14
T. Yu and G. E. W. Bauer
the magnon (annihilation) operator in the film and nanowire, respectively. The total
system Hamiltonian then reads
ˆ
H / =
k
ω k ˆ
α
†
k ˆ
α k +
k z
˜
ω k z ˆ
β
†
k z
ˆ
β k z
+
k
g k e
−ik y y 0 ˆ
α
†
k
ˆ
β k z + g
∗
k e
ik y y 0 ˆ
β
†
k z
ˆ
α k
,
(1.31)
where ω k and ˜
ω k z are the frequencies of spin waves in the film and nanowire and the
coupling
g k = F(k)
m
(k)∗
x , m
(k)∗
y
|k| ik y
ik y −k
2
y /|k|
˜
m
(k z )
x
˜
m
(k z )
y
,
(1.32)
with F(k) = −μ 0 γ
M s ˜
M s /Lφ (k). The form factor φ (k) = 2 sin(k y w/2)(1 −
e
−kd
)(1 − e
−ks
)/(k y k
2
) couples spin waves with wavelengths of the order of the
nanowire width (mode selection) and lim k→0 φ (k) = wsd. Pure exchange waves
are right-circularly polarized with m
(k y )
y
= im
(k y )
x
and their coupling is perfectly chiral since g −|k y | = 0 and g |k y | = 0.
Equations (1.11) and (1.13) give the spin-wave dispersion and amplitudes in the
thin film. The spin waves propagate in the nanowire along ˆ
z with amplitudes [6, 8]
˜
m
k z
x =
1
4D(k z )wd
, ˜
m
k z
y = i
D(k z )
4wd
,
(1.33)
where
D(k z ) =
H app + N xx ˜
M s + ˜
λ ex k 2
z
˜
M s
H app + N yy ˜
M s + ˜
λ ex k 2
z
˜
M s
.
(1.34)
H app and ˜
λ ex are the applied magnetic field and the exchange stiffness of the nanowire,
respectively. The demagnetization factors N xx w/(d + w) and N yy = d/(d + w)
[6] also govern the spin waves frequency
˜
ω k z = μ 0 γ
(H app + N yy ˜
M s + ˜
λ ex k 2
z
˜
M s )(H app + N xx ˜
M s + ˜
λ ex k 2
z
˜
M s ). (1.35)
When the magnetic field is antiparallel to the nanowire magnetization we require
H app
< min{N yy ˜
M s , N xx ˜
M s }. The ellipticity of the Kittel mode with k z = 0 is
strongly affected by the shape anisotropy when the applied field is sufficiently small
and the aspect ratio large: when d w, N xx → 1, N yy → 0, D is large and the mode
is nearly linearly-polarized. On the other hand, when d ≈ w, D → 1, and the Kittel
mode is circularly polarized. When d w, and the Kittel mode traces an elliptical
orbit. Figure 1.5 illustrates the chirality of the coupling parameter g k of the k z -Kittel
T. Yu and G. E. W. Bauer
the magnon (annihilation) operator in the film and nanowire, respectively. The total
system Hamiltonian then reads
ˆ
H / =
k
ω k ˆ
α
†
k ˆ
α k +
k z
˜
ω k z ˆ
β
†
k z
ˆ
β k z
+
k
g k e
−ik y y 0 ˆ
α
†
k
ˆ
β k z + g
∗
k e
ik y y 0 ˆ
β
†
k z
ˆ
α k
,
(1.31)
where ω k and ˜
ω k z are the frequencies of spin waves in the film and nanowire and the
coupling
g k = F(k)
m
(k)∗
x , m
(k)∗
y
|k| ik y
ik y −k
2
y /|k|
˜
m
(k z )
x
˜
m
(k z )
y
,
(1.32)
with F(k) = −μ 0 γ
M s ˜
M s /Lφ (k). The form factor φ (k) = 2 sin(k y w/2)(1 −
e
−kd
)(1 − e
−ks
)/(k y k
2
) couples spin waves with wavelengths of the order of the
nanowire width (mode selection) and lim k→0 φ (k) = wsd. Pure exchange waves
are right-circularly polarized with m
(k y )
y
= im
(k y )
x
and their coupling is perfectly chiral since g −|k y | = 0 and g |k y | = 0.
Equations (1.11) and (1.13) give the spin-wave dispersion and amplitudes in the
thin film. The spin waves propagate in the nanowire along ˆ
z with amplitudes [6, 8]
˜
m
k z
x =
1
4D(k z )wd
, ˜
m
k z
y = i
D(k z )
4wd
,
(1.33)
where
D(k z ) =
H app + N xx ˜
M s + ˜
λ ex k 2
z
˜
M s
H app + N yy ˜
M s + ˜
λ ex k 2
z
˜
M s
.
(1.34)
H app and ˜
λ ex are the applied magnetic field and the exchange stiffness of the nanowire,
respectively. The demagnetization factors N xx w/(d + w) and N yy = d/(d + w)
[6] also govern the spin waves frequency
˜
ω k z = μ 0 γ
(H app + N yy ˜
M s + ˜
λ ex k 2
z
˜
M s )(H app + N xx ˜
M s + ˜
λ ex k 2
z
˜
M s ). (1.35)
When the magnetic field is antiparallel to the nanowire magnetization we require
H app
< min{N yy ˜
M s , N xx ˜
M s }. The ellipticity of the Kittel mode with k z = 0 is
strongly affected by the shape anisotropy when the applied field is sufficiently small
and the aspect ratio large: when d w, N xx → 1, N yy → 0, D is large and the mode
is nearly linearly-polarized. On the other hand, when d ≈ w, D → 1, and the Kittel
mode is circularly polarized. When d w, and the Kittel mode traces an elliptical
orbit. Figure 1.5 illustrates the chirality of the coupling parameter g k of the k z -Kittel