Dionne G.F. Magnetic Oxides
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382 7 Magneto-Optical Properties
compounds render the specimens opaque in the frequency bands of interest. Fol-
lowing the model used by Dillon [18] for the reflected wave, we obtain the complex
Kerr rotation from the respective reflection coefficients r
˙
according to
K
D
K
C i
K
Di
r
r
C
r
C
C r
Di
N
N
C
N
C
N
1
; (7.50)
which can be expressed as
K
C i
K
D
k in
.n
n
C
k
k
C
1/ i .n
k
C
n
C
k
/
: (7.51)
After rationalization of the denominator, (7.51) becomes
K
C i
K
D
k
n
2
k
2
1
C n .2nk/
C i
k .2nk/ n
n
2
k
2
1
n
2
k
2
1
2
C 4n
2
k
2
:
(7.52)
where n and k are the averages defined below (7.47).
The real and imaginary parts of (7.52) are separate equations with two unknowns
that allow n and k to be solved for in terms of
K
and
K
, according to
n D
K
.2nk/
K
n
2
k
2
1
;
k D
K
n
2
k
2
1
K
.2nk/ ; (7.53)
and with the help of (7.48) the basic relations between the Kerr parameters and the
complex permittivity are stated as
"
0
1
D
K
k
3
3n
2
k C k
C
K
n
3
3nk
2
n
;
"
00
1
D
K
n
3
3nk
2
n
K
k
3
3n
2
k C k
: (7.54)
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384 7 Magneto-Optical Properties
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Chapter 8
Spin Transport Properties
Electrical conductivity in solids is traditionally associated with collective electron
metals, intermetallic compounds, and metal alloys. With outer shell s and p elec-
trons unbound in the sense of a Sommerfeld gas, the analysis of electrical properties
usually takes the form of a density of states calculation from a theory that assumes a
periodic lattice potential and applies Fermi statistics. The exercise leads to the cre-
ation of broadened energy states (bands) and the definition of a Fermi level to serve
as the zero energy reference. For transition metals with unfilled “inner” d -shells,
complications arise from the hybridization of “free” electron s states with states of
unpaired spins in the d shell that forms the collective electron band structure. The
first successful attempt to model ferromagnetism in collective electron metals was
developed by Stoner employing a phenomenological theory that explained the net
magnetization by introducing an exchange field that separated the d -spin popula-
tions into up .˛/ and down .ˇ/ bands [1], analogous to the Hund’s rule sorting of
spins in an individual ion, illustrated by Figs. 2.3 and 2.19. In subsequent years,
great strides were made in formalizing these concepts with the help of quantum
theory and other aspects of modern physics [2].
Despite the power and elegance of solid-state band theory, the basic assumption
of a periodic potential with Bloch-type functions can shroud the role of local inter-
actions that are important in insulator host compounds containing transition-metal
cations in varying concentrations. Randomly spaced effects such as lower symmetry
crystal-field splittings from electron-lattice vibronic interactions at individual sites,
variations in spin–orbit coupling, single-ion magnetoelastic effects, spin–lattice re-
laxation that controls spin resonance line shapes and spin-wave propagation, and to
some extent, the superexchange interactions that determine the type and stability of
magnetic ordering, are largely missing from the formalism. In select cases where
metallic properties have been observed in homogeneous ferromagnetic oxides, the
electrical properties have been interpreted by Hartree–Fock tight-binding approxi-
mations based on the spin-density-wave model of Overhauser [3] to split the d shell
into the ˛ (majority) and ˇ (minority) spin bands with the Fermi energy set by the
highest occupied state, in the manner of Stoner’s original model. However, in most
cases where electronic conduction in oxides occurs with spontaneous magnetism,
the charge transfer is from randomly dispersed polarons.
G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 8,
c
Springer Science+Business Media, LLC 2009
385
386 8 Spin Transport Properties
Charge transfer by electron hopping between mixed-valence cations of the same
atomic element is a mechanism of electrical conduction in ferrites, particularly
Fe
2C
$ Fe
3C
Ce
in the octahedral sublattice. The conductivity that results from
these random events generally has the temperature characteristic of an insulator or
semiconductor at room temperature. In other systems, however, charge transfer can
be metallic in temperature dependence and sometimes sufficiently coherent that su-
perconductivity can exist to temperatures greater than 130K. These phenomena fall
under the general class of polaronic motion, which is strongly influenced by the
state of spin ordering in the lattice. The general study of polarons was pioneered by
a number of workers, prominent among them being Fr¨ohlich [4]andMott[5]. For
metallic oxides, however, the molecular crystal approach of Holstein [6] and the in-
sightful review by Goodenough [7] are more relevant because of their conformance
with molecular-orbital framework on which the present discussion is based [8,9].
