Dionne G.F. Magnetic Oxides
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xii Contents
Appendix 5B T
2g
Solution for d
1
in an Exchange Field....................263
Appendix 5C Orbital States of d
5
in a Cubic Field .........................265
Appendix 5D Angular Dependence of Cubic Anisotropy Fields ...........267
References.......................................................................269
6 Electromagnetic Properties...................................................273
6.1 Magnetic Relaxation ...................................................274
6.1.1 Nonresonant Longitudinal Relaxation ........................274
6.1.2 Quantum Mechanisms of Spin–Lattice Relaxation...........278
6.1.3 Perturbation Theories of Spin–Phonon Interaction ...........286
6.2 Gyromagnetic Resonance and Relaxation.............................287
6.2.1 Paramagnetic Resonance.......................................288
6.2.2 Ferromagnetic Resonance .....................................292
6.2.3 Uniform Precession Damping .................................295
6.2.4 Inhomogeneous Resonance Line Broadening.................297
6.2.5 Fast-Relaxing Ion Effects ......................................300
6.2.6 The Exchange Isolation Effect.................................306
6.3 Exchange-Coupled Modes (Spin Waves)..............................307
6.3.1 Uniform Precession Decoherence (Degenerate
Spin Waves) ....................................................307
6.3.2 Instability Threshold (Classical Approximation) .............311
6.3.3 Instability Threshold (Nonlinear Spin Waves) ................315
6.3.4 Magnetostatic Modes ..........................................317
6.4 Permeability and Propagation..........................................318
6.4.1 Low-Frequency Longitudinal Permeability ...................318
6.4.2 High-Frequency Transverse Limits............................322
6.4.3 Snoek’s Law Considerations...................................324
6.4.4 Circular Polarization and Nonreciprocal Properties ..........327
6.4.5 Linear Polarization and Faraday Rotation.....................332
Appendix 6A Transverse Permeability Tensor ..............................333
Appendix 6B Classical Instability Threshold ...............................336
Appendix 6C Domain Wall Susceptibility Equation ........................338
References.......................................................................340
7 Magneto-Optical Properties ..................................................343
7.1 Infrared Exchange Resonance .........................................344
7.1.1 Classical Precession Model ....................................344
7.1.2 Quantum Spin Transition Model ..............................346
7.1.3 Experimental Exchange Spectra ...............................351
7.2 Combined Permeability and Permittivity..............................352
7.2.1 The ["] [] Tensor Solutions..................................352
7.2.2 Propagation Parameters and Faraday Rotation ...............353
7.3 Magneto-Optical Spectra ...............................................355
7.3.1 Electric-Dipole Transitions ....................................355
7.3.2 Yttrium Iron Garnet Spectra (Paramagnetic) .................360
Contents xiii
7.3.3 Iron Garnets with Bismuth Ions (Diamagnetic)...............366
7.3.4 Fe
3C
–Bi
3C
Hybrid Excited States ...........................371
7.3.5 Intersublattice Transitions and the S D 0 Rule.............376
Appendix 7A Magnetic Circular Birefringence and Dichroism ............381
References.......................................................................382
8 Spin Transport Properties ....................................................385
8.1 Polarons and Charge Transfer..........................................386
8.1.1 Transfer Among Equivalent Energy Sites (Small Polarons) .388
8.1.2 Transfer to Higher Energy Sites (Large Polarons) ............389
8.1.3 Transfer by Covalent Tunneling ...............................392
8.1.4 The Holstein Polaron Theory ..................................394
8.2 Metallic Oxides with Polarized Spins .................................396
8.2.1 Simple Oxides ..................................................397
8.2.2 Complex Oxides................................................397
8.2.3 Classical Resistivity–Temperature Model.....................400
8.3 Magnetoresistance in Oxides (CMR) ..................................401
8.3.1 Manganese-Ion Exchange Interactions........................402
8.3.2 Magnetoresistivity-Temperature Model .......................405
8.3.3 Dilute Magnetic Oxides........................................410
8.4 Superconductivity in Oxides ...........................................413
8.4.1 Classical Foundations ..........................................413
8.4.2 Zero-Spin Polarons and Magnetic Frustration ................419
8.4.3 Large-Polaron Superconductivity .............................423
8.4.4 Normal Resistivity and Critical Temperature .................426
