Dionne G.F. Magnetic Oxides

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342 6 Electromagnetic Properties

89. G.F. Dionne, D.E. Oates, D.H. Temme, and J.A. Weiss, Lincoln Lab. J. 9, 19 (1996)

90. G.F. Dionne, J.A. Weiss, and G.A. Allen, IEEE Trans. Magn. 24, 2817 (1988)

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

1962), p. 150

92. P.J.B. Clarricoats, Microwave Ferrites, (Wiley, New York, 1961), p. 63, with ˛ D 0

93. D.M. Pozar, Microwave Engineering, (Wiley, New York, 1998), p. 503

94. N. Bloembergen, Proc. IRE 44, 1259 (1956)

95. S. Chikazumi, Physics of Magnetism, (Wiley, New York, 1997), p. 348

96. P.J.B. Clarricoats, Microwave Ferrites, (Wiley, New York, 1961), p. 92

Chapter 7

Magneto-Optical Properties

In the previous chapters, the emphasis is placed on the electronic origins of local

and collective molecular magnetism in transition-metal oxides and their behavior

in alternating magnetic ﬁelds. Models of magnetic resonance based on precessing

magnetic moments provide a classical analog to quantum mechanical transitions

provided that the internal magnetic ﬁelds are large enough to produce the Zeeman

energy splittings for the particular frequency of interest. In the energy range that

can be easily reached by ﬁelds from laboratory electromagnets, electron paramag-

netic resonance (EPR) and ferromagnetic resonance (FMR) occur in the microwave

bands. However, resonances can also occur in magnetically ordered systems at the

energies of magnetic exchange. Since the exchange effects occur in the submillime-

ter and far-infrared bands, but have the properties of a magnetic-dipole stabilization,

this topic will serve as a transition to the subject of magneto-optics that is based on

magnetically polarized electric-dipole interactions with optical waves.

In the visible and ultraviolet bands, electric-dipole transitions can produce

magneto-optical phenomena without the need for large applied magnetic ﬁelds.

In this regime, the dielectric permittivity tensor with off-diagonal terms can pro-

duce nonreciprocal propagation at optical wavelengths analogous to those from

magnetic interactions with RF waves. Faraday rotation of the linear polarization of

plane-wave transmission and its complementary Kerr reﬂection effect are of ma-

jor importance for discrete ﬁber-optical technology. In later developments, optical

waveguides that simulate their microwave counterparts have shown promise for

integrated photonics technology that can beneﬁt from the nonreciprocal properties

of magneto-optical control devices. To remain within the scope of this volume, the

discussion of materials systems will be focused on the room temperature properties

of the garnet family of magnetic oxides, ﬁrst on the basic host compound yttrium

iron garnet and then on the dramatic effects of Bi

ion substitutions. The discus-

sion will review the work carried out at Lincoln Laboratory and the Department

of Physics of the Massachusetts Institute of Technology where the author was an

active participant, but is dawn heavily from the pioneering work of scientists at the

Mullard Research Laboratories in England and the Philips Research Laboratories

in Eindhoven, the Netherlands and Hamburg, Germany.

G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 7,

 Springer Science+Business Media, LLC 2009

343

344 7 Magneto-Optical Properties

7.1 Infrared Exchange Resonance

Before magnetooptical permittivity in transition-metal oxide compounds is ad-

dressed, the interionic magnetic resonance that originates from the molecular ex-

change stabilization of magnetic spins is an appropriate segue topic. Although the

exchange energies extend to the infrared bands, approaching those of electric-dipole

orbital energy transitions, the actual mechanism involves the reversal of magnetic

ion spins between parallel and antiparallel states of a covalently bonded molecule.

For convenience, it is treated phenomenologically as the result of a quasimagnetic

exchange ﬁeld. In reality, the exchange resonance effects are the result of neither

magnetic nor electric-dipole quantum transitions. Despite the fact that their fre-

quencies are typically several orders of magnitude greater than that of microwave

resonance, they greatly enhance the intensity of the microwave resonance.

7.1.1 Classical Precession Model

The phenomenological theory of gyromagnetic resonance has been applied to both

electron and nuclear magnetism represented generically as a magnetic moment m

precessing about a magnetic ﬁeld H directed along the z-axis of a Cartesian co-

ordinate system. The angular precession frequency !

D H

is derived from the

solution of the customary equation of motion (without damping) [1]

D .m  H /; (7.1)

where .Dgm

=„/ is the gyromagnetic ratio. Resonance takes place when the

frequency and sense of an electromagnetic wave circularly polarized in the x–y

plane matches those of the precession. In this basic form, !

is inﬂuenced only

by the value of H , while m determines the intensity of the observed resonance.

