12 Magnetoelastic Waves in Thin Films
295
ever, for thin films, this approximation is even valid for conductors as long as the
thickness of the film is sufficiently small with respect to the skin depth of the ferromagnet [27]. It should also be noted that in the derivations below, the dynamic
magnetization and the fields are averaged over the thickness of the film and are thus
uniform in the y-direction. This is a valid approximation when the wavelength is
much larger than the thickness of the film, i.e. kd 1. If this is not the case, it
is possible for thickness modes to arise which have varying amplitude along the
thickness [16]. These thickness modes will however not be considered here.
The magnetization is again defined as in (12.14) with the magnetization saturated in-plane along an external field H ext in the z-direction. The components of the
dynamic magnetization are in the x- and y-direction and form a plane wave, the spin
wave. The exchange field is not affected by the thin film boundaries and is given by
(12.13). As indicated by (12.8), the boundaries generate magnetic surface charges
and therefore act as a source of the dipolar field. Therefore, in contrast with the
exchange field, the dipolar field is affected by the boundaries.
For a thin film of finite thickness, the dipolar field can be approximated by [28–30]
h dip (r, t) = −[P
k · m
||k|| 2 k + (1 − P)(n · m)n]
(12.28)
= −
⎡
⎣
P sin
2
(θ )
0 P sin(θ ) cos(θ )
0
1− P
0
P sin(θ ) cos(θ ) 0
P cos
2
(θ )
⎤
⎦ m(r, t) (12.29)
with
P = 1 −
1 − e
−kd
kd
,
(12.30)
k
2
= k
2
x + k
2
z , as well as θ the angle between the static magnetization M 0 and
wavevector k. In the limit of an infinitesimally thin film, this simplifies to
lim
d→0
h dip (r, t) = −
⎡
⎣
0
1
0
⎤
⎦ m(r, t)
(12.31)
and thus only the out-of-plane magnetization component contributes to the spin
wave dipolar field. Hence, for a thin film of finite thickness, the spin wave behavior
is markedly different as compared to a thin film of infinitesimal thickness.
The linearized LLG equation (12.19) with the modified dipolar field in (12.28)
can then be written as
ω fx −iω
iω ω fy
m x
m y
= 0
(12.32)
with
295
ever, for thin films, this approximation is even valid for conductors as long as the
thickness of the film is sufficiently small with respect to the skin depth of the ferromagnet [27]. It should also be noted that in the derivations below, the dynamic
magnetization and the fields are averaged over the thickness of the film and are thus
uniform in the y-direction. This is a valid approximation when the wavelength is
much larger than the thickness of the film, i.e. kd 1. If this is not the case, it
is possible for thickness modes to arise which have varying amplitude along the
thickness [16]. These thickness modes will however not be considered here.
The magnetization is again defined as in (12.14) with the magnetization saturated in-plane along an external field H ext in the z-direction. The components of the
dynamic magnetization are in the x- and y-direction and form a plane wave, the spin
wave. The exchange field is not affected by the thin film boundaries and is given by
(12.13). As indicated by (12.8), the boundaries generate magnetic surface charges
and therefore act as a source of the dipolar field. Therefore, in contrast with the
exchange field, the dipolar field is affected by the boundaries.
For a thin film of finite thickness, the dipolar field can be approximated by [28–30]
h dip (r, t) = −[P
k · m
||k|| 2 k + (1 − P)(n · m)n]
(12.28)
= −
⎡
⎣
P sin
2
(θ )
0 P sin(θ ) cos(θ )
0
1− P
0
P sin(θ ) cos(θ ) 0
P cos
2
(θ )
⎤
⎦ m(r, t) (12.29)
with
P = 1 −
1 − e
−kd
kd
,
(12.30)
k
2
= k
2
x + k
2
z , as well as θ the angle between the static magnetization M 0 and
wavevector k. In the limit of an infinitesimally thin film, this simplifies to
lim
d→0
h dip (r, t) = −
⎡
⎣
0
1
0
⎤
⎦ m(r, t)
(12.31)
and thus only the out-of-plane magnetization component contributes to the spin
wave dipolar field. Hence, for a thin film of finite thickness, the spin wave behavior
is markedly different as compared to a thin film of infinitesimal thickness.
The linearized LLG equation (12.19) with the modified dipolar field in (12.28)
can then be written as
ω fx −iω
iω ω fy
m x
m y
= 0
(12.32)
with