8.4.5 Layered Cuprate Superconductors ............................430
8.5 Supercurrents and Magnetic Fields ....................................439
8.5.1 Supercurrent Formation ........................................439
8.5.2 Condensation Energy ..........................................442
8.5.3 London Penetration Depth .....................................443
8.5.4 Critical Magnetic Field ........................................445
8.5.5 Critical Current Density........................................447
8.5.6 Coherence Length ..............................................450
8.5.7 Type-II Superconductors .......................................452
Appendix 8A Magnetic Levitation...........................................455
References.......................................................................456
Index .................................................................................461
Chapter 1
Introductory Magnetism
The use of magnetic oxides in electronics technology has become so commonplace
that few systems can operate effectively without some form of them making a vital
contribution. Wherever magnetic materials with dielectric properties (or vice versa)
are required, there is likely to be an application for a ferrite or other transition-metal
oxide, from cores for inductors and transformers, to discs or tapes and read/write
heads for information storage, to thin films for high-density computer memories, to
nonreciprocal microwave control devices, to antennas for home electronics, to mi-
crowave antireflection coatings, to permanent magnets for automobile ignitions, to
isolator devices for fiber-optical laser sources, to rubberized refrigerator magnets.
In later years, exotic phenomena that jointly involve the magnetic and electrical
conductivity properties have been discovered in oxides that contain magnetic ions.
High-temperature superconductivity for low power loss and giant magnetoresis-
tance effects for magnetic field sensors and magnetic random access memories
(MRAMs) have been found in perovskite-based compounds. Hybrid combinations
of piezoelectric and magnetic compounds have spawned a growing interest in piezo-
magnetics and multiferroics. Magnetic oxides have also provided a molecular-scale
vehicle for fundamental investigations of the electronic and magnetic properties of
the important transition-metal and rare-earth elements of the Periodic table.
1.1 Fundamental Concepts and Definitions
Although the main focus of this book will be the molecular origins of magnetism
and its various manifestations in metal oxides, dielectric properties influence the
magnetic behavior of these materials and in many instances can determine the limits
of their applicability. To this end, we first review some of the basic relations of
electrostatics and extend them to magnetostatics.
G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 1,
c
Springer Science+Business Media, LLC 2009
1
2 1 Introductory Magnetism
1.1.1 Basic Electrostatics
In unrationalized electrostatic (Gaussian) units the electric field vector of a charge
q at a radial distance r is derived from Coulomb’s law
E D
q
r
3
r; (1.1)
where q would be labeled e for an electron and its value in these units is 4:8
10
10
esu. In the mks (SI) system of units, e D 1:6 10
19
C. For an electric dipole
of charge separation d, the dipole moment vector p D qd and
E
dip
Dr˝
e
; (1.2)
where the electric scalar potential ˝
e
D
p r
r
3
and r is the distance from the center
of the dipole.
In a dielectric material, the polarization vector P is proportional to E,
according to
P D
e
E; (1.3)
where
e
is the electric susceptibility (also defined as the electric polarizability ˛
when referred to an individual molecule). The associated charge displacement vec-
tor D of the Maxwell equation rD D 4 for a charge density is defined as
D D E C 4P D .1 C 4
e
/ E D "E ; (1.4)
where " is the electric permittivity or dielectric constant, which is scaled to the per-
mittivity of free space that is set to unity. The 4 factor is required for the convention
of Gaussian units to be used in this text. In SI or mks units the dielectric constant
becomes K
e
D "="
0
,where"
0
D 8:85 10
12
F=m. This basic definition of the
permittivity follows from the assumption that the E and P vectors are parallel. For
other cases, as with magneto-optical coupling discussed in Chap. 7, " is expressed
as a "
ij
tensor according to
D D Œ" E D
2
4
"
xx
"
xy
"
xz
"
yx
"
yy
"
yz
"
zx
"
zy
"
zz
3
5
0
@
E
x
E
y
E
z
1
A
: (1.5)
From (1.2), the dipolar interaction energy between two neighboring dipoles p
1
and
p
2
separated by r
12
is expressed as
E
dip
D
p
1
p
2
r
3
12
3.r
12
p
1
/.r
12
p
2
/
r
5
12
; (1.6)
1.1 Fundamental Concepts and Definitions 3
1.1.2 Basic Magnetostatics
To introduce the magnetic properties of matter, consider two principal areas where
basic magnetism entities occur: magnetic dipoles and their interactions with the
magnetic fields and each other, and the application of Maxwell’s equations to matter
containing magnetic dipoles.