In recent years, discoveries that involve the transport of electron spins have re-
energized the field of magnetic oxides. Among these initiatives are high-temperature
colossal magnetoresistance (CMR) and high-temperature superconductivity (HTS),
both dependent on polarized-spin transport in select transition-metal oxide com-
pounds. Promising materials for exploiting spin transport in magnetoresistance and
superconductivity are the manganite and cuprate perovskites, respectively. Metal-
lic conduction at room temperature has also been reported in other magnetic oxide
structures.
8.1 Polarons and Charge Transfer
A polaron is a charge carrier that resides in an energy trap created by the interaction
between the charge and its local crystalline surroundings in the manner of a dipole
with one of its charges mobile. The conventional trap is attributed to lattice distor-
tions that are induced by local electric fields and the associated differences in the
size of the ion that harbors the charge. Another term that is used to describe a polaron
is a “dressed” carrier because it travels together with its lattice accommodation. In
ionic compounds, polarons occur as a result of cation chemistry and mixed valence
induced by departures from stoichiometry, in contrast to the excitation necessary to
create an electron (or hole) carrier in a doped conventional band-gap semiconductor.
For electrical conduction, the carrier part of the dipole is not created by excitation,
but rather released from its trap by covalent tunneling or by phonons, thermal, or
otherwise. The participation of the polaron in an electrical current is usually de-
scribed as an activation of its mobility by overcoming the trap energy created by
the stabilizing effects of elastic, electrostatic, as well as magnetic exchange where
transition-metal ions with unpaired spins supply the charge carriers. Convenient ve-
hicles for studying these conduction mechanisms are the cubic ABO
3
and tetragonal
A
2
BO
4
perovskites shown in Fig. 8.1, emphasizing the octahedral B site at which
the polaron is centered. An important feature of this crystallographic family in re-
lation to spin exchange and transport are the 180
ı
B–O–B bond angles that form
B–O
2
chains along the z (or c) axis and B–O
4
planar layers in the x–y (frequently
called the a–b)plane.
8.1 Polarons and Charge Transfer 387
Fig. 8.1 Perovskite unit cells
cubic ABO
3
and tetragonal
A
2
BO
4
, highlighting the
octahedral B site
Consider the case of a mixed-valence complex transition metal perovskite
A
3C
B
3C
O
2
3
in which different ions of smaller valence charge A
02C
are sub-
stituted. To restore electrical neutrality (1) either a corresponding number of the
O
2
ions are converted to O
1
to create a partial peroxide A
3C
1x
A
0
2C
x
B
3C
O
1
x
O
2
3x
or (2) where possible a corresponding number of B
3C
are converted to B
4C
to
produce the mixed-valence in the B sublattice of A
3C
1x
A
0
2C
x
B
3C
1x
B
4C
x
O
2
3
. Because
larger ionic charges usually produce greater ionic bonding energies and higher sta-
bility, once the ionization potential and electron affinity tradeoffs are made between
cations and anions, the latter arrangement would likely result in a lower ionic lat-
tice energy. Relative to the neutral background, A
02C
is a stationary fixed negative
charge and B
4C
forms the positive half of a dipole that can occupy any one of the
several equivalent sites equidistant from A
02C
, as depicted in the two-dimensional
sketch of Fig. 8.2. Since the charge at the electronic hole labeled as B
C
is capable of
transferring to equivalent sites surrounding the A
0
fixed charge (in this ideal case
without a net change in energy) it is a polaron, and A
0
is the polaron source.
