Dionne G.F. Magnetic Oxides

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372 7 Magneto-Optical Properties

The MO eigenvectors for the two-level case with hybrid coefﬁcients are '

C c

and '



D c

 c

for the bonding and antibonding states,

respectively. There are two important results here (1) '

is of lower energy than '

by an amount E

 E

and (2) the volume densities of the initial wavefunctions

and '

hybridize in proportion to the values of c

and c

. For this generic

example, the fractional participation of '

in the bonding state is c

As discussed in the foregoing section, the enhanced magneto-optical effects are

believed to arise from the cooperative action of Fe

ions with degenerate excited

orbital terms that are split further by covalent interactions with bismuth. The excited

state splitting parameter  for individualions is determined principally by the eigen-

states of the operator LS .Thez-axis collinearity of the N magnetic moments can

occur through an applied ﬁeld H or an exchange ﬁeld H

. In both cases, the degree

of alignment follows a Brillouin function B,sothatN

eff

D N B .H; T /.Thereis,

however, an important distinction between these two situations – H inﬂuences both

L and S by a Zeeman effect, but H

can affect only the total spin of the ion by

ﬂipping the ground S

state.

The magnetization of the garnet is saturated in the z direction by superexchange

ﬁelds, causing B ! 1 and N

eff

! N . Through covalent interactions, the excited

states of both Fe

and Bi

should also have nondegenerate spin states. If the

exchange ﬁeld perturbationis applied, the lowest state S

value enters the calculation

as a multiplier when the orbital magnetic moment degeneracies of the MO states are

lifted by spin–orbit coupling L  S , which now reduces to L

. Although S

captured by H

, there is no competing crystal ﬁeld in this compound to prevent L

from maintaining its alignment with the z axis, so that the maximum value of S

for

the particular 2S C 1 manifold simply multiplies each 2L C1 state. For a

P state,

this situation is depicted in Fig. 7.20. Energies of the spin–orbit coupling manifold

in the hybrid state can be represented by





C 











Fig. 7.20 Exchange-ﬁeld spin quenching and spin–orbit splitting of a

P term

7.3 Magneto-Optical Spectra 373

D c

.

C c

.

; (7.44)

where S

and S

are the respective values of the lowest spin state. For the case





, L

D 0, ˙1 and S

D3=2;for





, L

D 0, ˙1 and

D1.Where

 

and c

is signiﬁcant, (7.44) can be simpliﬁed by

/  c

.

. The magnitude of the splitting then becomes 2 

ŒE

.C1/  E

.1/  c



Before the excited states of the Bi

molecule are examined, the energy

of the ground state must be determined. This energy may be estimated from the

electrostatic interactions between ions and outermost electrons. Here the binding

energy of the electrons of Fe

and Bi

can be computed from the algebraic

sum of the cation ionization potentials (IP) and the energy of the ﬁeld from the

negatively charged anion. For the outermost electron, this repulsive energy (RE) is

estimated by dividing the effective lattice energy (LE) by the ionic charge. For Fe

IP D54:8 eV and RE .from Fe

/ C25:5 eV [35]; for Fe

and Bi

,the

corresponding values are 45:3 and C15:8 eV, so that a common resultant ground

state ionic energy E

29 eV is estimated for both Fe

and Bi

. Destabiliza-

tion energies of the excited states are found by adding C29 eV to absorption energy

values from spectral data [25].

Interpreting the magneto-optical properties of Bi

is begun by analyzing

the covalent interactions between the excited

P term of Bi

in the dodecahedral

in the d and a sites. Since

excited state is about 4.2 eV above the

S ground state [27], hybridization is

possible with several Fe

excited states. Diagrams of the orbital terms [33, 36]as

functions of the crystal-ﬁeld strength parameter Dq for the d and a sites are shown

in Fig. 7.21.

Fig. 7.21 Crystal-ﬁeld sensitive excited terms for





in d and a garnet sites as functions of

crystal-ﬁeld parameter Dq. Energies were computed from Tanabe and Sugano matrices [33], with

B and C values based on the spectral analysis of Wood and Remeika [36]. Figure reprinted from

[26] with permission.

