Dionne G.F. Magnetic Oxides
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322 6 Electromagnetic Properties
6.4.2 High-Frequency Transverse Limits
Where gyromagnetic coupling produces the effects described, their appearance in
partially magnetized media generally offers an impediment to practical usage due to
the absorptive properties. Because the spectral nature of the interaction is unspecific,
only the frequency limits of the regimes are of interest. For the cubic systems, this
range is defined by
H
K
<!
r
<.H
K
C 4M
s
/: (6.96)
The resonance frequency from the magnetoelastic fields is the lower limit of
the multidomain absorption band, also known as the “low-field” loss region in the
microwave lexicon. From Table 5.7 for K
1
<0with <111> easy axes, the angular
resonance frequency is expressed as
!
r
D H
K
D
4
j
K
1
C
s
j
3M
s
4
j
K
1
j
3M
s
for 0: (6.97)
To this point in the discussion dynamic interactions, only materials of cubic lattice
symmetry have been included. From the earliest work in this area it was realized
that !
r
could be increased by the use of high-anisotropy hexagonal ferrites. The
most common of this family is the uniaxial .Ba; Sr/ Fe
12
O
19
(M -type) that features
a c-axis polar anisotropy field H
K
20 kOe. In polycrystalline form, this system
is categorized as a “hard” magnet because of its large coercive fields that result
from H
K
.D gK
u1
=M
s
/. As a consequence, the longitudinal drive field necessary
to cause wall movement is too great to allow any wall movement influence on
0
i
.
Beyond permanent magnets, these materials are of interest for gyromagnetism in
the millimeter-wave bands.
Above the frequency range of high longitudinal susceptibility, microwave proper-
ties come into focus. The gyromagnetic spectral transmission properties of ferrites
based on the imaginary part of the susceptibility are sketched in Fig. 6.28.To
the microwave device engineer, the issues are straightforward: the operating fre-
quency must be well out of the absorption or “low-field loss” regime. This means
that !>.H
K
C 4M
s
/ is a critical design criterion for partially magnetized fer-
rites. However, if the medium is magnetically saturated, ! can be above or below
the resonance frequency without incurring significant absorption losses. A special
case of this effect can be realized if uniaxial hexaferrite is utilized with its large
H
K
.or H
Ku
/ as a self-biasing field. As indicated in Fig. 6.28, low-loss transverse
susceptibility can be accessed either from below or above the resonance region as
was demonstrated in self-biased circulator devices at 31 and 73.5GHz [83,84].
Another form of hexagonal ferrite has been utilized to extend the dispersion
regime to frequencies above the range of cubic ferrites. In Fig.6.29,the spec-
tra for a cubic NiZn spinel ferrite and the “easy-plane” Co
2
Z hexagonal system
.Ba
3
Co
2
Fe
24
O
41
/ are compared. The higher frequency limit of the hexagonal sys-
tem can be explained as follows: in an easy-plane structure, the large H
K
forces
the M
s
vectors away from the c-axis and into the plane where they are subject to
6.4 Permeability and Propagation 323
Fig. 6.28 Spectral model of gyromagnetic interaction indicating regions of transmission and ab-
sorption. Figure reprinted from [80] with permission.
c
2003 by the IEEE
Fig. 6.29 Comparison of permeability dispersion for cubic nickel spinel ferrite and hexagonal
cobalt Z ferrite [80]. Figure reprinted from [80] with permission
c
2003 by the IEEE
the smaller in-plane azimuthal anisotropy field H
K
. H
K
/ field. As a result the
resonance frequency for a single-domain platelet following the Kittel theory [19]
discussed in Chap. 5 can be written as
p
H
K
H
K'
<!
r
<
q
.H
K
C 4M
s
/
H
K'
C 4M
s
: (6.98)
324 6 Electromagnetic Properties
Table 6.5 Initial susceptibility frequency limits
Mechanism
0
i
D
0
i
1!
min
!
max
Domain
rotation
2M
2
s
= jK
1
j
.
K
1
<0
/
4M
2
s
=3 jK
1
j
.
K
1
>0
/
– !
D 1=
Domain wall
movement
32M
2
s
=3 jK
1
j
.
K
1
<0
/
16M
2
s
= jK
1
j
.
jK
1
j >0
/
–
!
2
w
1=
2
2
w
1=2
Gyromagnetic
(cubic)
4M
s
=H
a
K
H
K
.H
K
C 4M
s
/
(multidomain)
ŒH
K
.
H
K
C 4M
s
/
1=2
(single-domain, planar)
Gyromagnetic 4M
s
=H
K
(polar, c-axis) H
K
.H
K
C 4M
s
/
(hexagonal) 4M
s
=H
K
(azimuthal, in
plane)
H
K
H
K
1=2
Œ
.
