Dionne G.F. Magnetic Oxides

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100 2 Magnetic Ions in Oxides

in which the single 3s electron of Na is transferred to ﬁll the single empty state in

the 3p shell of Cl. In the most straightforward approach, the energy of stabilization

among neutral atoms is determined by the resultant of two parts: (1) the Coulomb

energy, which arises from the mutual attraction between the positive and negative

ions, tempered by the shielding of ﬁlled shells, and (2) the algebraic sum of the Na

electron ionization potential energy .C5:14 eV/ and the Cl 3p

electron afﬁnity

energy .3:71 eV/ that may produce additional stabilization by the elimination of

partially ﬁlled shells. From the example worked out in Kittel [50]:

Na ! Na

C e



 5:14 eV;

Cl ! Cl



 e C3:71 eV; (2.51)

C Cl



! NaCl  1:43 eV C bond energy:

A ﬁrst approximation to the bond energy is obtained from a simple Coulomb at-

traction given by E

Coul

D e

(see Fig. 2.32), where R

.D2:81

A) is the ﬁnal

separation between ions as determined by minimization of the total energy that

encompasses the long-range ionic attraction opposed by the short-range repulsion

among the electron charge clouds, which begins to dominate where the nuclear

separation R<R

. This calculation yields E

Coul

of 5.1 eV, which is a factor of

about 1.5 less than the experimentally determined value of 7.9eV. The discrepancy

may be accounted for by a more complete calculation that includes summations over

more of the ionic lattice. Such methods are also described in Kittel where the vari-

ous ways of computing the Madelung energy constant E

Mad

are also described. For

the NaCl lattice, E

Mad

D 1:75, as compared with the 1.55 value determined from a

ﬁt to experiment.

The valence-bond analysis of two single-electron atoms designated a and b with

respective unpaired electrons (1) and (2) begins with the deﬁnition of the full

Hamiltonian that includes explicitly all of the interactions embodied in the screening

parameter h

of H

in (2.38)

H D H

C H

; (2.52)

where H

D H

C H

D

„







(2.53)

and H

D



: (2.54)

The next step is the formation of the two-electron degenerate wavefunctions '

.2/ '

.1/ and '

D '

.2/ '

.1/. The ﬁrst of these is the normal state with the

respective electrons on their host ions; the second is the energy equivalent state with

the electrons exchanged between the two atoms. As discussed in Sect. 1.3.3,the

eigenfunction solutions are linear combinations constructed according to

anti

Œ'

.1/ '

.2/  '

.2/ '

.1/ ; (2.55)

2.5 Covalent Stabilization 101

sym

Œ'

.1/ '

.2/ C '

.2/ '

.1/ :

The energy stabilization of the bond can be seen from the electron probability dis-

tributions expressed as

anti



C '

 2'



; (2.56)

sym



C '

C 2'



and illustrated in Fig.2.31. Bonding therefore can be explained by the reduction of

the repulsive forces between the nuclei due to the screening effect of the electrons.

For that reason '

sym

is the lower energy hybrid orbital state. These deﬁnitions will be

related to the concept of stabilization energy of the crystal ﬁelds when the molecular

orbital approach is introduced.

A basic principle of the valence bond theory is that the number of bonds in the

structure be equal to the number of available valence electrons that occupy the outer

shell. In the broader picture of the Periodic table, the ionic alkali halides may be

viewed as 1–7 compounds according to the number of occupied states in the outer s

and p shells of the neutral atoms. As the combinations compress toward the center

columns of the table, the distinction between cation and anion becomes less real

and covalent bonding similar to that of like atoms such as the H

molecule becomes

the preferred model. In the balanced case of the 4–4 of carbon, silicon, germanium,

and to a lesser extent their 3–5 gallium arsenide cousin, the bonds result from the

exchange of electrons among the s and p shells of neutral atoms. From the quar-

tet of four-electron orbital combinations designated by s

, the diamond lattice

structures of pure tetrahedral bonding (see Fig.2.5) are produced. With this con-

cept, therefore, the practice of bond formation by hybridization of atomic orbitals

was initiated.

Solutions to bonding problems by the valence-bond method are carried out from

ﬁrst principles by constructing a Hamiltonian that includes all of the nuclear and

electronic interactions necessary to determine the binding energy of the molecule

and its equilibrium atomic conﬁguration. It has therefore proven to be a power-

ful tool in determining molecular structures, even for complicated multielectron

systems. In spite of the overall effectiveness of the valence-bond method, however,

the model suffers from the shortcoming that it deemphasizes the electron transfer in-

teractions, that is, the “ionic” exchange terms, by relying on the Madelung energy to

account for these contributions. Even when hybridized, for example, s

,thewave

functions remain combinations of the orbitals within each atom, unperturbed by in-

teractions with neighboring charge clouds. As a consequence, the basic method also

A qualitative review of the valence-bond concept, featuring vivid illustrations of hybrid bond

formations, can be found in Chang [51].