Like electric fields, magnetic fields originate from electrical charges. Unlike elec-
tric fields, the electrical charges must be in motion relative to the frame of reference
of the observer, as in a wire carrying electric current, an electron orbiting a nucleus,
or a charged particle simply spinning. The basic unit of magnetism is not an inde-
pendent charge or pole, but rather a pair of poles, termed positive and negative – a
magnetic dipole with a moment defined as the vector quantity m. All magnetic fields
can be traced to some effective dipole moment as its source, from an individual or
a collection of moments. Our attention, therefore, is focused on the magnetic fields
and the energy associated with the interaction between fields and moments.
Simply stated, the field of a magnetic dipole is the gradient of the scalar potential
H
dip
Dr˝
m
; (1.7)
where the magnetic scalar potential ˝
m
D .m r=r
3
/. It can be determined by
inspection that the concept of magnetic scalar potential is analogous to those of
the electrostatic potential (charge/distance) of (1.1) and the gravitational potential
(mass/distance).
Isolated dipoles that occur in materials can couple to an applied magnetic field
and produce a collective magnetic moment. When expressed as a volume density it
is termed magnetic strength or magnetization M that is proportional to the magnetic
field that cause them to be polarized in the field direction, according to
M D
m
H ; (1.8)
where
m
is the magnetic susceptibility. Analogous to the electrostatic displacement
flux, the magnetic flux density is related through
B D H C 4M D .1 C 4
m
/ H D H ; (1.9)
where is the magnetic permeability that is scaled to the permeability of free space,
which is set to unity in the Gaussian system. In SI or mks units, the permeability of
free space is
0
D 4 10
7
Wb=A m and the relation in (1.9) is expressed in terms
of the relative permeability
r
D =
0
. Analogous to (1.4), this basic definition of
the permeability follows from the assumption that H and M are parallel. When this
is not true, as with gyromagnetic coupling discussed in Chap. 6, is expressed as a
[] tensor according to
B D ΠH D
2
4
xx
xy
xz
yx
yy
yz
zx
zy
zz
3
5
0
@
H
x
H
y
H
z
1
A
; (1.10)
where
ii
D 1 C 4
ii
and
ij
D 4
ij
.
4 1 Introductory Magnetism
The magnitude of
m
and
e
are constant for small densities of dipoles. As the
separation between dipoles decreases and the dipoles begin to interact with each
other through their fields, M and P approach saturation values. In materials with
a high density of magnetic moments that couple spontaneously through short-range
interactions related to the chemical bonding, a magnetization can exist without the
presence of an applied H in the form of ferromagnetism, antiferromagnetism, and
ferrimagnetism.
1.1.3 Demagnetization in Uniformly Magnetized Bodies
The significance of the aforementioned theory can be appreciated immediately in
the examination of magnetized bodies. Application of (1.9) produces insight into
properties of uniformly magnetized structures of specific geometric shapes. As il-
lustrated in Fig. 1.1, a magnetized body in a magnetic field H will develop magnetic
poles on the surfaces through which magnetic flux lines pass. In effect, the body
becomes a dipole that creates an external magnetic field, which is a property of
any permanent magnet. To satisfy the Maxwell equation rB D 0 and satisfy
Fig. 1.1 Two-dimensional sketch of an irregularly shaped body of magnetization M in a uniform
external magnetic field H . Internal demagnetization results from the presence of magnetic poles
on the surface that induce an opposing field of magnitude H
D
, which depends on the particular
geometric details
1.1 Fundamental Concepts and Definitions 5
the boundary condition that the normal component of B be continuous across any
interface, the surface poles must also create an internal demagnetizing magnetic
field H
D
that opposes the magnetization, so that the flux density becomes
B D H C 4M H
D
; (1.11)
where H
D
D N
D
.4M / with 0 N
D
1 represents the effective demagnetizing
factor that depends on the geometric details. [A frequently-used alternative has 4
absorbed into the demagnetizing factor so that 0 N
D
4.] Therefore, (1.11)is
commonly written as
B D H C .1 N
D
/4M: (1.12)
The flux patterns of this common situation are indicated by the sketch in Fig. 1.1.