H
K
C4M
s
/
H
K
C4M
s
1=2
a
In the present context, H
K
represents the effective maximum anisotropy field for the particular
physical situation. As discussed in the foregoing text, it is proportional to the ratio of K
1
=M
s
In comparing the magnitude of this range with that of the cubic case, one can de-
duce that the entire !
r
range can shift to higher frequencies by a ratio on the order
of
H
K
=H
K
1=2
, which can be at least a factor of ten in most cases, as suggested
by the measurement results in Fig. 6.29. A listing of the relevant relations for per-
meability and the various frequency limits may be found in Table 6.5.
6.4.3 Snoek’s Law Considerations
When (6.90)and(6.97) are combined into a product, the relation known as Snoek’s
law for unmagnetized materials is obtained [85]:
!
min
r
0
i
2
3
4M
s
: (6.99)
The implications of this relation for ac properties below the dispersion region where
most applications of ferrites are found can be seen graphically from Snoek’s data
for the NiZn spinel ferrite family shown in Fig.6.30. According to (6.99), Snoek’s
limit for an isotropic specimen can be characterized simply by the value of 4M
s
.
Consequently, the traditional strategy is to seek the highest magnetization. The room
temperature 4M
s
limit of high-resistivity ferrites appears to be about 5,000Gauss
(although reports of Fe
2C
-laden Fe
3
O
4
.1 x/ C ZnFe
2
O
4
.x/ combinations with
room temperature 4M
s
> 7;000 Gauss have been published [86]). One obvi-
ous approach would be to lower the operating temperature, for which the spinel
models discussed in Appendix 4B of Chap.4 could be applied. In addition, the gar-
net Gd
3
Fe
5
O
12
also offers high magnetizations if temperatures were reduced to the
liquid helium range, i.e., 4K.
6.4 Permeability and Propagation 325
Fig. 6.30 Comparison of permeability dispersion for a range of members of the Ni
ı
Zn
1ı
Fe
2
O
4
family, illustrating the origin of Snoek’s law. Original data are from Gorter [85]. Figure reprinted
from [80] with permission.
c
2003 by the IEEE
As pointed out above, the single-domain
0
i
can be greatly enhanced by wall
movement. However, practical considerations dictate that only domain rotation can
be active at the higher frequencies where gyromagnetic effects occur. For spinel
ferrites with 4M
s
D 5;000 Gauss, !
min
r
0
i
translates into 10 GHz for a system of
single-domain particles. Therefore, to obtain a permeability in the unmagnetized
state of at least 10, for example, the maximum frequency of operation is about
1 GHz, even without considering the losses from absorption.
Caution should be exercised when Snoek’s law is used to estimate the upper limit
of frequency for acceptable permeability in traditional nonresonant applications. As
stated, the frequency limit !
min
r
is derived from transverse coupling which is con-
figurationally independent of
0
i
. One must therefore recognize that the relation’s
validity can be justified only if the dispersion edge at !
D
1
exceeds the gyro-
magnetic lower limit of !
min
r
D H
K
. This apparent contrivance can be justified if it
is recognized that
1
also varies as H
K
through their common dependence on the
ratio of spin–orbit coupling to low-symmetry crystal-field splitting energies charac-
terized by = that appears in (6.36) from Mattuck and Strandberg [12], and in the
analysis of anisotropic g-factors by Dionne [87] in Sect. 5.1.1. As a consequence,
Snoek’s law should be used with the caveat that the upper frequency limit be !
or
!
min
r
, whichever is smaller.
Figure6.31 presents permeability data that illustrate how the variation in K
1
from
the substitution of fast-relaxing Co
2C
ions in .NiZnFe/
3x
Co
x
O
4
influences the
frequency limits [59]. At x D 0:03, the positive contribution of Co
2C
to K
1
that off-
326 6 Electromagnetic Properties
Fig. 6.31 Permeability spectra of Co
2C
-substituted NiZn spinel ferrite indicating the influence
of positive anisotropy contributions to the relaxation and anisotropy field effects on the frequency
limits. Original data are from [59]. Figure reprinted from [80] with permission.
c
2003 by the
IEEE
sets the negative anisotropy field, thereby allowing
i
to increase, while introducing
some local gyromagnetic effects at higher frequencies. When x is raised to 0.16, K
1
is strongly positive, causing
i
to drop (possibly extinguishing domain-wall move-
ment), while defining a higher frequency range of gryomagnetism because both the
high K
1
and coincidentally the faster
1
rate. Note also that the upper limit remains
dominated by the value of 4M
s
.