If the model of Fig. 8.2 is examined further, we can identify two parts to the
problem of analyzing charge movement in this model: first, the transfer of elec-
trons among sites of equivalent energy, depicted as periodic locations around the
rings and second, the transport of charges to sites farther from the source ion, de-
picted by the transitions to an outer ring of higher energy. In both cases, the gain in
binding energy is derived directly from the actual transfer of charge, and favors a
388 8 Spin Transport Properties
Fig. 8.2 Positive polaron
(hole) formation in
mixed-valence ABO
3
and the
concentric ring model
centered about a fixed
polaron source. Inner ring is
the radius of a small polaron,
outer rings of increasingly
higher electrostatic potential
energy are regions of large
polarons
large covalent exchange energy integral to stabilize a kinetic ground state. The stabi-
lization energy associated with the transfer is therefore critically related to the states
of spin polarization of the transfer ions because the larger the trap energy, the longer
the lifetime of the trapped carrier, and the lower the mobility becomes. For an ideal
small polaron, the initial and final states are identical and the probability of locating
the charge at any site around the inner ring is equal. When the dipole extends out-
side of the inner ring, the carrier becomes mobile, leading to the formation of a large
polaron. In this event, the energy of the final state is increased accordingly. The trap
is deepened by the increasing Coulomb potential of the dipole attraction and can
be examined according to electrostatic theory. Because of the increase in energy of
the receptor state, the density of large polaron carriers decays as a function of dis-
tance from the source analogous to the tail of an orbital wavefunction away from its
nucleus.
8.1.1 Transfer Among Equivalent Energy Sites (Small Polarons)
When an atom is ionized in free space, the removal (or addition) of an electron is
accomplished at the cost of an ionization energy of many electron volts. In a crystal
lattice, this energy (e.g., the electronic work function) is reduced to a few electron
volts because of the lower electrostatic fields due to polarizability of the dielectric
medium. In cases where covalent bonding is significant, however, a more efficient
mechanism is available by charge transfer between cations through interaction with
intermediaryanions. Where selection rules for energy-free transfer are satisfied, e.g.,
8.1 Polarons and Charge Transfer 389
spin conservation by S D 0, the transfer can take place spontaneously, and when
there is no net loss of energy to the lattice, the action can be termed “adiabatic.” The
activation energy from this double exchange transfer is therefore dependent on the
angle
ij
between adjacent ionic spins S
i
and S
j
.Where
ij
>0, an electron spin
flip is required to satisfy any intraorbital exchange (Hund’s rule) requirements on
the receptor ion,
1
and the spontaneous excitation-free sharing of the transfer spin
is allowed if sufficient energy is available to maintain the stabilization energy of
mobile exchange given by the relation examined previously [8]:
E
ex
ij
D2z
ij
J
ij
S
i
S
j
cos
2
ij
2
Dz
ij
J
ij
S
i
S
j
1 C cos
ij
; (8.1)
where J
ij
>0is the ferromagnetic exchange constant and z
ij
is the number of equiv-
alent neighboring ionic spins.
When
ij
D 0, E
ex
has a minimum value for spins that are parallel to satisfy
the S D 0 requirement for the charge transfer. If
ij
>0, E
ex
increases by an
amount U
ex
which becomes the energy needed to restore the spin alignment. In the
present context, U
ex
represents the loss in kinetic stabilization energy of spin transfer
between sites i and j . In the absence of quantum tunneling that would reduce the
effect of the trap on the transfer probability, U
ex
can also be equated a classical
thermal hopping energy according to
E
ex
hop
U
ex
D E
ex
ij
E
ex
.0/ D z
ij
J
ij
S
i
S
j
1 cos
ij
: (8.2)
It should be noted that
ij
values in the range 0 to are allowed by (8.2), and
that the theoretical maximum hopping activation energy from exchange is actually
E
ex
hop
D 2z
ij
J
ij
S
i
S
j
when the spins are antiparallel
ij
D
. For a ferromagnetic
to paramagnetic transition that occurs at the Curie temperature, the average angle
between spins becomes =2. The probability of mobility activation of small po-
larons among equivalent sites can then be written in the classical Boltzmann sense
as exp
h
E
0
hop
C E
x
hop
.
kT
i
,whereE
0
hop
U
0
is the corresponding activation
energy from the residual elastic trap energy when
ij
D 0.
8.1.2 Transfer to Higher Energy Sites (Large Polarons)
In the simplest case there are no spin polarization limitations, i.e., perfect ferromag-
netism, the small polaron ring described above would be likened to a giant molecule
1
The magnetic implications of a real spin transfer between a pair of covalently linked orbital states
arise when the accompanying states of the 3d shell are populated with unpaired spins aligned
according to Hund’s rule. These spin groups can favor parallel or antiparallel exchange interac-
tions. The ferromagnetic energy from the dynamic exchange could then be enhanced or reduced
depending on the details of the overall electronic spin structure.
