19 Magnetoelectric Near Fields
533
a
(e)
±
n
∝ R
d
0
L
j
(m)
s (z)
top
n
· d
ldz,
(19.12)
where d is the disk thickness. The edge magnetic current
j
(m)
s
top
is a persistent
current appearing due to the mesoscopic effect: the magnitude of such a resonant
current becomes appreciable when the parameters of the ferrite disk are reduced to the
scale of the dipole-dipole quantum phase coherence length of precessing electrons.
At the MDM resonances, one has the “spin-orbit” interaction between precessing
magnetic dipoles and a persistent orbital magnetic current. When the frequency of
the orbital rotation of MDM resonances in a ferrite disk is close to the ferromagnetic
resonance frequency, the precessing magnetic dipoles become strongly correlated
and one observers fermionization of the system composed of bosons. There are
the macroscopic quantum phenomena related to the collective motion of magnetic
dipoles coalescing into the same quantum state, described by a single coherent wavefunction of the condensate. This is a fundamentally distinctive feature from a boson
condensate created by small-scale exchange-interaction magnons [61].
It is worth noting that along with the circulation of the surface magnetic current
j
(m)
s
top
caused by the edge wave function, there is also the quadratic-form circulation due to this function. For the double-valued edge wave functions δ ± , we have
the following orthonormality condition on contour L = 2π R [55]:
R
2π
0
iμ a
∂(δ ± ) n
∂θ
δ
∗
±
n − (δ ± ) n
iμ a
∂(δ ± ) n
∂θ
∗
r =R
dθ
= Rμ a
(q ± ) n − (q ± ) n
2π
0
(δ ± ) n
δ
∗
±
n
r =R
dθ = 0
.
(19.13)
For mode n, there are the normalisation relations for the edge functions
2π
0
(δ + ) n
δ
∗
+
n
r =R
dθ = (N + ) n
(19.14)
and
2π
0
(δ − ) n
δ
∗
−
n
r =R
dθ = (N − ) n
(19.15)
where (N + ) n and (N − ) n are real quantities, which we characterize as surface power
flow density of the MDM mode. For a given direction of a bias magnetic field, the
533
a
(e)
±
n
∝ R
d
0
L
j
(m)
s (z)
top
n
· d
ldz,
(19.12)
where d is the disk thickness. The edge magnetic current
j
(m)
s
top
is a persistent
current appearing due to the mesoscopic effect: the magnitude of such a resonant
current becomes appreciable when the parameters of the ferrite disk are reduced to the
scale of the dipole-dipole quantum phase coherence length of precessing electrons.
At the MDM resonances, one has the “spin-orbit” interaction between precessing
magnetic dipoles and a persistent orbital magnetic current. When the frequency of
the orbital rotation of MDM resonances in a ferrite disk is close to the ferromagnetic
resonance frequency, the precessing magnetic dipoles become strongly correlated
and one observers fermionization of the system composed of bosons. There are
the macroscopic quantum phenomena related to the collective motion of magnetic
dipoles coalescing into the same quantum state, described by a single coherent wavefunction of the condensate. This is a fundamentally distinctive feature from a boson
condensate created by small-scale exchange-interaction magnons [61].
It is worth noting that along with the circulation of the surface magnetic current
j
(m)
s
top
caused by the edge wave function, there is also the quadratic-form circulation due to this function. For the double-valued edge wave functions δ ± , we have
the following orthonormality condition on contour L = 2π R [55]:
R
2π
0
iμ a
∂(δ ± ) n
∂θ
δ
∗
±
n − (δ ± ) n
iμ a
∂(δ ± ) n
∂θ
∗
r =R
dθ
= Rμ a
(q ± ) n − (q ± ) n
2π
0
(δ ± ) n
δ
∗
±
n
r =R
dθ = 0
.
(19.13)
For mode n, there are the normalisation relations for the edge functions
2π
0
(δ + ) n
δ
∗
+
n
r =R
dθ = (N + ) n
(19.14)
and
2π
0
(δ − ) n
δ
∗
−
n
r =R
dθ = (N − ) n
(19.15)
where (N + ) n and (N − ) n are real quantities, which we characterize as surface power
flow density of the MDM mode. For a given direction of a bias magnetic field, the