A comprehensive discussion of demagnetizing factors, including a table of N
D
values accurate for uniformly magnetized bodies of ellipsoidal shape, may be found
in Bozorth [1]. To calculate appropriate values of N
D
for use in the variety of ge-
ometrical shapes that are encountered in magnetic applications, numerical methods
must be applied wherever nonellipsoidal forms are involved. Although the magne-
tization is uniform throughout the body in these cases, the demagnetizing fields are
nonuniform and finite element or other methods that require computer aided design
must be employed. Where the shapes can be approximated by an ellipsoid, however,
analytical relations are available in the literature, for example, by Osborn [2]. For
ellipsoids of revolution, that is, prolate (elongated) or oblate (compressed) spheroids
with circular cross section, the following formulas apply:
Along the axis of symmetry for which the axis is greater than the diameter (pro-
late), that is, m
l
D l=d > 1,
N
D
l
D
1
2.m
2
l
1/
"
m
l
p
2
ln
m
l
C .m
2
l
1/
1=2
m
l
.m
2
l
1/
1=2
!
1
#
: (1.13)
Because the sum of the factors along the three major axes of the ellipse equals unity,
it follows that along any diameter,
N
D
d
D
1
2
1 N
D
l
: (1.14)
Along any diameter for which the diameter is greater than the length of the symme-
try axis (oblate), m
d
D d=l > 1,
N
D
d
D
1
2.m
d
2
1/
(
m
d
2
.m
d
2
1/
1=2
arcsin
"
.m
d
2
1/
1=2
m
d
2
#
1
)
(1.15)
and
N
D
l
D
1 2N
D
d
: (1.16)
6 1 Introductory Magnetism
Table 1.1 Demagnetizing factors N
D
for ellipsoids
of revolution with length l and diameter d
Axis Sphere Needle Disc
l D dl dl d
N
D
l
1/3 0 1
N
D
d
1/3 1=2 0
In most practical cases, there are three limiting situations listed in Table 1.1:a
sphere, for which N
D
D 1=3 isotropically, a cylindrical needle (acicular), for which
N
D
D 0 along its axis and 1/2 along any diameter, and a thin circular disc, for which
N
D
D 1 normal to its surface and 0 in its plane. For these and other less common
geometries, calculated design curves can be found in the literature [2].
Another concept that is important in the magnetism of materials is that of mag-
netic energy and some useful relations that may be found in any standard textbook
will be stated here for later reference. From the definition of H
dip
in (1.1)and
(1.2), the energy of interaction between two neighboring dipoles m
1
and m
2
in
each other’s respective magnetic field is
E
dip
D
m
1
m
2
r
3
12
3.r
12
m
1
/.r
12
m
2
/
r
5
12
; (1.17)
where r
12
is the distance between dipole centers. The energy of interaction between
a dipole moment with a magnetic field is simply
E
m
Dm H : (1.18)
The energy of a magnetized body in a magnetic field may be shown to be
E
m
D
1
2
Z
M H
D
dv
Z
M H dv; (1.19)
where the energy density is integrated over the volume of the magnet, first to account
for the self-energy of the magnet in its own demagnetizing field H
D
, and second to
add the energy of interaction with the external field H .
1.1.4 Domains in Partially Magnetized Bodies
When a body resides in an external field of strength insufficient to maintain the
magnetically saturated state, that is, when H<N
D
.4M
s
/, demagnetization be-
gins to take place through the formation of magnetic domains. It is a property of
spontaneously magnetic (ferromagnetic) materials. Theoretical analyses of domain
1.1 Fundamental Concepts and Definitions 7
structures may be found in most standard texts on magnetism. The basic physical
reason for their occurrence is to reduce the magnetostatic energy that results from
poles that are induced on the specimen surface in proportion to the strength of
the magnetization. As H is reduced, 4M decreases and the domains change
accordingly.