 1994 by the American Institute of Physics

374 7 Magneto-Optical Properties

To construct the hybrid states that represent the bands of half-width  between

the Bi





P excited state and the Fe

excited states that would produce

electron transition matrix elements with Fe

S ground states, consider the follow-

ing rationale: The candidates that satisfy the selection rule L

D˙1,thathave

symmetry transformation properties similar to the Bi

3C 3

P state, and are crystal-

ﬁeld (i.e., Dq) selective are the

states. Inspection of Fig. 7.21 reveals three

terms, associated with the

P ,and

F terms.

For the principal transitions that appear to cause the magneto-optical anomalies

at 2.6 and 3.15 eV [25], the obvious choice is the

P term, which has energies of

3.2 and 3.7 eV for the d and a sites, respectively, with corresponding Dq values of

0.065 and 0.15 eV. The bonding state hybrid function for the excited state is then

expressed as 

D c

C c

. If these quantities are used in the compu-

tations, values that agree with experiment are listed in Table 7.4 for the transition

energies



 E



from the ground state to the bonding states for c

D 0:26 and

0.33, and  D 0:03 and0.04forthed and a sites, respectively. The larger value

of  for the d site is consistent with its shorter bond lengths. To determine the 

parameters listed in Table 7.4,theFe

part of E

in (7.44) is treated as a baseline

and the small contribution of 

is ignored. With 

D 

and S

D1,themul-

tiplet energies E

 c



;0;c



, thereby yielding   c



.For



2 eV



17; 000 cm

1



[37]andc

<0:5,  values for d and a sites are 0.52 and

0.66 eV, respectively, in general agreement with the conclusions of experiment. Cal-

culated values of  are also in qualitative agreement with experimental results [25].

From the above calculations, Fig. 7.22 may be constructed to illustrate the origin

of the Faraday rotation peaks at 2.6 and 3.15 eV. In addition, this work presents an

opportunity to suggest that the weaker transition estimated to occur at 3.9eV, ini-

tially labeled as d -site, may result from a partial overlap cancellation of transitions

that originate from the MO states formed with the

F terms .E

 4:1 eV/ of Fe

ions in both d and a sublattices. This situation would explain that the location of a

second peak from the a sublattice should appear somewhere in this general energy

Table 7.4 Molecular-orbital parameters of Bi

Site E

–E

E

„! 



theor



exp

tet d 3.2 2.6 2.6 0.04 0:5 0:26 0.52 0:25

oct a 3.7 3.2 3.15 0.03 0:4 0:33 0.66 0:5

 4:2 eV for the Bi

P state inﬂuenced by Bi

–O

2

–Bi

interactions

29 eV, estimated from ionization potentials and lattice energies. The equivalence of

values for both Fe

and Bi

in this model is a coincidence

Half-bandwidth of bonding split band 

b=4

,whereb D

, agrees with exper-

imental estimate. The slightly larger value observed for the a site [25] may be due unresolved

structure

Based on 

, with 

 2 eV

These values were determined from estimates of Y

3x

with x D 0:25 [25]. Since only

part of the full complement of Bi

interactions was present, the effective  integrals would be

smaller, thereby resulting in smaller c

values and accounting for the above 

exp

values

7.3 Magneto-Optical Spectra 375

Fig. 7.22 Proposed molecular-orbital energy level diagram of Bi

referenced to a ground

state of E D 0. Oxygen interactions and all antibonding states have been omitted to reduce

confusion. Overlap and exchange integrals are treated semiempirically and represent resultants

of superexchange (Bi–O–Fe) and direct exchange (Bi–Fe) covalent interactions. Electric-dipole

transitions are presented as intraionic. Figure reprinted from [26] with permission.

 1994 by the

American Institute of Physics

regime. The experimental determination that the peak at 3.9 eV has the same sign as

the tetrahedral peak at 2.6 eV is consistent with the ratio (3:2) of d to a sites.

Based on the notion that magneto-optical effects involving excited states are

dependent on the product N

eff

, we may speculate on the effects of magnetic di-

lution and temperature variations. If the temperature T remains below the Curie

temperature T

, N

eff

should track with the magnetizations of the respective sublat-

tices as they apply to the individual transitions. Information about spin canting in

both the ground and excited states may be extracted from the peak intensities as

functions of Fe

content. For other ions with a .6s/

ground-state electron conﬁg-

uration similar to that of Bi

,i.e.,Pb

and Tl

, similar magneto-optical effects

should be expected.