390 8 Spin Transport Properties
Fig. 8.3 Two-cation molecular-orbital diagram for a polaron in a trap of energy U
p
in which the carrier would occupy all sites simultaneously in a quantum mechanical
sense. To explore this concept, a giant hybrid wavefunction based on the polaron
exchange integral b
p
must be constructed. In addition, U
p
is introduced to represent
the total trap energy from all sources, including the variable electrostatic force that
binds the mobile charge to its source in the lattice. It is therefore assumed that the
trap is of multiple origins, with the polaron carrier tethered to its nearby fixed charge
of opposite sign.
Estimates of the molecular-orbital functions '
a
and '
b
for a diatomic molecule
can be obtained by solving a two-level degenerate perturbation problem in the con-
ventional way based on the one-electron molecular-orbital diagram of Fig. 8.3.The
solution is worked out in terms of a polaron activation energy U
p
between the two
lattice cations M
a
and M
b
with corresponding electron energies E
a
and E
b
that
share an itinerant electron by following the procedures used previously. The elec-
tronic stabilization energy and hybrid eigenfunction coefficients may be found by
means of a self-consistent approach developed by Wolfsberg and Helmholtz [10].
The degenerate-state perturbation matrix equation for the diatomic molecule with
Hamiltonian H D H
a
C H
b
is expressed as
ˇ
ˇ
ˇ
ˇ
H
aa
E H
ab
E
H
ab
E H
bb
E
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
E
a
Eb
ab
E
b
ab
E E
b
E
ˇ
ˇ
ˇ
ˇ
D 0; (8.3)
which is derived from the secular equation
ˇ
ˇ
H
ij
E
ij
ˇ
ˇ
D 0,whereH
ab
D
h
'
a
j
H
j
'
b
i
and
ab
D
h
'
a
j
'
b
i
D 1 for a D b,and<1for a ¤ b.If
b
ab
D
h
'
a
j
H
j
'
b
i
is approximated by .E
a
C E
b
/
ab
and
2
ab
1 are used, the
solutions for the bonding .C/ and antibonding ./ states can be reduced to
E
˙
D
1
2
.E
a
C E
b
/ ˙
1
2
q
.E
a
E
b
/
2
C b
2
ab
; (8.4)
with corresponding normalized hybrid eigenfunctions given by
'
D
1
p
2
.c
ba
'
a
c
bb
'
b
/ antibonding
'
C
D
1
p
2
.c
aa
'
a
C c
ab
'
b
/ bonding: (8.5)
8.1 Polarons and Charge Transfer 391
After the standard solutions [11] are applied to (8.5) for the case of interest b
2
ab
.E
a
E
b
/
2
,
'
1
p
2
"
1
E
a
E
b
b
ab
1
2
'
a
1 C
E
a
E
b
b
ab
1
2
'
b
#
;
'
C
1
p
2
"
1 C
E
a
E
b
b
ab
1
2
'
a
C
1
E
a
E
b
b
ab
1
2
'
b
#
: (8.6)
For the limiting cases of small and large polarons, b
p
D b
ab
and U
p
D E
a
E
b
,
where both are negative energies, and (8.4) then reduces to
E
˙
E
F
˙
1
2
q
U
2
p
C b
2
p
: (8.7)
where E
F
D .E
a
C E
b
/=2is the mean energy of the free ion states.
In the small-polaron case where b
2
p
U
2
p
, and the bonding-state solution of
(8.4) simplifies to
E
C
E
F
C
U
p
2
q
1 C b
2
p
=U
2
p
E
F
C
U
p
2
C
b
2
p
4U
p
;
E
C
E
a
C
b
2
p
4U
p
: (8.8)
The b
2
p
=4U
p
term represents the increment of stabilization energy that the hybrid
state provides for tunneling by the electrons not involved in a thermal hopping
process.
For the corresponding large-polaron approximation b
2
p
U
2
p
,and
E
C
E
F
C
b
p
2
q
1 C U
2
p
=b
2
p
E
F
C
b
p
2
C
U
2
p
4b
p
;
E
C
E
a
C
U
2
p
4b
p
: (8.9)
In this limit, U
2
p
=4b
p
is the additional stabilization of the single spin in the bonding
state. It can be readily recognized that if U
p
D 0, there is no barrier or trap for the
transfer of the spin between '
a
and '
b
because their coefficients in the hybrid orbital
state are equal. Therefore, U
2
p
=4b
p
is the energy that must be gained to eliminate the
trap. The effective classical activation energy for general polaron charge transport
becomes
E
hop
D
U
2
p
4b
p
; (8.10)
where U
p
D U
0
C U
0
.r/ C U
ex
which now includes a variable electrostatic energy
term that is introduced in Sect. 8.1.3.