In a paramagnet, however, where m represents the combined moment of a collec-

tion of independent isolated moments, the intensity is determined by the degree

of saturation, which is a function of ﬁeld and temperature. This situation is af-

fected dramatically when the independence of the separate moments is removed by

magnetic exchange coupling through molecular bonding that becomes the dominant

aligning agent and creates a second resonance condition.

In a system where the individual spins form a magnetization vector M in a

molecular or exchange ﬁeld H

, the result is a treated as a Larmor-type preces-

sion usually in an infrared band. For the two magnetically opposed sublattices of a

The reader is advised that sign conventions vary in the literature of this subject. Wherever possi-

ble, the convention from Chap. 6 will be continued.

7.1 Infrared Exchange Resonance 345

Fig. 7.1 Microwave and

exchange resonance modes in

a ferrite. Precession

directions are opposite for the

two modes

ferrite (see Fig. 7.1), (7.1) is elaborated to account for the additional stabilization by

[2]

D 

ŒM

 .H C H

C H

ex1

/ ;

D 

ŒM

 .H C H

C H

ex2

/ ; (7.2)

where H

K1;2

are the respective sublattice anisotropy ﬁelds, and the corresponding

exchange ﬁelds are H

ex1;2

D N

1;2

with N

.<0/ as the intersublattice anti-

ferromagnetic molecular-ﬁeld coefﬁcient. The intrasublattice molecular ﬁelds N

and N

are ignored in this approximation.

Simplifying (7.2) for our present purposes, we assume zero-orbital angular mo-

mentum and let 

D 

D  and H

D H

D 0.From(7.2), two solutions, one

for microwaves (RF) and another for infrared waves (IR), emerge with precession

frequencies:

D H

D 

H C H

D ŒHC N

 M

; (7.3)

which represent the sketches in Fig. 7.1 in their most primitive forms. If the ferrite

is now treated as ferromagnetic with magnetization M D

 M

,whichisa

Note that although the anisotropy and exchange terms are presented in vector notation as magnetic

ﬁelds, only the anisotropy energy in its role as a demagnetizing ﬁeld that modiﬁes the applied ﬁeld

is legitimate in the Maxwell sense. H

has its origins in the chemical bond of the molecule

and draws its value and sense from the implications of the indistinguishability of the electron as

manifested by the Pauli Exclusion principle. As a consequence, it should be considered as a scalar,

separate from the sequence of perturbations affecting an individual cation, and would not inﬂuence

the removal of degeneracies in the electronic energy level structure.

346 7 Magneto-Optical Properties

conventional approach to treating ferrimagnetism classically, (7.2) simpliﬁes to a

single magnetic lattice:

D ŒM  .H C H

/ D ŒM .H C N M / : (7.4)

The dispersive parts of the undamped permeabilities for right- and left-hand circular

polarization modes (C and ) can be expressed as [3]



RF˙

D 1 C

!  !

D 1 C

!  H

;



IR˙

D 1 C

!  !

D 1 C

!  .HC NM/

; (7.5)

where !

D 4M and  D 1:76  10

rad s

1

Inspection of (7.5) indicates that the amplitude of both resonances is propor-

tional to the net magnetization, which is saturated by a large exchange energy that

stabilizes the spins into the collective magnetization vector. Conﬁrmation that 

follows a Brillouin–Weiss thermomagnetization characteristic was reported by Tin-

kham [4]. The question that arises is how the applied ﬁeld H that is responsible for

the polarization of the combined moment can produce an independent precession if

it adds vectorially to the exchange ﬁeld. The reasoning applied to the equations of

motion [(7.2)] is that both H and H

.DNM / are vectors that combine in creating

and controlling the net M , but at the same time are independent when producing

separate precessions of the same M . If the exchange energy is the result of a real

magnetic ﬁeld, then it follows that there should be no separate RF precession be-

cause the two ﬁelds would not combine as scalars, leaving only the exchange ﬁeld

with magnitude enhanced slightly by H . On the other hand, if the exchange

stabilization is not the result of a magnetic ﬁeld, there should not be an IR preces-

sion. Only the RF precession should occur. In essence, the classical model predicts

independent reversals of magnetic moments, i.e., spin “ﬂips” through 180

, from the

common ground state into identical excited quantum states by means of plane-wave

excitations of radically different wavelengths, one from microwave interactions and

the other of lattice vibration origins. To explore further this apparent paradox, it is

necessary to consider the quantum mechanical description of the physical situation.

7.1.2 Quantum Spin Transition Model

If the two ferrimagnetic precessions of opposite sense in (7.3) are replaced by an

equivalent ferromagnetic model sketched in Fig. 7.2, with both precessions in the

same direction, the question of interaction between RF and IR resonance effects

can be viewed from a quantum mechanical perspective [5]. A hybrid energy di-

agram of a single electron spin .S D 1=2/ in an exchange ﬁeld of parameter J

7.1 Infrared Exchange Resonance 347

Fig. 7.2 Ferromagnetic model for the two precession modes of a ferrite, shown in Fig. 7.1 Extra

arrows on the IR precession orbit indicate that S>1=2and the angular velocity is higher. Figure

reprinted from [5] with permission.