Hansen et al. [38, 39] reported comprehensive data and analysis of magnetic

and magneto-optical properties that revealed thermomagnetic behavior based on the

molecular-ﬁeld approach described in Chap. 4. These results veriﬁed the role of su-

perexchange of the spin system in establishing the alignment of electric dipoles

necessary for the observed collective magneto-optical effects at room tempera-

ture. Their investigations also included anomalous Faraday rotation enhancement

at 633 nm wavelength from Pb

.6s/

ions substituted in c sites. With smaller 

values, however, the effects of the  splitting in these alternative ions might be

reduced from those of Bi

376 7 Magneto-Optical Properties

7.3.5 Intersublattice Transitions and the S D 0 Rule

In the foregoing analysis of the hybrid electric-dipole excitations, the electron spin

has been largely ignored. In Sect. 7.1, the role of superexchange in freezing the

direction on the ionic spins in a collective magnetically ordered system was mod-

eled. It was therefore assumed that magnetic-dipole transitions with selection rule

S

D˙1 do not occur in the optical frequency bands. If this assumption is valid, it

follows that Laporte’s rule for an orbital angular momentum transition L

D 0, ˙1

must be augmented by the rule S

D 0 to deny spin reversals and conserve parity.

As concluded in Sect. 7.3.4, however, the transitions that would explain the bismuth

Faraday rotation anomaly are

S !

P and

P (hybrids), thereby seemingly in vio-

lation of this requirement. This issue is examined in the context of

1a;d

1d;a

intersublattice transfers within the antiferromagnetically paired Fe

O

2

Fe

molecule depicted in Fig. 7.23 and will be reviewed in this section.

The earliest indication that neighboring Fe

ions from opposing sublattices par-

ticipate jointly in magneto-optical transitions was reported by Wood and Remeika,

who concluded that the intensities of the spectral lines increased approximately as

the square of the combined Fe

concentration in Y

5x

[36]. Similar

observations were made by other workers [20,40,41]. In addition to the dramatic en-

hancement of the off-diagonal permittivity tensor element "

at selected energies by

substitution for Y

, moderate increases in the Curie temperature [42]andthe

diagonal element "

[21,22,43] also occur. These latter effects suggest that Bi

also

augments covalent interactions in the Fe

O

2

Fe

ground state, which further

stabilize antiparallel spin order and boost the intersublattice transition probabilities

by increasing the off-diagonal element "

of the permittivity tensor. The magnitude

of the element is directly dependent on the orbit and spin selection rules for the

Fig. 7.23 Reproduction of Fig. 7.22 but with electric-dipole transitions presented as interionic

7.3 Magneto-Optical Spectra 377

transition. In theory, a nonzero off-diagonal matrix element can occur only between

even and odd parity orbital wavefunctions where the operator is odd. Therefore,

L D˙1 for an electric-dipole operator, e.g.,

˙1

. If the spin operator

and functions are included, that contribution must be unchanged under the transition,

thereby dictating that S D 0 and S

D 0.ForFe





ions in octahedral

/ and tetrahedral .O

/ coordinations, the orbital term energies formed within the

3d shell are presented as a function of the crystal-ﬁeld parameter Dq in Fig. 7.21.

Note that none of the spin multiplicities in the spectral energy range of interest will

satisfy the S D 0 requirements by an intrasublattice transition. Furthermore, oc-

tahedral

would also violate the parity rule because both ground and

excited states have centers of inversion symmetry and would be even functions.

In later work, Allen showed that independent dilution of Œa and .d / sites by In

and Al

, respectively, in a

2:53

0:47

ŒFe

.Fe

/ O

host produced almost

identical changes in the Bi

-enhanced complex permittivity "

, thereby conﬁrm-

ing that both sublattices are involved [23,43]. The spectra of "

and "

shown in

Fig. 7.24 indicate a decrease in magnitude of about 25% across the energy band

from 2.0 to 4.0 eV for both compositions. The amounts of In

and Al

were

selected to reduce the respective sublattice Fe

concentrations by approximately

Fig. 7.24 Experimental comparison "

of fY

2:53

0:47

.

host with equal fractions

of In

and Al

substituted into the [a]and(d) sites, respectively. Images are reproduced from

Allen’s thesis [23]. Figure reprinted from [43] (2004) with permission.