102 2 Magnetic Ions in Oxides

does not address directly the effects of crystal ﬁelds on the partially ﬁlled d electron

states, which provide a critical part of the electrostatic stabilization that determines

the magnetic properties. For this analysis, wavefunctions that represent the shared

electrons of the molecule must be obtained. Appendix 2B presents examples of how

solutions molecular-orbital hybrid eigenfunctions can be used to reﬁne the results

of a valence-bond analysis.

Appendix 2A Homonuclear Molecule Ion

Because the starting wavefunctions of a molecular orbital analysis of necessity lack

orthogonality, it is important to consider the implications of these conditions. To this

end, we review the special case of homonuclear diatomic molecule. From (2.44),



1  2h



.1  

/ C b

.2h  1/

.1  

: (2.57)

For a homonuclear diatomic molecule, U D 0, b  hE

 and (2.57) can simpliﬁed

further as



1  2h



˙ E

.2h  1/ 

.1  



1  2h

˙ 2h  



.1  

: (2.58)

Factoring of the numerator and canceling of common factors yields the results for

the antibonding and bonding energies



.1  2h/

1  

;

.1 C 2h/

1 C 

: (2.59)

Since E

is chosen to be negative relative to the zero reference energy of ions in their

uncombined “free” state, it is evident that E

will be the ground state provided that

h>0, that is, the symmetric orbital bonding state that would require antiparallel

spins to satisfy the Pauli principle requirement of combined asymmetry, discussed

in Chap. 1. Based on the sign convention adopted, this means that the transfer energy

b is negative, which is expected for attractive cross-ionic electrostatic potentials V

and V

in (2.36). Conversely, where the mutual electron repulsion is large, h<0and

the antisymmetric orbital antibonding state would be lower, offering the possibility

of ferromagnetism.

Further study of (2.59) reveals also that the destabilization of E



is greater than

the stabilization of E

, which leaves the solution with an increase in overall en-

ergy that violates the invariance of the diagonal sum, as an apparent consequence

Appendix 2B 103

of the lack of orthogonality of the starting wavefunctions. If one now considers

the total energy when both levels are populated according to the unnormalized

eigenfunctions [43],



D .1  / .antibonding/;

D .1 C / .bonding/: (2.60)

Then the total net energy becomes

D E

C E



D E

.1  2h/ C E

.1 C 2h/ D 2E

; (2.61)

which is unchanged. It is therefore concluded that only the unnormalized functions

can restore the energy balance. This suggests that a more appropriate normalization

condition is '

C '



D 2.

Appendix 2B Valence-Bond Diatomic Molecule

The usefulness of the molecular orbital approximation and how it augments the

valence-bond method (including the Madelung ionic bond limit) can be appreciated

by applying it to the case of a two-electron diatomic molecule. The exercise begins

with the construction of two-electron products of the single-electron hybrid molecu-

lar orbitals of (2.35) that now become '

D '

.1/ '

.2/ and '



D '



.1/ '



.2/.

In expanded form, the bonding and antibonding states are expressed as



D N



Œ'

.1/  



.1/ Œ'

.2/  



.2/ ;

D N

Œ'

.1/ C 

.1/ Œ'

.2/ C 

.2/ : (2.62)

If the bonding state is expanded by multiplication, ionic and covalent contributions

can be separated according to

D '

.ionic/ C '

.covalent/; (2.63)

where

.ionic/ D N



.1/ '

.2/ C 

.1/ '

.2/



;

.covalent/ D N



Œ'

.1/ '

.2/ C '

.2/ '

.1/ : (2.64)

The normalizing relation for (2.64)isN



1 C 

C 2





1=2

,where D

From (2.64) the different components of the chemical bond can now be exam-

ined. 

(ionic) is the part of the wavefunction that accounts for the two electrons

104 2 Magnetic Ions in Oxides

occupying the bonding orbital of the same atom, thereby converting either the a

or b atom into a negatively charged ion. This part is overlooked in the valence

bond method as described in the text of Sect. 2.5.1 (although later versions have

attempted to introduce it through a variational perturbation [52]). For N

D 1=2

and 

D 1,'

(covalent) will be recognized as the Heitler–London H

valence

bond function '

sym

that combines the '

and '

terms in (2.55).

