88 3. FINITE FERRITE SAMPLES
that N
eqz
D 0 since (3.29) does not include
N
M
s
D M
s
Oz and N
eqx
D N
eqy
D N
eq
D Dk
2
=M
s
. In
turn, the corresponding magnetic susceptibility is given by (3.9), and the spin-wave resonance
frequency is given by (3.13), which reads as follows:
!
K
D
q
!
0
C !
m
N
eqx
C
!
0
C !
m
N
eqy
D
0
H
0
C
.
0
M
s
/
Dk
2
=M
s
D
0
H
0
C Dk
2
:
(3.31a)
We must keep in mind that (3.30) does not include anisotropy effects or the dipole-dipole
interaction. Recall that from (3.22b), the anisotropy field is approximately constant in time,
and for a Oz-biased ferrite, it is also Oz-directed,
N
H
anis
D .2k=M
s
/Oz [26]; that is, it is parallel to
N
H
DC
D H
0
Oz and it can be directly added to it, yielding a modification in !
0
as
!
0
! !
0˛
D
0
.
H
0
C H
anis
/
D
0
.
H
0
C 2k=M
s
/
: (3.31b)
For the above approximation to be valid, that is, to ignore dipole-dipole interaction or,
equivalently, ignore the demagnetization fields, the wavelength of the spin wave
n
must be at
least a few times smaller than the sample dimensions (`
S
). An approximate limit [12] to ignore
the boundary conditions at the sample air interface is that jkj > k
min
and k
min
D 2=
n min
10=`
S
. According to [12], above this limit, the pole (magnetic charges
m
) distribution induced
by Nm
K
e
j
N
kNr
at the sample-air interface alternates signs very rapidly. In turn, the field created by
these poles (dipole-dipole interaction) does not reach an appreciable distance inside the sample,
(vanishing at negligible distance from the interface). It can in turn be considered a zero magnetic
charge density or r
N
M D 0. Requiring (3.27) to have zero divergence yields:
r
N
M D j
k
x
Ox C k
y
Oy C k
z
Oz
Nm
k
D 0 $ k
x
D k
y
D 0: (3.32)
It is obvious that, due to symmetry, it is required to have k
x
D k
y
D 0, and the gyroscopic
precession cone is circular. A representative spin wave is given in Figure 3.4 [20].
k
̅
H
,
M
s
,
Figure 3.4: A spin wave with ignored dipolar contributions and
N
k=
N
M
s
[20].
3.11 SPIN WAVES INCLUDING DIPOLAR INTERACTIONS
It has been noted in the previous sections that spin waves have a short wavelength and can
be treated as plane waves affected only by exchange and anisotropy fields. Moreover, they are
3.11. SPIN WAVES INCLUDING DIPOLAR INTERACTIONS 89
independent of demagnetization fields. At the end of the spectrum, magnetostatic modes are
independent of exchange interactions, which are short-range forces and dependent mostly on
demagnetization fields. However, there is a gray area between these two states, which is defined
for 0
C
< k < k
min
.
According to Suhl [23], spin waves with wavelengths comparable to sample dimensions
are not normal modes of the system even to first order. In order to clarify this, a short overview
of the basic principles of demagnetization energy will be given. Starting from the divergence of
the magnetic flux density
N
B D
0
.
N
H C
N
M / inside the sample, an equivalent magnetic charge
density, (or pole density),
m
.Nr/ can be defined. From
r
N
B.Nr/ D 0 $ r
0
N
H .Nr/ C
N
M .Nr/
D 0 (3.33a)
we have
r
N
H .Nr/ D r
N
M . Nr/ D
m
.Nr/ or
m
D r
N
M : (3.33b)
Assuming a perfectly lossless ferrimagnetic material, free electric charges do not ex-
ist. us, the electric current density can be set equal to zero J D 0. Furthermore, the fre-
quency of these spin waves may be considered low enough to ignore the displacement current
("
@
N
E
@t
D j!"
N
E), at least as far as the generation of the magnetic field is concerned. erefore,
following [24, 25] and [27], the magnetic field curl can be set equal to zero for the induced
dipolar field .
N
H
d
/, which can then be expressed in terms of a static magnetic potential ˆ
m
.Nr/:
r
N
H
d
D 0 $
N
H
d
.Nr/ D rˆ
m
.Nr/: (3.34)
Substituting back (3.33) reads as follows:
r
2
ˆ
m
.Nr/ D
m
.Nr/: (3.35)
Moreover, in addition to the volume magnetic charge density
m
.Nr/, the discontinuity
of magnetization at the sample-air interface produces an effective magnetic surface-charge den-
sity
m
.Nr/. is can be easily expressed in terms of
N
M .Nr/ by imposing the boundary condition,
(integrating r
N
M ), at the interface [27],
m
.Nr/ D On.Nr/
N
M . Nr/; (3.36)
where On.Nr/ is the outward unit normal at any surface point . Nr/.
e resulting magnetic potential is also given in [27] as
ˆ
m
.Nr/ D
1
4
•
v
r
0
N
M .Nr
0
/
j
Nr Nr
0
j
dv
0
C
1
4
II
s
On.Nr
0
/
N
M . Nr
0
/
j
Nr Nr
0
j
ds
0
: (3.37)
It is important to note that when magnetization can be considered uniform through spec-
imen volume v, the first term of (3.37) vanishes since r
N
M becomes zero. In contrast, the
90 3. FINITE FERRITE SAMPLES
surface-charge density term always exists for finite samples; it is what we have already called
“shape anisotropy,” accounting for the demagnetization factors ŒN . is is exactly the case
in which uniform magnetization is considered and shape anisotropy is included in Eqs. (3.1)
through (3.12). Moreover, the analysis of the previous section and hence Eq. (3.31), is valid for
a uniform magnetization since it does not account for any magnetic-charge densities. Always
remember that isolated magnetic charges (monopoles) cannot exist, so they always appear as
pairs of their opposite.
A qualitative explanation of the physical phenomena involved in the demagnetization field
may prove valuable in understanding spin and magnetostatic waves. To begin with, let us define
magnetization energy [24] as:
W
d
.Nr/ D
1
2
•
v
N
M
N
H
d
dv D
Œ
X
e
2
•
v
ˇ
ˇ
N
H
d
ˇ
ˇ
2
dv 0: (3.38)
It is clear that W
d
is always positive, but the basic principle of energy minimization re-
quires that the demagnetizing field distributes itself in such a way as to avoid the creation of vol-
ume as well as surface magnetic charge densities. e above is known as the “pole avoidance prin-
ciple.” Concerning volume charge, it is avoided when magnetization is uniformly (r
N
M D 0)
aligned in a specific direction. In contrast, the avoidance of surface poles requires that
N
M must be
parallel to the specimen surface to achieve (On
N
M D 0). It is obvious that there is a contradiction
in these two tendencies since the fulfillment of one excludes the other. us, demagnetization
energy cannot be equal to zero. A compromise of the two requirements of minimizing W
d
will
be achieved, as shown, for example, in Figure 3.5 [24].
M
M
(a) (b)
Figure 3.5: e pole avoidance principle causes demagnetization energy minimization by align-
ing the magnetization parallel to the surface and maintaining it as uniform as possible in the
interior: (a) parallelepiped and (b) spherical specimen [24].
From the above, and particularly from Figure 3.5, it is obvious that demagnetization is
a long-range mechanism since it affects practically the whole specimen volume and its surface.
However, the above consideration is accurate only at magnetostatics, e.g., when frequency is low
enough where r
N
H D 0 is valid, or the specimen’s dimensions are so small that this mechanism
becomes strong or even dominant.
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