 2004 by the American

Institute of Physics

378 7 Magneto-Optical Properties

the same fraction of 1=6 .17%/. Since the measurements were made at 300 K,

spin canting also contributed to the decrease in transition probability. The canting

was caused by reductions in the exchange ﬁelds that also decline with dilution, and

were ﬁtted successfully by a modiﬁcation of the molecular-ﬁeld theory for diluted

magnetic garnets [44]. Following the suggestion of Wood and Remeika [36], joint

participation of Œa and .d / sites proportional to the product of the Fe

occupation

densities N

˙a

 N

˙d

 N

was adopted in the model [43].

The intersublattice charge-transfer concept based on the ionic nature of

 O

2

 Fe

molecule evolved from a mechanism proposed for MnF

[45] that was later applied to the magnetic oxides [28,46,47]. The theories used to

explain the Bi

enhancements [25, 26] that considered internal ionic transitions

1a;d

of the Fe

and Fe

ions are extended to include the interionic

charge transfer. The concept is illustrated in Fig. 7.25. Photon absorption probabil-

ities are determined not only by the oscillator strengths of the tensor elements, but

also by the efﬁciency of the two-electron superexchange that now insures S D 0

transitions.

Fig. 7.25 Schematic diagrams of the intersublattice pair transfer between states of magnetically

opposed octahedral and tetrahedral sites

7.3 Magneto-Optical Spectra 379

Interionic charge transfer by actual ionization is required to create local Fe

a;d

d;a

hole–electron pairs (excitons, possibly invoking a O

1

peroxide), with co-

incident magnetic-dipole agents in the form of magnons made possible by inter-

sublattice superexchange. These interrelated events allow electric-dipole transitions

without a net change in spin value and permit the mutual

1a;d

1d;a

inter-

sublattice transfers within the antiferromagnetically coupled Fe

 O

2

 Fe

molecule. For this to occur (1) there must be a sufﬁcient concentration of Fe

ions

to create the necessary molecular pairings; (2) the relevant orbital overlap integrals

must be large enough for the intersite hybrid wave functions to have a transfer prob-

ability that exceeds that of the competing “forbidden” internal transitions; and (3)

a coincident electron “back” transfer to an empty orbital state must occur through

delocalization exchange to recover the 3C valence states from the Fe

and Fe

ions created as part of the transfer processes





C Fe





! Fe





C Fe





„!

 .IP/;





CFe





! Fe





CFe





.„!

C E

/ C.IP/:

These processes result in





C Fe

/ ! Fe





C Fe

/  .„!

C„!

where .IP/ is the ionization potential energy expended to create the interim 4C

and 2C states, and E

.D„!

/ is the restabilizing energy permitted by the cova-

lent exchange mechanism. These actions are depicted in Fig. 7.25, where the cross

transfers are diagrammed. A beam of photons of energy „!

or „!

will cause

both

and

to be excited, in the manner of two coupled harmonic oscillators

driven at their resonance frequencies by external stimulation of only one of them.

As suggested by the spectral data, the orbital bonding mechanisms are based on

the unmixed linkages 3t

2ga

–2p–3t

and 3e

–2p–3e

between the sites of the op-

posing sublattices. Since both

and

excited terms would occur through

rearrangement of spins within the respective 3d

manifolds, the “virtual” spin ex-

change of the static ferrimagnet would become “real” from the absorption of the

photon beam energy.

With the growth of integrated optics and the burgeoning ﬁeld of photonics by

thin-ﬁlm deposition of optical layers onto semiconductor substrates, the need for

Faraday rotation (FR) media for ﬁber-optic isolators at 1:55 m has prompted

the search for new magneto-optical oxide compounds. In discrete devices, Bi-

substituted magnetic garnets have proven to be greatly successful [48]. Due to

lattice-mismatch considerations, however, the garnet crystal structure is not com-

patible with either that of conventional Si or GaAs semiconductors or the available

dielectric buffer layers available for such integration purposes. Ferrimagnetic spinel

layers grow well as thin ﬁlms, but were found to be unsatisfactory for low-loss

transmission despite high Faraday rotation [49]. Perovskites provide another struc-

ture that can be grown with good quality on a range of substrates, and that can show

380 7 Magneto-Optical Properties

magnetic properties. For example, BaTiO

substituted with Fe revealed unexpected

ferromagnetic properties, but little FR [50]. The FR of orthoferrite perovskites, with

formula

ŒFe O

, have been explored at shorter wavelengths for bulk materials

[19], but there are few data on thin ﬁlms [51].

