12 Magnetoelastic Waves in Thin Films
289
elastodynamic equations of motion are described. Finally, the fundamental properties
of magnetoelastic waves in thin films with finite thickness are derived and illustrated
both analytically and graphically for different magnetization orientations.
12.2 Spin Waves
Spin waves are collective excitations of the magnetization in magnetic materials.
The properties of spin waves are strongly affected by the geometry and the dominant
interactions inside the material. Hence, the relevant magnetic interactions will be
shortly introduced, followed by the derivation of the properties of spin waves in bulk
and thin film ferromagnets using the plane wave method.
12.2.1 Magnetic Interactions and Magnetization Dynamics
The magnetization dynamics in a ferromagnet can be described by the Landau–
Lifshitz–Gilbert (LLG) equation [18, 19]
dM
dt
= −γ μ 0 (M × H eff ) +
α
M s
M ×
dM
dt
(12.1)
with γ the absolute value of the gyromagnetic ratio (s
−1 T
−1 ), μ 0 the vacuum permeability (TmA
−1 ), α the Gilbert damping constant, M s the saturation magnetization
(Am
−1 ), and H eff the effective magnetic field (Am
−1 ). The first term in the LLG
equation describes the precession of the magnetization around the effective magnetic
field. The second term in the LLG equation leads to the damping of the magnetization
precession towards the direction of the effective magnetic field.
Multiple magnetic interactions and effects exist that influence the magnetization
dynamics such as the exchange interaction, dipolar interaction, magnetocrystalline
effect, magnetoelastic effect, etc.. It is possible to derive a magnetic field that corresponds to every interaction via
H = −
1
μ 0
δU (M)
δM
and U (M) =
V
E(M)dV
(12.2)
with E(M) the corresponding energy density of that interaction. The total effective
magnetic field H eff , which is governing the magnetization dynamics in the LLG
equation, is given by the sum of all individual magnetic fields, including externally
applied fields. Below, the dipolar and exchange interaction are explained in more
detail since these lead to spin waves. Fully elaborated discussions of spin waves and
their properties can be found in [16, 20, 21].
289
elastodynamic equations of motion are described. Finally, the fundamental properties
of magnetoelastic waves in thin films with finite thickness are derived and illustrated
both analytically and graphically for different magnetization orientations.
12.2 Spin Waves
Spin waves are collective excitations of the magnetization in magnetic materials.
The properties of spin waves are strongly affected by the geometry and the dominant
interactions inside the material. Hence, the relevant magnetic interactions will be
shortly introduced, followed by the derivation of the properties of spin waves in bulk
and thin film ferromagnets using the plane wave method.
12.2.1 Magnetic Interactions and Magnetization Dynamics
The magnetization dynamics in a ferromagnet can be described by the Landau–
Lifshitz–Gilbert (LLG) equation [18, 19]
dM
dt
= −γ μ 0 (M × H eff ) +
α
M s
M ×
dM
dt
(12.1)
with γ the absolute value of the gyromagnetic ratio (s
−1 T
−1 ), μ 0 the vacuum permeability (TmA
−1 ), α the Gilbert damping constant, M s the saturation magnetization
(Am
−1 ), and H eff the effective magnetic field (Am
−1 ). The first term in the LLG
equation describes the precession of the magnetization around the effective magnetic
field. The second term in the LLG equation leads to the damping of the magnetization
precession towards the direction of the effective magnetic field.
Multiple magnetic interactions and effects exist that influence the magnetization
dynamics such as the exchange interaction, dipolar interaction, magnetocrystalline
effect, magnetoelastic effect, etc.. It is possible to derive a magnetic field that corresponds to every interaction via
H = −
1
μ 0
δU (M)
δM
and U (M) =
V
E(M)dV
(12.2)
with E(M) the corresponding energy density of that interaction. The total effective
magnetic field H eff , which is governing the magnetization dynamics in the LLG
equation, is given by the sum of all individual magnetic fields, including externally
applied fields. Below, the dipolar and exchange interaction are explained in more
detail since these lead to spin waves. Fully elaborated discussions of spin waves and
their properties can be found in [16, 20, 21].