 2009 by the American Institute of Physics

Fig. 7.3 Energy level model for an S D 1=2 system coupled ferromagnetically in an exchange

“ﬁeld.” Molecular environment stabilizes the spin by a large energy, but leaves Kramers doublets

as degenerate levels. Occupation probability of spin ground state is determined by the exchange

energy, which has the effect of lowering the temperature of the spin system and permit FMR to

be strong up to the Curie temperature. Figure reprinted from [5] with permission.

 2009 by the

American Institute of Physics

and real magnetic ﬁeld H is shown in Fig. 7.3. It must ﬁrst be recognized that the

exchange energy that stabilizes the

S spin ground state is not the result of pertur-

bation in a spin-Hamiltonian equation, but rather the result of interionic molecular

bonding. Relative to the isolated ion states, the molecular state is stabilized by

 S



D˙2J



in the Ising approximation, with S

representing the

348 7 Magneto-Optical Properties

source of the exchange acting on S. Because the exchange coupling of the spin

vectors is not the result of a true magnetic ﬁeld interaction, both S D˙1=2 states

remain as degenerate Kramers doublets.

The two magnetic transitions indicated in Fig. 7.3 correspond to the IR and RF

precessions of Fig. 7.2. The paradox is immediately apparent from the energy level

diagram. In the quantum model, the distinction between H and H

becomes a

meaningful issue. First, the inability of H

to split the S

D˙1=2 Kramers degen-

eracy allows the Zeeman effects of energy gm

(where g  2 for a spin and

is the Bohr magneton) to become the source of the RF precession and ferromag-

netic resonance of the individual ions even when the real magnetic ﬁeld H saturates

the already exchange-coupled spins. The point here is that the paramagnetic en-

ergy level structure and resulting microwave spectral frequencies are unaffected by

the exchange stabilization. What is affected, however, is the Boltzmann population

probabilities of the spin ground state that is now determined by the combined split-

ting energy 4J

C gm

HS

of the system, and not just by the paramagnetic

term that is orders of magnetic smaller than the exchange effect.

If the Boltzmann population ratio is applied to the molecular-orbital states sepa-

ratedinenergyby4J

in Fig. 7.3,

D exp







; (7.6)

where n

C n

D n, the population density of like magnetic ions participating in

the distribution. For the energy levels of the Zeeman splitting from the applied ﬁeld

H in the absence of exchange, i.e., J

D 0, the population ratio is



D exp





HS



D exp







; (7.7)

where S

D 1 in accord with the magnetic-dipole selection rule. When the ex-

change and Zeeman energies are combined as scalars,



D exp





C gm



: (7.8)

An underlying assumption of the Boltzmann theory of population distribution

among the various energy states, e.g., vibrational modes of a CO

molecule, is the

noninteraction between individual members. Multiple energy levels of isolated mag-

netic ions induced by a common magnetic ﬁeld (paramagnetism) can also qualify

unless their individuality is lost by the spin ordering stabilization imposed by chem-

ical bonding. As a consequence, (7.7) cannot be applied as stated unless J

D 0.

Only (7.8) is a valid application of the Boltzmann law because it involves energy

changes within the constraints of the collective magnetization that are then respon-

sible to Fermi statistics dictated by the Pauli principle. Because the ground-state

population n

is common to both (7.7)and(7.8), it follows that the population

7.1 Infrared Exchange Resonance 349



D n



in order to avoid the impossibility of the same excited state S

D –1=2

with different Boltzmann populations at two widely different energies. However, the

question remains as to the actual expressions for the exchange-stabilized Kramers

doublet when the physical reality of n



D n



is imposed. Equating (7.7)and

(7.8), we deﬁne an effective “spin” temperature T

in terms of the lattice tempera-

ture T



C gm





C H





C H



for S

D 1=2; (7.9)

where H

D 2J

=gm

The conclusion to be reached from (7.9) is that a stabilization energy that com-

mits the spin alignment to its neighbors rendering them dependent in a collective

system will have the effect of increasing the ground state population, thereby cre-

ating a lower effective spin temperature and resulting in a higher intensity of the

Zeeman resonance effect. It is for this reason that microwave FMR signals retain

their strength in ferrites at room temperature, as indicated by the presence of !

in the classical relation for the permeability in (7.5). The approximate fractional

population difference derived from (7.7)and(7.8)isexpressedas[6]

 n



M.T/

M.0/

D tanh



C gm

2kT



D tanh



1=2

C gm

2kT



F for S

D 1=2: (7.10)

From (7.6), F D n

=n D



1 C exp



4J

=kT



1

, which approximates to 1

for J

 kT at low T . These results are also consistent with the Brillouin–

Weiss function B

for S D 1=2 [7]:

1=2

.x/ D tanh



C gm



D tanh



1=2

C gm

2kT



; (7.11)

where x is the argument of the tanh function. This general concept was used to

interpret the M vs. T characteristics of the highly magnetoresistive (La,Sr)MnO

manganites [8].