The domain pattern is usually initiated by the appearance of 180
ı
domains of
reverse magnetization separated by narrow regions (Bloch walls) in which the mag-
netic moments rotate to accomplish the reversal in direction. As the magnetizing
field is reduced further the reverse domains grow at the expense of the existing
magnetized volume until equilibrium is reached. To remove the surface poles and
suppress the attendant magnetostatic energy, 90
ı
closure domains are created at the
specimen edges.
In Fig. 1.2 the two-dimensional model of rectangular geometry illustrates the
concept of how the 90
ı
closure domain walls are set up at 45
ı
angles to confine
the flux within the specimen, with a simple and then more complex pattern. The
actual arrangement is determined by balancing the magnetostatic energy with the
anisotropy energy contained in the domain walls.
1
Within the domains themselves,
the magnetization vectors can rotate further in their anisotropy fields to complete the
demagnetization process. The phenomena of magnetic anisotropy and hysteresis are
explored in Chap. 5.
Fig. 1.2 The elimination of surface poles by 90
ı
closure domains: (a) magnetized specimen with
surface poles, (b) simplest example of two 180
ı
domains with 90
ı
closure domains, and (c) domain
pattern more typical of an actual situation
1
Where hysteresis is significant, as in permanent magnets, there are internal bias fields arising
from magnetocrystalline anisotropy that must be overcome, for example, the coercive field H
c
,
before the domain walls can move. When the coercive field (and its related anisotropy field H
K
/ is
taken into account, the onset of demagnetization would not begin until H C H
c
<N
D
.4M
s
/.
8 1 Introductory Magnetism
1.2 Induced Magnetism
Magnetic moments occur in all states of matter, but will be limited to solids for
the discussion at hand. The fundamental types have been traditionally character-
ized as follows: (1) diamagnetism, from magnetic dipoles that must be induced and
aligned by an applied magnetic field, (2) paramagnetism, with permanent magnetic
moments that may be aligned by an applied magnetic field and are randomized by
increasing temperature, (3) spontaneous magnetism in which the moments align
collinearly (ferromagnetism), antiparallel (antiferromagnetism), and the important
case of combined ferromagnetism and antiferromagnetism termed ferrimagnetism,
all of which exist without the presence of an applied field. A fourth class of mag-
netic materials with unusual magnetic and electrical properties are superconductors,
which now includes metal oxides.
1.2.1 Diamagnetism and Paramagnetism
All materials have the potential for diamagnetism. According to Lenz’s law, the
magnetic dipole moment of a current loop induced by a magnetic field will align
so as to produce a magnetic field that “opposes the field producing it.” Unbound
electrons of any material can be arranged to form magnetic dipoles that give a mag-
netization defined by (1.4) with a temperature-dependent diamagnetic susceptibility
dia
<0. A sampling of diamagnetic susceptibilities expressed in emu is given
in Table 1.2. In the discussions of sold-state magnetism presented in later chapters,
ions with closed-shell electronic structures are diamagnetic. In the general literature,
they are commonly referred to as nonmagnetic.
In certain of these materials, the atoms or ions already have magnetic moments
that generally result from unfilled quantum states with unpaired electron spins to
provide a net moment, sometimes coupled to moments associated with the electron
orbital motion. The magnitude of the susceptibility of typical paramagnets is on the
order of 10
4
, which would dominate the susceptibilities of diamagnets by more
than an order of magnitude. The origin of paramagnetism in solids begins with elec-
trons of charge e and mass m
e
, which possess magnetic moments that arise from
both spin and the moment generated by the simulated current orbital motion about
Table 1.2 Diamagnetic susceptibilities
Element
dia
10
6
Element
dia
10
6
Ag 31 Cl 15:6
B 7 F 20:1
Bi 192 I 44:6
C 6 N 5:51
P 26:3 O 4:61