the

strength of the J

interaction and in the J-T assisted ferromagnetism of J

in the

range 0  x  0:5.

References

1. W. Heitler and F. London, Z. Physik 44, 455 (1927)

2. L.E. Orgel, Introduction to Transition-Metal Chemistry: Ligand-Field Theory, (John Wiley,

New York, 1959)

3. G.F. Dionne, Magnetic Interactions and Spin Transport, A. Chtchelkanova, S. Wolf, and

Y. Idzerda, eds., (Springer, New York, 2003), Chapter 1

4. C.J. Ballhausen, Molecular Electronic Structures of Transition Metal Complexes,

(McGraw-Hill International, Chatham, Great Britain, 1979), pp. 84–89

5. E. Cartmell and G.W.A. Fowles, Valency and Molecular Structures, (Butterworths, London,

1961), Chapter 8

6. P.W. Anderson, Phys. Rev. 115, 2 (1959)

7. J.B. Goodenough, Prog. Solid State Chem. 5, 145 (1972), Section IID

8. A.H. Morrish, The Physical Principles of Magnetism, (John Wiley, New York, 1965), p. 279

9. K. Yosida, Theory of Magnetism, (Springer, New York, 1996), p. 54

10. R.M. White, Quantum Theory of Magnetism, (Springer-Verlag, New York, 1983), Chapter 2

11. P.W. Anderson, Solid State Phys. 14, 99 (1969)

12. J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1962); A277, 237 (1964); A281, 401 (1964);

A285, 542 (1965); A296, 82, 100 (1966)

13. B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics, (McGraw-Hill, New York,

1962), p. 65

14. H.B. Kramers, Proc. Amsterdam Acad. Sci. 33, 959 (1930); also H.B. Kramers, Physica 1, 182

(1934)

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

Chapter 3

16. C. Zener, Phys. Rev. 82, 403 (1951)

17. P.-G. de Gennes, Phys. Rev. 118, 141 (1960)

18. P.W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 (1955)

19. T. Nagamiya, K. Yosida, and R. Kubo, Adv. Phys. 4, 1 (1955)

20. M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954)

21. T. Kasuya, Prog. Theor. Phys. 16, 45 (1959)

22. K. Yosida, Phys. Rev. 106, 893 (1957)

23. J.B. Goodenough, New Developments in Semiconductors, P.R. Wallace, R. Harris, and

M.J. Zuckermann, eds., (Nordhoff International Publishing, Leyden, 1973), pp. 145–151

24. J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959)

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

Chapter 3, p. 213

26. J.B. Goodenough and A.L. Loeb, Phys. Rev. 8, 391 (1955)

27. J.C. Slater, Phys. Rev. 35, 509 (1930)