For hexaferrite specimens of planar geometry, provided that they are single-
domain or comprise crystallographically oriented single-domain particles with mag-
netization vectors and domain rotation is confined to the plane, Snoek’s law can be
modified to produce a more favorable frequency limit. Based on the expressions for
0
i
and !
min
listed in Table6.5:
!
min
r
0
i
D 4M
s
s
H
K
H
K
: (6.100)
For Co
2
Z ferrite, H
K
D 13;000 Oe, H
K
D 112 Oe, and 4M
s
D 3;350 Gauss.
Therefore, the Snoek’s frequency limit should be increased by a factor of about ten
in comparison to a cubic Ni spinel ferrite of equivalent magnetization. The curves
in Fig. 6.31 support the reasonable accuracy of this model.
A qualitative summary of this analysis is presented schematically in Fig. 6.32,
where it is suggested that the tradeoff between permeability and high-frequency
operations is also dependent on the spin–lattice relaxation time. Low-wall damping
(large
w
and &
w
) promises superior low-frequency properties, but with the disad-
vantage of an earlier onset of dispersion in multiple-domain specimens. Above the
6.4 Permeability and Propagation 327
Fig. 6.32 Schematic diagram of permeability variations as a function of frequency caused by
various magnetic damping actions. Figure reprinted from [80] with permission.
c
2003 by the
IEEE
domain wall damping edge, the reduced permeability of single-domain rotation is
still present provided that !
>!
d
. Amidst these conflicts lies the influence of K
1
which is also dependent on !
.Large usually means smaller K
1
, higher
i
,and
the earlier appearance of gyromagnetic loss effects because of a reduced value of
the anisotropy field H
K
.
In conclusion, the issue of domains walls for unmagnetized materials can be
resolved into a simple rule: multiple domains, low K
1
,high for low frequencies;
single domain, high K
1
,low for high frequencies. For microwaves, where ! is
above the dispersion region, the upper edge of the gyromagnetic resonance region
becomes a concern, such that the low K
1
,high combination can be preferred,
unless very high H
K
hexaferrites are desired for millimeter waves.
6.4.4 Circular Polarization and Nonreciprocal Properties
To begin the discussion of microwave propagation in magnetically ordered systems
(ferrites), we must first point out that only one circular component of the z-axis-
directed rf signal has been considered to this point. In Chap. 1, it was recognized
that only the mode that is in phase with the Larmor precession can produce the con-
tinuous rotation torque necessary for magnetic resonance absorption. However, the
counterrotating or anti-Larmor mode (termed negative by convention) is also im-
portant in rf propagation, and a relation for its susceptibility needs to be established.
The complex susceptibility tensor is derived in Appendix 6A. For our immediate
328 6 Electromagnetic Properties
purposes, a shortcut to the relations for the two modes
˙
D
0
˙
i
00
˙
can be ar-
rived at if we presume the anti-Larmor solution to be analogous to (6.54), but with
the vector direction (sign) of ! reversed. For off-resonance conditions, approxima-
tions to the two modes can be expressed as
0
C
Š M
H
r
!
.H
r
!/
2
C 4.!/
2
00
C
Š M
2!
.H
r
!/
2
C 4.!/
2
.Larmor/; (6.101)
0
Š M
H
r
C !
.H
r
C !/
2
C 4.!/
2
00
Š M
2!
.H
r
C !/
2
C 4.!/
2
.anti-Larmor/; (6.102)
where the bracketed factors represent the resonance line-shape functions that mod-
ify the dc susceptibility M=H
r
at the resonance field. The rf subscripts have been
dropped from . The frequency variation of the two modes can be appreciated by
the graphical presentation in Fig. 6.33. In effect, the permeability of the magnetic
medium can be radically different for the two senses of circular polarization, and
the corresponding effects on propagation are the origin of nonreciprocity.
From Appendix 6A, the rf susceptibility tensor is written as
ΠD
2
4
xx
xy
0
yx
yy
0
000
3
5
; (6.103)
from which a permeability tensor can be constructed:
ΠD 1 C 4 ΠD
2
4
i0
Ci0
001
3
5
: (6.104)
where D 1 C 4
xx;yy
,and D i4
xy
D i4
yx
. The respective scalar
permeability solutions for the two circular polarization plane-wave modes
.1=2/
H
x
rf
H
y
rf
are the eigenvalues of the determinant, which can be seen
by inspection to be
˙
D ˙ for the x y plane after diagonalization is carried
out. The desired relations are then obtained from (6.102)and(6.103)as
0
˙
D 1 C 4M
H
r
!
.H
r
!/
2
C 4.!/
2
;
6.4 Permeability and Propagation 329
Fig. 6.33 Sketch of the real and imaginary parts of the ferrite complex permeability at an operat-
ing frequency ! D 10 GHz, illustrating the nonreciprocal features that arise from the different
solutions for the two circular modes of a linearly polarized plane wave identified by the C
and subscripts. The resonance frequency is varied by the internal magnetic field according to
!