There are two special cases to examine:

(1) Homonuclear molecule, 

D 1: This is the H

molecule case, for which the

bonding state becomes



D N

Œ

.1/ 

.2/ C 

.1/ 

.2/ C Œ

.1/ 

.2/ C 

.2/ 

.1/

;

(2.65)

where the normalizing relation reduces to N

D .2 C 2/

1=2

. The second

term is precisely the result of the valence bond analysis for '

sym

given by

(2.55). Most notable in this molecular orbital function is the inclusion of the

“ionic” term, which would represent the H



and H



possibilities that

are given equal weight to the covalent terms. It has been pointed out that al-

though the valence bond solution errs in its exclusion of these ionic terms, the

molecular orbital theory exaggerates their importance. In either case, however,

the magnitude of the orbital overlap integral  maintains its inﬂuence.

(2) Ionic molecule, 

>> 1 (for a nearly pure ligand orbital): This situation could

represent the NaCl molecule with '

D '

and '

D '

. The bonding state is

then expressed as

Š N





.1/ '

.2/ C 

.1/ '

.2/





Œ'

.1/ '

.2/ C '

.2/ '

.1/



; (2.66)

where N







C 2





1=2

. Because   1, N

 

1

in the particular

case of NaCl, (2.66) can be further simpliﬁed to

Š '

.1/ '

.2/ C 

1

Œ'

.1/ '

.2/ C '

.2/ '

.1/ ; (2.67)

where the covalent term can now be treated as a perturbation that is inversely pro-

portional to 

. Note that the dominant term in the ionic part of (2.64) is the one that

represents both electrons occupying the 3p orbital shell of Cl, that is, the Na



component. To a ﬁrst approximation then,

Š '

.1/ '

.2/ : (2.68)

The potential energy of the different contributions to the bonding energy of NaCl

is sketched as a function of the distance between nuclei [53]inFig.2.36.Notein

particular the difference between the covalent and ionic curves at the minimum

energy and at larger separations between the two nuclei, above the crossover of the

References 105

Fig. 2.36 Bonding energy contributions to the stabilization of the NaCl molecule as a function of

separation

two energy curves. In this regime the combination of neutral atoms NaCCl becomes

more stable than the separated ions Na

C Cl



by the amount of the net ionization

energy 1.43 eV. In purely covalent compounds this correction does not apply.

References

1. M.J. Buerger, Elementary Crystallography, (John Wiley, New York, 1956)

2. W. Borchardt-Ott, Crystallography, (Springer, New York, 1995)

3. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962)

4. H.L. Schl¨afer and G. Gliemann, Basic Principles of Ligand Field Theory, (Wiley-Interscience,

New York, 1969)

5. J.S. Grifﬁth, The Theory of Transition-Metal Ions, (Cambridge University Press, London,

1961)

6. J.H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, (Oxford University

Press, London, 1932)

7. E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra, (Cambridge University Press,

London, 1963)

8. H.B. Kramers, Proc. Amsterdam Acad. Sci. 32, 1176 (1929)

9. J.H. Van Vleck, Phys. Rev. 41, 208 (1932)

10. W.G. Penney and R. Schlapp, Phys. Rev. 41, 194 (1932); R. Schlapp and W.G. Penney, Phys.

Rev. 42, 666 (1932)

11. W. Low, Paramagnetic Resonance in Solids, (Academic Press, New York, 1960), p. 15

12. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962), p. 93

13. W. Low, Paramagnetic Resonance in Solids, (Academic Press, New York 1960) p. 22

14. M.H.L. Pryce and W.A. Runciman, Disc. Faraday Soc. 26, 34 (1958)

15. G.F. Dionne and B.J. Palm, J. Magn. Reson 68, 355 (1986)

106 2 Magnetic Ions in Oxides

16. M.T. Hutchings, Solid State Phys. 16, 227 (1964)

17. J.H. Van Vleck, Discuss. Faraday Soc. 26 90 (1958)

18. K.W.H. Stevens, Proc. Phys. Soc. A65, 209 (1952)

19. M.S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory, Applications to the Physics of

Condensed Matter, (Springer, 2008)

20. H.A. Bethe, Ann. Phys 3, 133 (1929)

21. R.S. Mulliken, J. Chem. Phys. 3, 375 (1935)

22. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962), p. 100

23. L.E. Orgel, J. Chem. Phys. 23, 1004 (1955)

24. H.L. Schl¨afer and G. Gliemann, Basic Principles of Ligand Field Theory, (Wiley-Interscience,

New York, 1969), Chapter 1

25. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962), p. 20

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

27. G. Racah, Phys. Rev. 62, 438 (1942); G. Racah also Phys. Rev. 63, 367 (1943)

28. K.R. Lea, M.J.M. Leask, and W.P. Wolf, J. Chem. Phys. Solids 23, 138 (1967)

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

New York, 1959)

