Dionne G.F. Magnetic Oxides
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References 271
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Chapter 6
Electromagnetic Properties
In the previous chapters, the emphasis is placed on the electronic origins of static
magnetism in electrically insulating compounds, beginning with isolated moments
of transition metal ions coupled paramagnetically and progressing to the various
spontaneous magnetic systems. Apart from occasional references to their applica-
tion in magnetic recording by the switching of magnetic domains and their use as
permanent magnets in applications requiring electrical insulation, few indications
are given that relate to their important applications in time-varying magnetic fields
and as electromagnetic transmission media.
The most common usage of ferrimagnetism is for high permeability cores of
inductors and transformers that typically operate at audio frequencies, i.e., below
20 kHz, and most commonly 60 Hz for power applications. Despite their generally
lower magnetization when compared with ferromagnetic metal alloys, the reduc-
tion of eddy current losses makes these oxide materials an attractive alternative to
laminated magnetic steel in many cases.
In the partially magnetized state, the frequency dependence of the permeability
results from three causes (1) magnetization rotation, (2) magnetic domain wall reso-
nance, and (3) gyromagnetic resonance within individual domains. The former two
arise from longitudinal coupling between the alternating magnetic field and the do-
main magnetization vectors, and usually occur in the frequency range below 1 GHz.
The third is the same transverse interaction effect that is exploited in magnetic res-
onance, where a magnetic field renders the material single domain. It can set in
below 1 GHz, usually after the longitudinal permeability has been reduced to small
values by damping effects, and can cause large absorption over a band that reaches
to several GHz, depending on the anisotropy and geometrical demagnetizing fields.
Because of their excellent dielectric properties at microwave frequencies, ferrites
can transmit electromagnetic waves with relatively low absorption losses. Above
the absorption band mentioned earlier, significant permeability can remain in the
tail of the dispersion curve with relatively small magnetic losses. For this rea-
son, the microwave properties of ferrites have been investigated extensively and
their frequency-dependent propagation in bands from 1 to 100 GHz have found
widespread application in communications and radar technology.
In all of these frequency regimes, the strength of the coupling between the
magnetization and the crystal lattice is crucial in determining the efficiency of
G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 6,
c
Springer Science+Business Media, LLC 2009
273
274 6 Electromagnetic Properties
transmission and the frequencies at which the material has useful permeability.
The results of theoretical analyses point ultimately to the magnetocrystalline
anisotropy, and particularly to the energy transfer between spins and lattice vi-
brations (phonons), which will be the central theme of this chapter.
6.1 Magnetic Relaxation
To begin the examination of the dynamic properties of magnetic systems, it is ap-
propriate to assume a priori that the switching rate of magnetic moments between
equilibrium states is limited by the relaxation time. Because of its importance,
magnetic relaxation in relation to gyromagnetic resonance was introduced phe-
nomenologically in Chap. 1. For our present purposes, however, some additional
concepts must be reviewed. Specifically, the system of magnetic moments, fre-
quently called the spin system because of the crystal field quenching of orbital
moments in the 3d
n
ion transition group, relaxes to an equilibrium state with the
phonon system of the lattice. Except for special situations where the ground state
retains a significant orbital component, the relaxation rate is determined by a spin–
lattice time constant
1
10
7
s at room temperature, which increases to 10
2
sin
the cryogenic range where the phonon density is much lower.
A much shorter time constant
2
10
10
s exists for the coherence lifetime
among the individual spins in a typical paramagnetic system. Within the timescale
of the spin–lattice relaxation, disturbances of the spin system can generally be ig-
nored. Moreover, since the interaction between spins is essentially dipolar, with no
significant influence from the lattice, there is little temperature dependence. The
spin–spin relaxation rate, however, increases in proportion to the volume concen-
tration of spins. Where spin alignments become spontaneous, however, the issue
of spin coherence in a magnetically ordered ferromagnetic system is moot because
strong exchange coupling tends to lock the spins in collective unison and
2
!1.
