12 Magnetoelastic Waves in Thin Films
319
τ fm =
2
α(ω fx + ω fy )
.
(12.83)
The lifetime at GHz frequencies is typically of the order of ns in metallic ferromagnets, such as Ni considered in this chapter, and of the order of μs for low-damping
magnetic insulators such as Yttrium Iron Garnet (YIG). On the other hand, much
less is know for elastic waves at GHz frequencies although estimates suggest that
the lifetime is similar to that of spin waves. Experimentally, it is typically found that
the mean free path of (surface) elastic waves at these frequencies is somewhat larger
than these of spin waves [40, 51, 52, 59], however, the topic still requires further
research.
In the case of magnetoelastic waves, analytical derivations of the lifetimes and
decay lengths are rather complex. In the quasi-elastic regime, it is clear that the
lifetime is strongly determined by the lifetime of the elastic wave. The energy of
quasi-elastic waves is almost completely stored in the elastic system, with only a
negligible part in the magnetic system. Hence, the dissipation due to the magnetic
loss has negligible influence on the overall dissipation. An analogous argument can
be made for the quasi-magnetic regime, where magnetic properties and lifetimes
should determine the decay of the magnetoelastic waves.
In the strongly coupled magnetoelastic regime, i.e. near the anticrossing, no simple
conclusion can be drawn. In this regime, the energy is distributed between magnetic
and elastic domains and is transferred forth and back during propagation. Therefore
both magnetic and elastic losses contribute to the total energy dissipation. One may
expect in such a case that the lifetime of a magnetoelastic wave is given by a suitable weighted average of the lifetimes of magnetic and elastic waves. In general, the
lifetime depends on multiple parameters, such as the orientation of the static magnetization, the interaction coefficient, the wavenumber, etc.. Further work is required to
fully understand in particular the effect of the magnetoelastic interactions on the lifetime of strongly coupled magnetoelastic waves. By contrast, the group velocities of
magnetoelastic waves are well understood and can be calculated from the dispersion
relations, so the assessment of mean free paths is straighforward once the lifetime is
known.
12.5 Conclusion
The first part of this chapter presented a review of magnetic and elastic interactions
as well as the formation of magnetic (spin) and elastic waves. It has been shown
that the dynamic behavior of the magnetization and displacement can be seen as an
eigensystem with eigenvalues corresponding to the dispersion relations and eigenstates describing the polarization and ellipticity of the resulting waves. Based on this
formalism of eigensystems, both magnetic and elastic waves have been studied in
bulk and thin film materials. For magnetic (spin) waves, different regimes have been
identified and their correlation with Maxwell’s equations has been explained.
319
τ fm =
2
α(ω fx + ω fy )
.
(12.83)
The lifetime at GHz frequencies is typically of the order of ns in metallic ferromagnets, such as Ni considered in this chapter, and of the order of μs for low-damping
magnetic insulators such as Yttrium Iron Garnet (YIG). On the other hand, much
less is know for elastic waves at GHz frequencies although estimates suggest that
the lifetime is similar to that of spin waves. Experimentally, it is typically found that
the mean free path of (surface) elastic waves at these frequencies is somewhat larger
than these of spin waves [40, 51, 52, 59], however, the topic still requires further
research.
In the case of magnetoelastic waves, analytical derivations of the lifetimes and
decay lengths are rather complex. In the quasi-elastic regime, it is clear that the
lifetime is strongly determined by the lifetime of the elastic wave. The energy of
quasi-elastic waves is almost completely stored in the elastic system, with only a
negligible part in the magnetic system. Hence, the dissipation due to the magnetic
loss has negligible influence on the overall dissipation. An analogous argument can
be made for the quasi-magnetic regime, where magnetic properties and lifetimes
should determine the decay of the magnetoelastic waves.
In the strongly coupled magnetoelastic regime, i.e. near the anticrossing, no simple
conclusion can be drawn. In this regime, the energy is distributed between magnetic
and elastic domains and is transferred forth and back during propagation. Therefore
both magnetic and elastic losses contribute to the total energy dissipation. One may
expect in such a case that the lifetime of a magnetoelastic wave is given by a suitable weighted average of the lifetimes of magnetic and elastic waves. In general, the
lifetime depends on multiple parameters, such as the orientation of the static magnetization, the interaction coefficient, the wavenumber, etc.. Further work is required to
fully understand in particular the effect of the magnetoelastic interactions on the lifetime of strongly coupled magnetoelastic waves. By contrast, the group velocities of
magnetoelastic waves are well understood and can be calculated from the dispersion
relations, so the assessment of mean free paths is straighforward once the lifetime is
known.
12.5 Conclusion
The first part of this chapter presented a review of magnetic and elastic interactions
as well as the formation of magnetic (spin) and elastic waves. It has been shown
that the dynamic behavior of the magnetization and displacement can be seen as an
eigensystem with eigenvalues corresponding to the dispersion relations and eigenstates describing the polarization and ellipticity of the resulting waves. Based on this
formalism of eigensystems, both magnetic and elastic waves have been studied in
bulk and thin film materials. For magnetic (spin) waves, different regimes have been
identified and their correlation with Maxwell’s equations has been explained.