73
C H A P T E R 3
Finite Ferrite Samples
3.1 DEMAGNITIZATION FACTORS AND FERRITE
SAMPLES
In the analysis in Chapter 2, an infinitely extending ferrite material, biased by a DC magnetic
field, was considered. In turn, the permeability tensor expressions either with or without losses
are only valid for the internal fields within the ferrite and only when the alternating fields are
uniform throughout all space, e.g., Hogan [1]. However, in practice, a finite ferrite specimen is
used. e specimen can be in the form of small spheres, rods or posts, thin disks or plates as well
as in the form of thin or thick films. Films are of particular interest in the quest for miniaturiza-
tion and the integration process. Moreover, the ferrite permeability tensor is expressed in terms
of the DC biasing field
N
H
0
D
N
H
i
internal to the ferrite. e question in the finite sample case
is how this internal field
N
H
i
is related to the corresponding externally applied one
N
H
a
(which is
more readily measured), that is, the DC magnetic field in the air surrounding the ferrite sample.
e answer in principle is straightforward. Fields
N
H
a
and
N
H
i
are related through the boundary
conditions at the ferrite-air interface. However, this is not so simple since the ferrite perme-
ability involved depends on
N
H
i
. A formulation accounting for this field should be established.
A classical approach given by Kittel [2] is the introduction of demagnetization factors into the
magnetic susceptibility or permeability tensor. e demagnetization factors can be defined by
applying the boundary conditions at the ferrite-air interface. Let us first examine two special
cases of the DC biasing field orientation with respect to the ferrite sample surface. First, we as-
sume a planar sample (Figure 3.1a) transversely magnetized at saturation (
N
M D
N
M
s
) as shown in
Figure 3.1b. e continuity of the magnetic flux density normal to the planar sample component
(B
n
) yields:
B
n
D
0
H
a
D
0
.
H
i
C M
s
/
: (3.1)
In addition, the internal field becomes
H
i
D H
a
M
s
: (3.2)
Equation (3.2) shows that M
s
reduces the internal field with respect to its value in the
air region. In contrast, when a longitudinal saturation magnetization is applied, the continuity
of the tangential magnetic field
N
H
t
retains an internal magnetic field identical to the externally
applied one
N
H
a
, as shown in Figure 3.1c. In other words,
N
H
t
D
N
H
a
D
N
H
i
: (3.3)
74 3. FINITE FERRITE SAMPLES
Perpendicular
Parallel
Air
Ferrite
FerriteH
0
H
a
H
a
H
a
M
s
H
0
M
s
Air
Air
Air
Parallel
H
a
(a) (b) (c)
Figure 3.1: (a) Directions of internal
N
H
i
D
N
H
0
and external
N
H
a
magnetic fields for a thin pla-
nar ferrite sample magnetized at saturation. (b) transversely—normal and (c) longitudinally—
tangential.
In general, the internal DC magnetic field normal to the sample surface is reduced by
the projection of the saturation magnetization along the direction of each component. For such
an arbitrary bias orientation, the two continuity conditions (3.2) and (3.3) can be rewritten as
follows:
Normal components: On
N
H
i
D On
N
H
a
N
M
s
(3.4)
Tangential components: On
N
H
i
D On
N
H
a
; (3.5)
where On is the outward unit vector normal to the sample surface of the ferrite.
When an AC external magnetic field is also applied, this gives rise to an AC magnetization
N
M . is magnetization causes a reduction in the corresponding internal AC field component
normal to the sample surface. By symbolizing the total (including DC) internal and external
magnetic fields with
N
H
i
;
N
H
e
, respectively, and the total magnetization with
N
M D
N
M
ac
C
N
M
s
,
the internal magnetic field can be written as [3, 4],
N
H
i
D
N
H
e
ŒN
N
M ; (3.6)
where ŒN D ŒN
x
; N
y
; N
z
is the demagnetization factor. According to the above analysis, the
elements N
x
; N
y
, and N
z
are obviously dependent on the sample’s shape and are independent
of the DC bias direction (this only defines the direction or components of
N
M
s
). By definition,
we have,
N
x
C N
y
C N
z
D 1: (3.7)
e demagnetization factors are accurately calculated only for ferrite samples of an ellipsoidal
shape and are given in [4]. Note that for all other (non-ellipsoidal) shapes, magnetization is non-
uniform, and hence the macroscopic field varies from point to point within the sample. ere-
fore, there is no macroscopic field that can be defined for the sample as a whole [1]. Suitable
ellipsoidal shapes approximate practically useful samples such as rods and disks. eir demag-
netization factors can be defined by providing that the sample dimensions are small compared
3.1. DEMAGNITIZATION FACTORS AND FERRITE SAMPLES 75
to the wavelength. ese are (see Figure 3.2):
Rod along the z-axis: N
x
D N
y
D 1=2; N
z
D 0 (3.8a)
Sphere: N
x
D N
y
D N
z
D 1=3 (3.8b)
in disk or plate with
a perpendicular z-axis: N
x
D N
y
D 0; N
z
D 1: (3.8c)
x
z
y
Nx = Ny = 0
Nz = 1
Nx = Ny = 1/2
Nz = 0
Nx = Ny = Nz = 1/3
Infinite Plane
Infinite Rod
Sphere
Figure 3.2: Elements N
x
, N
y
, and N
z
of three different shapes of ferrite samples.