30. E.C. Stoner, Proc. Leeds Phil. Soc. 2, 391 (1933)

31. M.R. Ibarra, R. Mahendiran, C. Marquina, B. Garcia-Landa, and J. Blasco, Phys. Rev. B 57,

R3217 (1998 II)

32. O.G. Holmes and D.S. McClure, J. Chem Phys. 26, 1686 (1957)

33. Y. Tanabe and S. Sugano, J. Phys. Soc. Japan 9, 753 (1954)

34. J.M. Baker, B. Bleaney, and K.D. Bowers, Proc. Phys. Soc. (London) 69, 1205 (1956); also

A.L. Kipling, P.W. Smith, J. Vanier, and G.A. Woonton, Can. J. Phys. 39, 1859 (1961)

35. M.M. Schieber, Experimental Magnetochemistry, (John Wiley, New York, 1967), p. 250

36. A. Abragam and M.H.L. Pryce, Proc. R. Soc. A205, 135 (1951)

37. H.A. Jahn and E. Teller, Proc. R. Soc. (London) A161, 220 (1937)

38. H.A Jahn, Proc. R. Soc. (London) A164, 117 (1938)

39. J.H. Van Vleck, Phys. Rev. 57, 426 (1940)

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

Chapter III, Sections IE and IF

41. J.B. Goodenough, J. Phys. Chem. Solids 25, 151 (1964)

42. M. Kaplan and B. Vekhter, Cooperative Phenomena in Jahn–Teller Crystals, (Plenum, New

York, 1995)

43. E. Cartmell and G.W.A. Fowles, Valency and Molecular Structure, (Butterworths, London,

1961), Chapter X

44. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962), p. 161

45. C.J. Ballhausen and H.B. Gray, Molecular Electronic Structures, an Introduction,

(Benjamin/Cummings, Reading, MA, 1980)

46. M. Wolfsberg and L. Helmholz, J. Chem Phys. 20, 837 (1952)

47. G.F. Dionne, Covalent Electron Transfer Theory of Superconductivity, (MIT Lincoln

Laboratory Technical Rept. 885, 1992), NTIS No. ADA2539757

48. G.F. Dionne, J. Appl. Phys. 99, 08M913 (2006)

49. L. Pauling, The Nature of the Chemical Bond, (Cornell University Press, New York, 1960)

50. C. Kittel, Introduction to Solid State Physics, (Wiley, New York, 1966), p. 90

51. R. Chang, Chemistry in Action, (Random House, New York, 1988), Chapter 9

52. E. Cartmell and G.W.A. Fowles, Valency and Molecular Structure, (Butterworths, London,

1961), pp. 92–94

53. C.J. Ballhausen and H.B. Gray, Molecular Electronic Structures, an Introduction,

(Benjamin/Cummings, Reading, MA, 1980), p. 84

Chapter 3

Magnetic Exchange in Oxides

In the previous chapter, quenching of cation orbital angular momentum by the

anion charges and the origin of the energy stabilization by covalent bonding was

introduced by elementary crystal ﬁeld and molecular orbital theory. In magneti-

cally dilute compounds, the isolated 3d

ions are inﬂuenced next by the weakened

spin-orbit coupling perturbations and multiplet structures that determine the mag-

netoelastic properties to be examined in Chap. 5. These effects are initially local

but can become cooperative when concentrations increase to levels where perco-

lation can occur, e.g., a cooperative Jahn-Teller effect. However, because the spin

alone is the agent of magnetic ordering in this series, the multiplet energies can be

largely ignored in the discussion of spontaneous magnetism. Consequently, the next

important effective ﬁeld in a ligand lattice to be addressed is the magnetic exchange

ﬁeld that arises from the transfer integral linking magnetic cations. Magnetic ex-

change, therefore, is the term used to describe the energy stabilization gained from

spin ordering (parallel or antiparallel) of atoms or ions covalently coupled in an

ionic crystal lattice.

Although the magnetic behavior is determined by the disposition of unpaired

electron spins, the underlying mechanisms are of electrostatic origin. In Chap. 1,

the concepts of direct magnetic exchange were outlined for the traditional case of

metallic elements of the transition groups. The basis for that discussion was the

origin of spontaneous magnetism from unﬁlled shells within collective electron sys-

tems, and the formalism is based on the Heitler–London solution of the H

molecule

[1] that opened the way for the valence-bond approach to chemical bonding outlined

in Sect. 2.5.