Figure.7.4 shows a comparison between the d

–d

case (S

;SD 1=2)ofTi

or Cu

and the multispin d

–d

case (S

;S D 5=2)ofFe

encountered in

ferrites and other magnetic oxides. For the latter case, S

D 5=2 in (7.10), and

the S

D 1 FMR transition is S

DC5=2 !C3=2. If the spin temperature

concept of (7.9) is applied

350 7 Magneto-Optical Properties

Fig. 7.4 Comparison of S D 1=2 with S D 5=2 spin systems based on the model of Fig. 7.3 Note

that S

D 5 in the exchange transition. Figure reprinted from [5] with permission.

 2009 by

the American Institute of Physics

5=2

 n

3=2

5=2

C n

3=2

.T /

.0/

D tanh



5=2

C gm

2kT



D tanh



1=2

C gm

2kT



F: (7.12)

where J

5=2

 J

1=2

=5 from the general relation J

q=2

 J

1=2

=q,andq.5/ is

the number of half-ﬁlled orbital states [9]. As J

=kT ! 0 near the Curie

temperature, F ! 1=2 because half of the total population will begin to ﬁll

the upper molecular-orbital state when the exchange energy splitting reduces to

zero and the system becomes paramagnetic. To obtain additional reﬁnement in the

high-temperatureregime, both molecular states would have to be included in the for-

malism. A more general solution would broaden to the case of unlike magnetic ions,

e.g., S ¤ S

, where it must also be remembered that g varies among the different

transition metal ions.

Further consideration of (7.12) provides the answer to a long-standing question

of why ferromagnetic Zeeman transitions in ferrimagnets are intense well above

room temperature. A paramagnetic specimen with a comparable number of mag-

netic ions usually requires not only cryogenic temperatures but also a spectrometer

of much greater sensitivity. As the magnetization dwindles with increasing tem-

perature, the resonance intensity decreases until the Curie temperature (or Neel

temperature in some situations) falls below the operating temperature T

.Atthis

point, the exchange alignment is frustrated by thermal energy and the ferromag-

netic system becomes paramagnetic. It is appropriate to comment that without spin

ordering by exchange, microwave magnetic devices would not perform at room

temperature.

7.1 Infrared Exchange Resonance 351

Any stabilizing energy that organizes an otherwise independent population of

spins into a collective magnetic moment should be examined in the context of an

effective spin temperature when thermal properties are interpreted. Another situa-

tion where Boltzmann ground state populations can be increased is by a cooperative

magnetostrictive strain (see Sect.8.3.3).

7.1.3 Experimental Exchange Spectra

Exchange resonance is a unique effect in that it is derived from the cation–cation

molecular bonding, mediated by oxygen anions. Hence, there follow the beginnings

of a bonding/antibonding band gap included in Fig. 7.3 as a function of J

.Ex-

amples of exchange resonance are somewhat sparse in the literature. One reason is

the lack of power sources in the submillimeter and far-infrared wavelength bands

that conﬁne device applications to passive roles. In many instances, the source is

limited to a ﬁxed-frequency low-power laser beam commonly drawn from the rota-

tional or vibrational modes of simple gas molecules such as H

O, CO

,orCH

(formic acid).

For a comprehensive review of the early work in this area, the reader is di-

rected to Lax and Button [10]. Among the contributions of particular signiﬁcance is

the comprehensive analysis by Geschwind and Walker in which antiferromagnetic

resonance is explained in detail [2]. Tinkham and his colleagues have provided ex-

perimental results for rare-earth iron garnets and some common antiferromagnetic

compounds, FeF

, MnO, and NiO, listed in Table 7.1. Later work on exchange res-

onance has been focused on the millimeter and submillimeter bands with a goal set

toward device applications.

In subsequent sections, magneto-optical phenomena in transition-metal oxides

are described. Before these issues are addressed for speciﬁc molecular systems,

however, the classical phenomenology of magnetic circular birefringence (MCB)

will be reviewed in a context consistent with radiofrequency gyromagnetism.

Table 7.1 Exchange resonance data

Compound T

Tesla



1



Ref.

MnF

67.7 54 – [57, 58]

307 210 – [58–60]

MnTiO

65 100 –[61]

FeF

78.4 54 52.7 [62]

MnO 120 100 27.5 [63]

NiO 523 400 36.6 [4,64]

YbIG – – 14.1 [4]

ErIG – – 10.0 [4]

SmIG – – 33.5 [4]