0
D gm
B
H
0
. Note that the differential between
0
C
and
0
is smaller when !
0
>!
00
˙
D 4M
2!
.H
r
!/
2
C 4.!/
2
: (6.105)
The Maxwell equations for the respective magnetic and dielectric fields for a fixed
value of z are
rH D"
@E
@t
D i!"E
rE D
@H
@t
Di!H ; (6.106)
from which E can be eliminated to produce
rrH D !
2
"ΠH : (6.107)
A general analytical solution for this expression can be found in Lax and Button
[88]. For the transverse electric and magnetic plane wave (TEM) in a semi-infinite
medium, the permeability enters the propagation constant
p
. In standard electro-
magnetic notation
p
D ˛
p
C iˇ
p
in terms of the attenuation and phase constants,
respectively. In a linearly polarized plane wave, the two equal counterrotating cir-
cular modes of initial amplitude H
rf
=2 are depicted in Fig. 6.34. As a function of
time and distance, the rf field vector (confined in the x–y plane) can be expressed
in terms of its components,
330 6 Electromagnetic Properties
Fig. 6.34 Tutorial diagram illustrating the origin of Faraday rotation of the rf signal axis of a
linearly polarized plane wave under the influence of a magnetized medium with nonreciprocal
propagation properties
H
rf
.t; z/ D
1
2
.H
rfC
i C H
rf
j /
1
2
H
rf
e
Ci!t
p˙
z
i C e
i!t
p˙
z
j
D
1
2
H
rf
Œe
pC
z
.cos !tiCsin !tj/Ce
p
z
.cos !ti sin !tj/:
(6.108)
where cos!t D H
x
rf
=H
rf
and sin !t D H
y
rf
=H
rf
. Accordingly, separate propagation
constants
p˙
are assigned to each circular polarization mode. By the usual conven-
tion, right-hand circular polarization (RHCP) is designated as the clockwise rotation
.C!/ of the vector .H
rfC
/ when viewed along the direction of propagation .Cz/.It
is the RHCP mode that drives the Larmor precession of the magnetic moment into
resonance.
The corresponding
p˙
for the case of both H and the propagation directed lon-
gitudinally to the z-axis can be treated as scalars and expressed in terms of the
complex permeabilities [88] according to
3
2
p˙
D
˛
p˙
C iˇ
p˙
2
D
!
c
2
".˙ / ; (6.109)
3
Two other propagation situations have been analyzed in a medium of semiinfinite cross section,
where H is directed perpendicular instead of parallel to the z-axis. In this case, however, the H
rf
of
a TEM wave can vary from parallel to perpendicular with respect to H . The former produces zero
gyromagnetic interaction between H and H
rf
k
D 1
, and the latter gives an averaged result
?
2
2
:
6.4 Permeability and Propagation 331
from which
˛
2
p˙
ˇ
2
p˙
D
!
c
2
"
0
˙
;
2˛
p˙
ˇ
p˙
D
!
c
2
"
00
˙
(6.110)
and
˛
p˙
D
!
c
r
"
2
q
0
2
˙
C
00
2
˙
0
˙
1=2
; (6.111)
ˇ
p˙
D
!
c
r
"
2
q
0
2
˙
C
00
2
˙
C
0
˙
1=2
:
For the important high-frequency case where ! H
r
;
00
˙
=
0
˙
1,and(6.111)
can be simplified to
˛
p˙
!
c
p
"
2
00
˙
p
0
˙
;
ˇ
p˙
!
c
p
"
0
˙
p
0
˙
: (6.112)
It is important to examine the physical significance of the ˇ
p˙
parameter.
Equation (6.109) expresses that each circular mode has a different propagation
velocity provided that the medium is in a magnetized state. From (6.105)for
! 4M and H
r
(the frequency far above resonance) where absorption losses
are negligible .! jH
r
!j/, ˛
p˙
,andˇ
p˙
can be simplified to
˛
p˙
!
p
"
2c
2!
M
!
.H
r
!/
2
D
!
p
"
c
.4M /!
.H
r
!/
2
; (6.113a)
ˇ
p˙
!
p
"
c
1 C
1
2
4M
H
r
!
!
p
"
c
1
1
2
4M
!
(6.113b)
and the corresponding propagation velocities are then
v
p˙
!
ˇ
p˙
c
p
"
1 ˙
1
2
4M
!
: (6.114)
For a fixed M vector, a circularly polarized wave will produce a phase shift
˙
D ˇ
p˙
z that is either delayed or advanced relative to the empty space value,
depending on the sign of the mode. In further contrast to nonmagnetic media, the
phase shift of a TEM wave propagating parallel to the M vector is dependent on
its direction, which means that if the signal traverses the medium and is then re-
flected back to its point of origin along the same path with polarization unaltered,