For oxides and other insulating compounds, however, direct exchange is usually

not an important contributor to magnetic properties because the metal ions are too

far apart for the mutual electron repulsion term e

to establish an antibonding

state with a Hund’s rule parallel spin alignment. If the bonding is dominated by

cation–anion interactions, the unpaired cation spins couple through the mediation

of the anion orbital states to create indirect or “superexchange” that observes the

Pauli exclusion requirement of antiparallel alignment.

This chapter is an overview of some underlying theoretical concepts of magnetic

exchange between two transition-metal cations with unpaired electron spins, and

the relation between the bonding stabilization and antibonding destabilization in

G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 3,

 Springer Science+Business Media, LLC 2009

107

108 3 Magnetic Exchange in Oxides

determining the type of spontaneous magnetic order. More speciﬁcally, the case of

3d electrons linking covalently through the mediation of 2p ligand orbitals will be

emphasized.

3.1 Interionic Magnetic Exchange

In an elementary context, the spontaneous magnetism in a crystal lattice can take

two forms: parallel spins (ferromagnetism) and antiparallel or paired spins (antifer-

romagnetism). These two extremes are analogous to the high and low-spin states

of the individual ions, and the interactions that select one in preference to the other

are essentially the same. In Fig.3.1, schematic diagrams indicate how the different

spin-couplings evolve as two magnetic cations are related through a chemical bond,

beginning with (a) paramagnetism when the spins fully localized to their parent

Fig. 3.1 Orbital-lobe diagrams for the different magnetic interaction and bonding situations

3.1 Interionic Magnetic Exchange 109

nuclei align in a polarizing H ﬁeld with an inverse square dependence on sepa-

ration. In Fig. 3.1b, direct exchange ferromagnetism arises from mutual electron

repulsion that stabilizes an antibonding state. Initially, the ground state spin distri-

butions are parallel, correlated into an S D1 high-spin state as dictated by Hund’s

rule.

As the two nuclei are moved closer together, electron–nuclear cross-term at-

traction energies can overcome the e

energy. In Fig. 3.1c, antiferromagnetism

is stabilized when the electron–nuclear attraction becomes dominant in forming a

bonding ground state, and the orthogonality becomes compromised by the over-

lapping orbital functions of the now-delocalized spins. As the transfer integral b

increases and the splitting between the bonding and antibonding states becomes

larger, the orbital energy stabilization favors the low-spin SD0 antiparallel con-

ﬁguration in conformance with the Pauli’s exclusion principle. In the case of a

metal–oxygen–metal bond in Fig. 3.1d, the antiferromagnetic ordering of the two

localized metal spins is mediated indirectly by orbital overlaps to a common anion

(termed superexchange). Analogous in some respects to the one d -electron model

used to display the spin occupancies of the crystal-ﬁeld ground states determined

by the Aufbau principle, these bonding diagrams can be used as rough guides to the

spin alignments in certain cases.

3.1.1 Molecular-Orbital Exchange Approximation

For a transition-metal ion (M) of the 3d series in an octahedrally coordinated

oxygen (O) or ligand (L) site, one-electron energy-level models are useful plat-

forms for semiempirical estimates of the crystal-ﬁeld stabilization energy [2,3]. The

ground-state electron spin distributions are presumed to ﬁll the levels sequentially

across the series (Aufbau method), in accordance with Hund’s rule and the Pauli’s

principle. If the 3d-electron orbital basis functions are hybridized with the 2p states

of the O

2

ligands, a molecular-orbital energy level model (Fig. 3.2) can represent

the contribution of the covalent bonding in these primarily ionic compounds. For

a single orbital state of a metal–ligand diatomic molecule (M–L), the stabilization

that occurs for an orbital transfer energy b

D h



C H



,asdeﬁned

in terms of the Hamiltonian energies H

and H

. By the Wolfsberg–Helmholtz ap-

proximation with h D 1 for oxides from (2.38), b

D .E

C E

/

,where

and E

are the in-lattice energies of the respective electrons before covalent

perturbations are applied, and the orbital overlap integral 



.For



<< 1, the bonding (+) and antibonding (–) state energy solutions reduced from

(2.39)are

This initial ferromagnetic bias arises from the electrostatic repulsion of two electrons that is not

included in the one-electron molecular–orbital Hamiltonian (2.38) and is implicit in the screening

parameter h. It is convenient to treat it as a high-spin state analogous to the d

electron models

that retain parallel spin alignments until stronger crystal ﬁelds (or larger b integrals) overcome

the mutual repulsion and allow antiparallel spin alignment to stabilize in the lower orbital states

producing low-spin conﬁgurations.