3.2. SPIN WAVES AND MAGNETOSTATIC WAVES 77
!
r
D
0
H
eff
. A physical reasoning may prove very helpful in understanding these phenomena.
When the circular frequency (!) of the applied AC field is far below !
0
, the AC magnetization
components are very small and there is a very small difference between the magnitudes of inter-
nal
N
H
i
and external
N
H
e
magnetic fields. As frequency ! is increased, approaching !
0
(that is,
! !
0
), magnetization reaches infinity (maximized in the lossy case) and, following Eq. (3.6),
the internal field
N
H
i
tends to zero (see Hogan [1], or Waldron [5]). is Polder’s condition is
rather physically paradoxical or presents a mathematical singularity. at is because the inter-
nal field
N
H
i
generates magnetization through internal magnetic susceptibility as
N
M D ŒX
N
H
i
.
erefore, when X tends to infinity and
N
H
i
tends to zero, a singularity occurs and magnetization
N
M is indefinable. According to Waldron [5], in the Polder’s condition, the microwave energy is
excluded from the ferrite specimen and magnetization also remains small. From a different point
of view, this situation can be understood through the boundary conditions as follows. e mag-
netic field source outside the specimen acts on it, producing a very high permeability at ! D !
0
,
which, in turn, reflects this AC field or does not allow any AC energy penetration. When losses
are considered, the internal field to be built is just enough to make up for the energy loss of the
electron spins’ precessional motion [1].
Returning to Kittel’s resonance, as the frequency of the applied field is increased above
!
0
.! > !
0
/, the phase of magnetization (
N
M ) reverses with respect to the externally applied
field
N
H
e
. If we also consider the minus sign in Eq. (3.6), then the magnitude of the internal field
starts to increase until it reaches its maximum value at Kittel’s resonance, !
r
D
0
H
eff
, given
in Eq. (3.14) [1]. Under this condition, the internal field .
N
H
i
/ becomes theoretically infinite (ig-
noring losses) or practically maximum (including losses), and magnetization
N
M is maximized
as well. However, under Kittel’s condition, the large magnetization observed is not due to the
gyromagnetic resonance, but due to the concentration of microwave energy inside the speci-
men [5]. In view of the above analysis, microwave losses under Polder’s resonance condition are
small due to the negligible energy concentration inside the sample. In contrast, at Kittel’s reso-
nance there is heavy absorption (high losses) of microwave energy caused by the large fields in
the specimen [5].
3.2 SPIN WAVES AND MAGNETOSTATIC WAVES
Up to this point, we have seen cases in which a ferrite medium was placed in a uniform magnetic
field strong enough to produce saturation. e exchange field forces act to align all magnetic
dipoles parallel to one another, and all are lined up with the applied DC field. It was then
considered that the whole ferrite medium behaves as a large magnetic dipole precessing about
the DC magnetic field. From a different point of view, all the individual spins precess in phase.
When an additional uniform, circularly polarized (CP), high-frequency magnetic field acts on
the ferrite, the spins’ precession angle is increased or decreased depending on the circular right-
(RHCP) or left (LHCP)-handed polarization. Furthermore, for the above uniform field, all
spins still precess in phase. If the field is non-uniform, for example if it is applied to one part of
78 3. FINITE FERRITE SAMPLES
a ferrite sample, then the precession angle will be increased (assume an RHCP field) in the area
where this field is applied. e internal exchange fields will tend to align their neighboring spin
dipoles; in other words, they will act in such a way as to swing their neighbors into the largest
precession angle [7]. e above will happen with some delay. In this manner, the largest precession
angle disturbance will travel through the crystal lattice in the form of a wave. is is called a
“spin-wave” and appears to have both phase and amplitude to change from dipole to dipole [8]. If
a high-frequency field with LHCP is applied to one part of the ferrite sample, the disturbance
would be a smaller precession angle traveling again through the sample as a spin-wave. e
wavelength of low-order spin-wave modes can be very long and can be expected to occur in
the very low microwave range, even as “magnetostatic waves.” However, high-order spin waves
may have very short wavelengths. In any case, it is important to know how these wavelengths
are compared with the dimensions of the ferrite specimen. When the wavelength is quite larger
than the sample dimensions, propagation effects may be ignored, and the corresponding waves
are “magnetostatic waves.” In contrast, when the wavelength is comparable or shorter than the
ferrite specimen, propagation must be taken into account, and the waves are “spin waves.”
To generalize the above example, a spatially non-uniform high-frequency magnetic field
can potentially excite spin waves. is non-uniform magnetic field could come from the cor-
responding externally applied magnetic field. Two mechanisms are usually responsible for this
internal non-uniformity and, as a result, for the excitation of spin waves. e first one concerns
the ferrite’s non-linear behavior when high-power microwave fields are applied to it [
4, 5]. e
second mechanism concerns different anisotropy fields often caused by geometrical irregulari-
ties [7, 8].
Starting with the first mechanism, when the microwave magnetic field exceeds a certain
critical value, the magnetization in the direction of the biasing DC field (e.g., M
z
for rods and
disks, where z is the axis of symmetry) becomes unstable. Assume, for example, a transversely
biased disk with .
N
H
DC
D H
0
Oz/ for which the magnetization component M
z
D M
s
C m
z
(m
z
is the microwave part) decreases locally, perhaps due to thermal vibration. is, in turn, corre-
sponds to an increase in angle- between the electrons’ spin axes and the Oz-direction of the po-
larizing field. If M
z
is stable, it will increase, by a reduction of , to its initial value. In contrast, if
M
z
is unstable, it will continue to decrease (increase of ) until a new steady state is reached [5].
is local disturbance of spin precessional motion will affect the neighboring electron spins
through exchange forces. In this manner, the disturbance will propagate in all directions within
the ferrite specimen as a spin-wave. Something worth keeping in mind is that saturation mag-
netization (M
s
) is a major part of M
z
. us, the characteristics of spin waves are expected to
depend strongly on M
s
as well as on the exchange forces. In addition, this phenomenon may be
accompanied by strong demagnetization.
e second mechanism occurs mostly in normally biased thin ferrimagnetic films. In this
case, a uniform microwave field may excite long wavelength (i.e., magnetostatic) spin waves.
Actually, the electron spins on the film’s surface observe different anisotropy fields than those of
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