12 Magnetoelastic Waves in Thin Films
309
Figure 12.2 clearly shows that the two branches of the magnetoelastic wave dispersion relations do not cross each other. If the transversal elastic waves and the spin
waves were not interacting, their dispersion relations would intersect. However, due
to the magnetoelastic interaction, this crossing is avoided, leading to a gap between
the two curves. This so-called anticrossing behavior of the dispersion relations is a
typical characteristic of interacting waves [15, 16].
The gap formation is also visible in the equation of the dispersion relations,
i.e. (12.72). At the point in reciprocal space where the dispersion relations of the
uncoupled waves would intersect, i.e. (ω cross , k cross ), the term ω
2
fm − ω
2
H vanishes.
At this condition, the interaction coefficient J k
2
ω fy has a strong influence on the
dispersion relation. When J k
2
ω fy ω
2
fm − ω
2
H , the interaction between the magnetic
and elastic system is strong, leading to the formation of coupled magnetoelastic
waves. As a result, the anticrossing is formed, with a frequency gap that quantifies
the strength of the interaction. This frequency gap is, to first order, given by
ω(k cross ) ≈
J k 2
cross ω fy
ω cross
=
γ B 2 ω fy
C 44 M 0
(12.73)
where the relation ω cross =
C 44
ρ
k cross was used. Note that ω fy also depends on k cross
and that this approximation is only valid when ω(k cross ) < ω cross . On the other hand,
when J k
2
ω fy ω
2
fm − ω
2
H , the interaction term can be neglected, leading to nearly
uncoupled elastic and magnetic waves. In this regime, the waves are called quasielastic or quasi-magnetic [15, 16]. Hence, the interaction between the elastic and
magnetic waves is strongest when they are (nearly) degenerate, resulting in coupled
magnetoelastic waves. By contrast, quasi-noninteracting waves are obtained when
their frequencies and/or their wavelengths differ strongly.
The wavenumber at the crossing, k cross , can be found by equalizing the dispersion
relations of the noninteracting systems, i.e. ω H (k cross ) = ω fm (k cross ), and solving for
k cross . For the geometry considered here, the noninteracting dispersion relations are
equal when
v t k cross = (ω 0 + ω M λ ex k
2
cross )
2
+ ω M
ω 0 + ω M λ ex k
2
cross + ω M (1 − P)P
,
(12.74)
which needs to be solved iteratively. Note that P is also a function of k cross according
to (12.30). Once k cross is determined, the interaction coefficient J k
2
ω fy and the gap
amplitude can be calculated. In general, the coupling increases strongly for higher
wavenumbers k cross . This originates from the behavior of the magnetostriction and
the Villari effect: a shorter wavelength leads to larger gradients of both displacement
and magnetization. As a result, the magnetoelastic body force in (12.60) and the
magnetoelastic field in (12.65) increase, leading to stronger interactions for higher
k cross values. This behavior opens possibilities to control the interaction strength by
external parameters. For example, increasing an external magnetic field shifts the
309
Figure 12.2 clearly shows that the two branches of the magnetoelastic wave dispersion relations do not cross each other. If the transversal elastic waves and the spin
waves were not interacting, their dispersion relations would intersect. However, due
to the magnetoelastic interaction, this crossing is avoided, leading to a gap between
the two curves. This so-called anticrossing behavior of the dispersion relations is a
typical characteristic of interacting waves [15, 16].
The gap formation is also visible in the equation of the dispersion relations,
i.e. (12.72). At the point in reciprocal space where the dispersion relations of the
uncoupled waves would intersect, i.e. (ω cross , k cross ), the term ω
2
fm − ω
2
H vanishes.
At this condition, the interaction coefficient J k
2
ω fy has a strong influence on the
dispersion relation. When J k
2
ω fy ω
2
fm − ω
2
H , the interaction between the magnetic
and elastic system is strong, leading to the formation of coupled magnetoelastic
waves. As a result, the anticrossing is formed, with a frequency gap that quantifies
the strength of the interaction. This frequency gap is, to first order, given by
ω(k cross ) ≈
J k 2
cross ω fy
ω cross
=
γ B 2 ω fy
C 44 M 0
(12.73)
where the relation ω cross =
C 44
ρ
k cross was used. Note that ω fy also depends on k cross
and that this approximation is only valid when ω(k cross ) < ω cross . On the other hand,
when J k
2
ω fy ω
2
fm − ω
2
H , the interaction term can be neglected, leading to nearly
uncoupled elastic and magnetic waves. In this regime, the waves are called quasielastic or quasi-magnetic [15, 16]. Hence, the interaction between the elastic and
magnetic waves is strongest when they are (nearly) degenerate, resulting in coupled
magnetoelastic waves. By contrast, quasi-noninteracting waves are obtained when
their frequencies and/or their wavelengths differ strongly.
The wavenumber at the crossing, k cross , can be found by equalizing the dispersion
relations of the noninteracting systems, i.e. ω H (k cross ) = ω fm (k cross ), and solving for
k cross . For the geometry considered here, the noninteracting dispersion relations are
equal when
v t k cross = (ω 0 + ω M λ ex k
2
cross )
2
+ ω M
ω 0 + ω M λ ex k
2
cross + ω M (1 − P)P
,
(12.74)
which needs to be solved iteratively. Note that P is also a function of k cross according
to (12.30). Once k cross is determined, the interaction coefficient J k
2
ω fy and the gap
amplitude can be calculated. In general, the coupling increases strongly for higher
wavenumbers k cross . This originates from the behavior of the magnetostriction and
the Villari effect: a shorter wavelength leads to larger gradients of both displacement
and magnetization. As a result, the magnetoelastic body force in (12.60) and the
magnetoelastic field in (12.65) increase, leading to stronger interactions for higher
k cross values. This behavior opens possibilities to control the interaction strength by
external parameters. For example, increasing an external magnetic field shifts the