Returning to Eq. (3.6), magnetization
N
M is related to the internal magnetic field
N
H
i
through the magnetic susceptibility X previously given from Eqs. (2.15), (2.18), and (2.19)
as:
N
M D ŒX
N
H
i
: (3.9)
By substituting
N
H
i
from Eq. (3.6) into Eq. (3.9), a new relation is obtained between
N
M
and the magnetic field outside the sample at an infinitesimal distance from its surface. is
relation is
N
M D ŒX
N
H
e
ŒX ŒN
N
M
or
f
ŒI CŒX ŒN
g
N
M D ŒX
N
H
e
:
(3.10)
Equation (3.10) can be written in the form
N
M D ŒX
e
N
H
e
, where ŒX
e
is called “the exter-
nal susceptibility tensor.” at is because X
e
expresses magnetization in terms of the field just
outside the sample (for details, see [3] or [4]). To examine the effects of the demagnetization
phenomenon on the gyromagnetic resonance, let us repeat the external susceptibility expression
76 3. FINITE FERRITE SAMPLES
for the simple case in which the DC bias is applied along the z-axis [3]:
M
x
D X
e
xx
H
xe
C X
e
xy
H
ye
D
X
xx
1 C X
yy
N
y
X
xy
X
yx
N
y
D
H
xe
C
X
xy
D
H
ye
(3.11a)
M
y
D X
e
yx
H
xe
C X
e
yy
H
ye
D
X
yx
D
H
xe
C
X
yy
.
1 C X
xx
N
x
/
X
yx
X
xy
N
x
D
H
ye
(3.11b)
M
z
D 0 (3.11c)
D D
.
1 C X
xx
N
x
/
1 C X
yy
N
y
X
yx
X
xy
N
x
N
y
: (3.11d)
Note that expressions (3.11a), (3.11b), (3.11c), and (3.11d) relate the external AC mag-
netic field (outside the sample) to the corresponding AC components of the magnetization.
Futhermore, the internal susceptibility expression ŒX is already given in Eq. (2.17). e im-
portant point to stress here is that the gyromagnetic resonance of an infinitely extending ferrite
medium occurs at the frequency where the denominator of the internal susceptibility ŒX van-
ishes, that is, at !
r
D ! D !
0
, when losses are ignored. At this frequency, the elements of the
permeability tensor and k become infinite. us, it is also called “Polder’s resonance” [5]. In
contrast, the gyromagnetic resonance of a finite ferrite sample occurs at the frequency where
the denominator of the external susceptibility ŒX
e
vanishes, namely, at D D 0. us, ignoring
losses, Eq. (3.11d) reads as follows:
1 C
!
0
!
m
N
x
!
2
0
!
2
1 C
!
0
!
m
N
y
!
2
0
!
2
!
2
!
2
m
N
x
N
y
!
2
0
!
2
2
D 0: (3.12)
Solving Eq. (3.12) for the gyromagnetic resonance frequency of a z-biased sample, we
have:
!
r
D ! D
q
.
!
0
C !
m
N
x
/
!
0
C !
m
N
y
: (3.13)
In (3.13), we substitute frequencies !
m
D
0
M
s
and !
0
D
0
H
0
, where H
0
D H
iz
D
H
a
N
z
M
s
is the internal DC biasing field in the z-direction. An expression, known as “Kittel’s
equation,” [1, 2, 5, 6] results for the resonance frequency called “Kittel’s resonance”:
!
r
D
0
q
Œ
H
a
C
.
N
x
N
z
/
M
s
H
a
C
N
y
N
z
M
s
D
0
H
eff
: (3.14)
It is important to realize from Eq. (3.14) that the gyromagnetic resonance of a finite
sample depends on its demagnetization factors (N
x
; N
y
, and N
z
) and, in turn, on its shape.
e appearance of H
eff
and, consequently, the dependence of Kittel’s resonance on saturation
magnetization (M
s
) are also shown in Eq. (3.14). Recall that at the gyromagnetic resonance of a
lossless ferrite, magnetic susceptibility becomes infinite, driving magnetization to infinity. How-
ever, when losses are taken into account, the above quantities are finite but take their maximum
value. As stated by Hogan [1], a serious confusion may result when considering what actu-
ally happens in a finite ferrite sample at the two resonance frequencies !
r
D !
0
D
0
H
0
and
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