318
F. Vanderveken et al.
the thickness becomes comparable to the wavelength, different thickness modes
can arise. In the magnetic domain, these are called perpendicularly standing spin
waves and in the elasticity domain, these are called Lamb waves. The magnetoelastic
coupling of such waves is beyond the scope of this chapter and will generally require
numerical calculations.
12.4.3 Damping of Magnetoelastic Waves
So far, all waves have been considered to be lossless and their intrinsic damping was
neglected. However, in real systems, magnetoelastic waves are expected to decay
during propagation. Since their decay length is of great practical interest, we will
present in this last part a brief introduction on the damping of magnetoelastic waves.
More detailed discussions can be found in [54–58].
Several different energy loss mechanisms exist, which dampen the magnetization and displacement dynamics. In the semi-classical continuum theory used in this
chapter, it is common to subsume all different loss mechanisms in a single phenomenological damping term, which is then included in the equation of motion.
The damping of the magnetization dynamics is captured by the damping term in the
LLG equation, characterized by the phenomenological Gilbert damping parameter
α. Analogously, for elastic waves, damping can be introduced into the equations of
motion via phenomenological complex stiffness constants.
The addition of the damping terms to the equations of motion results in energy
dissipation of the dynamic system. As a consequence, the amplitude of the plane
waves considered above decays in time and space. Therefore, the plane wave ansatz
to solve the equations of motion needs to be modified by adding an exponential decay
factor. This damping factor can be seen as originating from the complex frequency,
i.e.
w(r, t)e
i((ω r +iω i )t+k·r)
= w(r, t)e
−t/τ e
i(ω r t+k·r)
= w(r, t)e
−x/δ e
i(ω r t+k·r) (12.82)
with τ = 1/ω i the lifetime, δ = v g τ the mean free path or attenuation length, and
w(r, t) = [u x , u y , u z , m x , m y ]
T the dynamic components of the wave. Note that the
lifetime characterizes the decay of the wave in time and the mean free path characterizes the attenuation of the wave in space.
To determine the decay characteristics of a wave, the imaginary part of its frequency needs to be assessed. This can be achieved within the above approach, which
is based on finding nontrivial solutions of homogeneous linear systems by calculating the roots of their determinants. The real part of the resulting frequency still
represents the dispersion relation, whereas the imaginary part originates from the
additional damping terms and represents the inverse of the lifetime.
For spin waves in a ferromagnetic medium, the lifetime can be found by solving
the LLG equation and is given by
F. Vanderveken et al.
the thickness becomes comparable to the wavelength, different thickness modes
can arise. In the magnetic domain, these are called perpendicularly standing spin
waves and in the elasticity domain, these are called Lamb waves. The magnetoelastic
coupling of such waves is beyond the scope of this chapter and will generally require
numerical calculations.
12.4.3 Damping of Magnetoelastic Waves
So far, all waves have been considered to be lossless and their intrinsic damping was
neglected. However, in real systems, magnetoelastic waves are expected to decay
during propagation. Since their decay length is of great practical interest, we will
present in this last part a brief introduction on the damping of magnetoelastic waves.
More detailed discussions can be found in [54–58].
Several different energy loss mechanisms exist, which dampen the magnetization and displacement dynamics. In the semi-classical continuum theory used in this
chapter, it is common to subsume all different loss mechanisms in a single phenomenological damping term, which is then included in the equation of motion.
The damping of the magnetization dynamics is captured by the damping term in the
LLG equation, characterized by the phenomenological Gilbert damping parameter
α. Analogously, for elastic waves, damping can be introduced into the equations of
motion via phenomenological complex stiffness constants.
The addition of the damping terms to the equations of motion results in energy
dissipation of the dynamic system. As a consequence, the amplitude of the plane
waves considered above decays in time and space. Therefore, the plane wave ansatz
to solve the equations of motion needs to be modified by adding an exponential decay
factor. This damping factor can be seen as originating from the complex frequency,
i.e.
w(r, t)e
i((ω r +iω i )t+k·r)
= w(r, t)e
−t/τ e
i(ω r t+k·r)
= w(r, t)e
−x/δ e
i(ω r t+k·r) (12.82)
with τ = 1/ω i the lifetime, δ = v g τ the mean free path or attenuation length, and
w(r, t) = [u x , u y , u z , m x , m y ]
T the dynamic components of the wave. Note that the
lifetime characterizes the decay of the wave in time and the mean free path characterizes the attenuation of the wave in space.
To determine the decay characteristics of a wave, the imaginary part of its frequency needs to be assessed. This can be achieved within the above approach, which
is based on finding nontrivial solutions of homogeneous linear systems by calculating the roots of their determinants. The real part of the resulting frequency still
represents the dispersion relation, whereas the imaginary part originates from the
additional damping terms and represents the inverse of the lifetime.
For spin waves in a ferromagnetic medium, the lifetime can be found by solving
the LLG equation and is given by