Magnetic relaxation will be discussed first as a general effect under nonresonant
conditions. In sections that follow, we examine more closely the phenomenon of
magnetic resonance and introduce the quantum mechanical point of view, which
arises from specific conditions dictated by the electronic energy-level splittings in a
magnetic field (Zeeman effect) combined with an incident electromagnetic wave of
matching quantum energy.
6.1.1 Nonresonant Longitudinal Relaxation
Because of the spin–lattice interaction, a phase lag between a magnetization vec-
tor M
ac
and an alternating longitudinal drive field H
ac
of angular frequency ! will
occur. If the time dependent component is parallel to the static component H (longi-
tudinal as opposed to the transverse excitation H
rf
of conventional radio-frequency
6.1 Magnetic Relaxation 275
magnetic resonance), the total magnetic field can be represented by
H.t/D H C H
ac
cos !t; (6.1)
and the magnetization with a phase lag angle becomes
M.t/D M CM
ac
cos .!t '/ D M CM
ac
.cos !t cos ' sin !t sin '/ : (6.2)
Inspection of (6.2) leads to extraction of the real and imaginary parts of the com-
plex susceptibility. From the definitions of Sect. 6.1.2, the static susceptibility is
m
D M=H. By analogy, we can then separate the in-phase (real) and orthogonal
(imaginary) components according to
0
ac
D
M
ac
cos '
H
ac
;
00
ac
D
M
ac
sin '
H
ac
; (6.3)
with the phase angle defined by tan D
00
ac
=
0
ac
.IfH is expressed in conventional
complex notation as H C H
ac
exp .i!t/,(6.2) can be expressed as
M.t/D M C
ac
H
ac
exp .i!t/; (6.4)
where the longitudinal ac susceptibility
ac
D
0
ac
i
00
ac
. One useful relation involv-
ing the imaginary
00
ac
that can be derived by calculation of the energy integral over
one cycle
R
M dH reveals that the power absorbed per unit volume is given by
P D
1
2
!
00
ac
H
2
ac
: (6.5)
Because of the association of
00
ac
with power absorption, the parameter tan D
00
ac
=
0
ac
already defined is often referred to as the loss tangent. A nonzero
00
ac
oc-
curs because of a damping effect that is introduced as a relaxation time similar
to a capacitance-resistance time constant that can lead to a dispersion/absorption
centered at an angular frequency defined by !
D 1=. However, the underlying
physical situation involving magnetic moments comprising excited spins is more
complicated because of the various interactions present. In sketches of Fig. 6.1,this
longitudinal dispersion is contrasted with the transverse resonance introduced in
Chap. 1.
Among the earliest studies of relaxation phenomena in paramagnetism was the
thermodynamic analysis of [1]. In the general context of magnetic interactions of
all types, this approach is also referred to as “nonresonant” relaxation. Their ini-
tial model comprised separate spin and lattice systems, each in thermal equilibrium
at respective temperatures T
spin
and T
latt
and subject to spin–lattice relaxation time
(here characterized by for the macroscopic case), as depicted schematically in
276 6 Electromagnetic Properties
Fig. 6.1 Schematic diagram contrasting the physical origins and permeability effects of longi-
tudinal and transverse interactions between alternating magnetic drive fields and magnetization
vectors. Figure reprinted from [80] with permission.
c
2003 by the IEEE
Fig. 6.2. In an alternating field of frequency much less than the relaxation rate
1
,
T
spin
T
latt
, because the heat exchange response between the two systems can
follow the changes in field and magnetization without a significant phase lag. At
higher frequencies, however, the two systems can no longer remain in thermal equi-
librium and in the limiting case there is no transfer of heat (Q 0) between them.