The choice of cubic perovskites, however, remains attractive because of their

crystallographic compatibility with preferred substrates and buffer layers. In this

paper, the underlying physics for the design of two magneto-optically active per-

ovskites are described. Both are charge-ordered transition-metal compounds with

mixed alternating cations that satisfy in theory the various requirements for a room-

temperature Faraday rotator at 1:55-m wavelength.

From the development of the

Y; Bi

class of garnet rotators at infrared

(IR) wavelengths, much understanding of the fundamental physics of magnetically

aligned electric-dipole transitions is available. For oxide systems of other lattice

structures that could be more compatible with integrated photonic applications,

some of the main requirements for a successful design of a Faraday rotator can

be summarized as follows:

1. To align the orbital angular momentum vectors of the individual ion elec-

tric dipoles (through spin–orbit coupling), the material must be spontaneously

magnetic (ferro- or ferrimagnetic) with reasonably low anisotropy ﬁeld while

maintaining a Curie temperature T

> 300 K.

2. Electric-dipole transitions of the magnetic ion must satisfy orbital and spin se-

lection rules L

D 0, ˙1 (Laporte’s rule) and S

D 0, respectively.

3. Wavelengths of interest must fall in the wings of the Lorentzian-shaped line to

avoid the high absorption loss near its center frequency. Because optical spec-

tra are not tunable as in the case of some magnetic-dipole transitions, i.e., by a

Zeeman effect, this narrows further the choice of possible candidates.

To satisfy the above conditions in a magnetic insulator, two approaches with pos-

sible room-temperature net magnetization have been considered [52]. Both are

“double” perovskites of generic formula





with octahedral-site B

and B

cations of different magnetic moments and/or ionization states. If the mo-

ments differ and antiferromagnetic spin ordering is stabilized, a quasi-ferrimagnet

will be the result, thereby raising the possibility of interionic transitions to satisfy

the S

D 0 rule. If, however, the B and B

ions differ by their valence charge,

delocalization exchange can create ferromagnetic ordering [53, 54], which offers

the possibility of spin-preserved intraionic transitions if the orbital term structure is

suitable.

Although somewhat beyond the scope of this volume, the properties of magneto-

optically active 4f

series rare-earth ions in various compounds are of substantive

importance to the understanding of the basic physics of these effects. The readers

are encouraged to consult the high magnetic ﬁeld work on these systems, particu-

larly the investigation of paramagnetism that produces Faraday rotation reported by

Guillot, Le Gall, Ostorero, and others at magnet laboratories in France [55,56].

Appendix 7A 381

Appendix 7A Magnetic Circular Birefringence and Dichroism

In the near-IR and higher energy bands, the luxury of ignoring attenuation effects

can no longer be enjoyed. For that reason, the Faraday rotation parameter is ex-

pressed as a complex quantity 

D 

C i

,where

is termed “ellipticity”

because it represents the difference in amplitude between the circular polariza-

tion modes caused by unequal k



and k

. The real and imaginary parts of 

are frequently called magnetic circular birefringence (MCB) and magnetic circular

dichroism (MCD). To describe these effects analytically, we adopt a development

based on that of Dillon [18], but with the sign conventions already chosen in Chap.6

for the microwave bands:



C i







 N





Œ.n



 n

/  i .k



 k

/ ;





.n  ik/ (7.45)

From the earlier deﬁnitions N

D "

˙ "

and "

D "

 i"

˙ "

D n

 k

 i2n

(7.46)

and



 n







 k





;

D .n

 n



/: (7.47)

If the average of n D .n

C n



/=2 and k D .k

C k



/=2 are introduced, (7.47)

can be expressed as

Dnn C kk;

Dnk  kn: (7.48)

When (7.45)and(7.48) are combined, the key relations between permittivity, Fara-

day rotation, and ellipticity, and refractive index and extinction coefﬁcient are

obtained:

D





.n

 k

D





.k

C n

/: (7.49)

An alternative to transmission measurements is the Kerr effect. Reﬂection

measurements become necessary if intrinsic properties such as electrical con-

ductivity or microstructure imperfections in ceramic forms of normally transparent