For these conditions, isothermal and adiabatic susceptibilities are defined, respec-
tively, as
T
D
@M
@H
T
;
Q
D
@M
@H
Q
: (6.6)
The corresponding specific heats at constant M and H are defined as C
M
D
.dQ=dT/
M
and C
H
D .dQ=dT/
H
, respectively, and are related by
C
H
C
M
D
T
Q
: (6.7)
6.1 Magnetic Relaxation 277
Fig. 6.2 Block diagram of thermodynamic concept of nonresonant relaxation. When the spin
and lattice systems have different temperatures, the mechanism that restores thermal equilibrium
is characterized by the relaxation time
1
The analysis begins with the recognition that the heat transfer rate from spin to
lattice baths is proportional to the difference temperature between the systems, ac-
cording to
dQ
dt
D˛
Q
T
spin
T
latt
D˛
Q
T: (6.8)
where ˛
Q
is the coupling constant between the spin and lattice baths. The ex-
panded solution of this differential equation in thermodynamic formalism with dQ
expressed in terms of entropy may be found in standard references [2,3]. Here only
some of the key steps will be stated. From the basic relation
dQ D C
M
dT T
@H
@T
M
dM; (6.9)
(6.8) can be expressed as
˛
Q
T dt D C
M
dT T
@H
@T
M
dM: (6.10)
Time and frequency dependence is introduced with the assumption that T fol-
lows the same exponential function as the magnetic field and magnetization, and
the susceptibility relations of (6.6) are substituted at the appropriate stage in the
derivation. The resulting relations for the frequency-dependent parts of the complex
susceptibility are:
0
ac
D
Q
C
T
Q
1 C .!/
2
;
278 6 Electromagnetic Properties
Fig. 6.3 Characteristic susceptibility and power solutions as a function of ! plotted in arbitrary
units
00
ac
D
T
Q
!
1 C .! /
2
; (6.11)
where D C
H
=˛
Q
.SinceC
H
is the specific heat at constant H , there is an im-
plicit temperature dependence which will be examined as we consider the quantum
mechanical aspects of this phenomenon. The relaxation time in this theory is an
average of all contributing mechanisms. In that sense it differs from
1
and
2
which
represent specific paths back to the ground state. It might therefore be expected that
for the longitudinal case would be shorter than spin–lattice relaxation time
1
of a
transversely excited ferromagnet.
In Fig. 6.3,
0
ac
,
00
ac
, and P are plotted from (6.11)and(6.5) as a function of ! to
illustrate the dispersion effect that occurs at ! D 1. Note the appearance of the two
susceptibility limits of
0
ac
,andthe
00
ac
peak at the center of the dispersion region,
indicating a condition of maximum loss tangent.
6.1.2 Quantum Mechanisms of Spin–Lattice Relaxation
To this point, the concept of relaxation has been used to rationalize the existence of
a phase lag that leads to a complex susceptibility in an alternating magnetic field.
It has also been seen as a controlling influence in the transfer of energy between
spins and the lattice within a thermodynamic system. Ultimately, we seek to de-
scribe the origin of
1
in quantum mechanical terms. To initiate the analysis of spins
transferring between a lower energy E
1
and higher level at E
2
,weconsideratwo-
level system comprising N magnetic moments (m D gm
B
S
z
) per unit volume at a
6.1 Magnetic Relaxation 279
Fig. 6.4 Schematic comparison between equilibrium and excited Boltzmann populations of spins
in a two-state gyromagnetic system
temperature T . Referring to Fig. 6.4, we relate the equilibrium populations N
1
and
N
2
, respectively, to the relaxation rates:
N
1
12
D
N
2
21
; (6.12)
where
12
and
21
are the corresponding relaxation times, i.e., inverse transition
probabilities. If these populations are expressed as a Boltzmann ratio for an energy
splitting ı,
N
2
N
1
D
21
12
D exp
ı
kT
1
ı
kT
(6.13)
and
N
1
N
2
N
1
C N
2
D
12
21
12
C
21
tanh
ı
2kT
(6.14)
for ı kT , thereby expressing a difference between the upward and downward
relaxation frequencies. The ground state E
1
population is decreased by the amount
of increase of the excited state E
2
population, establishing a transient N
0
1
and N
0
2
which can be defined by a Boltzmann function of higher temperature depicted in
Fig. 6.4. If we assume that
12
and
21
are independent of the level populations, the
rate of change of N
0
1
can be expressed as
dN
0
1
dt
D
dN
0
2
dt
D
N
0
2
21
N
0
1
12
: (6.15)
From (6.15) it follows that
d
N
0
1
N
0
2
dt
D 2
N
0
2
21
N
0
1
12
: (6.16)
280 6 Electromagnetic Properties
Recalling that N D N
1
C N
2
D N
0
1
C N
0
2
, we can express (6.16)as
d
N
0
1
N
0
2
dt
D .N
1
C N
2
/
1
12
1
12
N
0
1
N
0
2
1
12
C
1
21
: (6.17)
If N
1
C N
2
is expressed in terms of N
1
N
2
from (6.14), (6.17) then reduces to
d
N
0
1
N
0
2
dt
D
.N
1
N
2
/
N
0
1
N
0
2
1
12
C
1
21
: (6.18)
When multiplied by m, the factors (N
1
– N
2
)and(N
0
1
N
0
2
) become magnetiza-
tions M and M
0
, respectively. Equation (6.18) reduces to the relaxation relation for
precessing magnetization in Fig. 1.14
dM
0
dt
D
M M
0
1
; (6.19)
where the spin–lattice relaxation rate is defined as the combined relaxation rate
1=
1
D 1=
12
C 1=
21
.
The next task is to examine the temperature dependence of
1
, which will in-
troduce spin–orbit–lattice interactions. To this end, we recall the relation for the
creation probability of a phonon of energy ı
p.ı/D
1
exp .ı=kT / 1
: (6.20)
Since the spins are coupled to the lattice vibrations by a time-dependent relaxation,
perturbation theory described in Sect. 6.1.3 can be used to relate spin and phonon
transition probabilities according to
1
12
D KŒp .ı/ C 1;
1
21
D Kp.ı/; (6.21)
where K is a temperature-independent rate parameter that carries units of s
1
.If
(6.20)and(6.21) are combined, the effective relaxation rate becomes
1
1
D Kcoth
ı
2kT
K
2kT
ı
D AT for ı 2kT: (6.22)
This linear relation for
1
as a function of temperature corresponds to the direct
process of one spin transition producing one phonon of the same frequency. This re-
lation contains a magnetic field dependence if the splitting is the result of a Zeeman
effect where ı D 2gm
B
HS, according to the model of Fig. 6.4.
6.1 Magnetic Relaxation 281
Fig. 6.5 Schematic diagram comparing one- and two-phonon relaxation processes
The above analysis can be extended to an indirect two-phonon process by in-
troducing a third energy level separated from E
2
by energy . Although only a
special case of multiphonon relaxation usually referred to as a Raman process, it
is particularly relevant to certain fast-relaxing ion situations encountered frequently
in ferrimagnetic resonance. Two-phonon relaxation occurs first by an absorption of
a phonon to excite a magnetic state from level j2i up to level j3i, followed by the
emission of a phonon through relaxation of the excited state to the ground state j1i,
with the net increase ı to the phonon spectrum energy. The transition sequences of
one- and two-phonon processes are compared in Fig. 6.5.
In the following discussion of the paramagnetic two-level case the nomenclature
employed by [4] is used. Relations similar to (6.21) can be written as
1
13
D K
1
p.ıC /
1
31
D K
1
Œp .ı C / C 1 (6.23)
and
1
23
D K
2
p./
1
32
D K
2
Œp ./ C 1: (6.24)
For this process, we assume that there is no relaxation between levels j1i and j2i,
and then express
d
N
0
1
N
0
2
dt
D K
1
N
0
3
31
N
0
1
13
K
2
N
0
3
32
N
0
2
23
: (6.25)