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F. Vanderveken et al.
mous attention, which has led to the introduction of commercial magnetic memory
technologies. However, the efficient energy conversion from the electric to the magnetic domain and vice versa still remains challenging. Current magnetic memories
are based on spin-transfer or spin-orbit torques to switch the magnetization [1, 2];
however, these mechanisms depend on the current density in the device and require
typically energies on the order of (10 s of) femtojoules to reverse the magnetization of
a nanomagnet, despite a much lower intrinsic energy barrier of the order of attojoules.
While femtojoule switching energies are promising for nonvolatile memories, they
are not competitive in other spintronic applications, such as spintronic logic circuits.
Therefore, much research has been devoted to developing more efficient transducers
between electric and magnetic subsystems of spintronic devices.
One of the most promising devices to efficiently couple electric and magnetic
properties in a spintronic system is the magnetoelectric transducer. Magnetoelectric
transducers consist of composite materials, which comprise piezoelectric and magnetostrictive layers [3–7]. Applying a voltage to the piezoelectric layer(s) leads to
the formation of strain in the compound. This strain introduces an effective magnetic
anisotropy field in the magnetostrictive ferromagnetic component, leading to an coupling between voltage (electric field) and magnetization. The coupling is bidirectional
since rotating the magnetization of a magnetostrictive layer also induces strain in the
compound and consequently a polarization in the piezoelectric. Hence, such magnetoelectric schemes provide indirect coupling between electricity and magnetism
mediated by elastodynamics. Since generating large electric fields in the piezoelectric layers can be energy efficiency, capacitive magnetoelectric transducers promise
a much higher energy efficiency than their current-based counterparts.
The coupling scheme of a magnetoelectric transducer can be split into two parts,
(i) piezoelectric coupling between electric and elastic domains, and (ii) magnetostrictive coupling between elastic and magnetic domains. Here, we investigate the second
part and focus especially on the behavior at GHz frequencies that are relevant for fast
electronic devices. In this frequency range, elastic waves (hypersound) interact with
magnetic waves (spin waves), forming hybrid magnetoelastic waves under resonant
conditions. The physics of the magnetoelastic resonance and the resulting magnetoelastic waves have been described in bulk and infinitesimally thin films decades
ago [8–17]. However, modern applications of magnetoelastic waves in magnetoelectric and spintronic devices are based on nm-thick films. The finite thickness of
these magnetic films alters the dynamic dipolar field with respect to infinitesimally
thin films. Consequently, also the magnetoelastic coupling and the behavior of the
magnetoelastic waves change due to the finite film thickness. In this chapter, the
magnetoelastic theory and equations are extended to describe magnetoelastic waves
in thin films of finite thickness.
The chapter begins by introducing basic magnetic interactions and by reviewing the properties of spin waves in bulk and thin film ferromagnets. The waves are
described by a general formalism to calculate the eigensystem. The effect of the
finite film thickness is then incorporated in this formalism. In the second part, linear
elasticity and elastic waves in thin films are discussed. In the third part, the magnetoelastic interactions together with the combination of the magnetodynamic and
F. Vanderveken et al.
mous attention, which has led to the introduction of commercial magnetic memory
technologies. However, the efficient energy conversion from the electric to the magnetic domain and vice versa still remains challenging. Current magnetic memories
are based on spin-transfer or spin-orbit torques to switch the magnetization [1, 2];
however, these mechanisms depend on the current density in the device and require
typically energies on the order of (10 s of) femtojoules to reverse the magnetization of
a nanomagnet, despite a much lower intrinsic energy barrier of the order of attojoules.
While femtojoule switching energies are promising for nonvolatile memories, they
are not competitive in other spintronic applications, such as spintronic logic circuits.
Therefore, much research has been devoted to developing more efficient transducers
between electric and magnetic subsystems of spintronic devices.
One of the most promising devices to efficiently couple electric and magnetic
properties in a spintronic system is the magnetoelectric transducer. Magnetoelectric
transducers consist of composite materials, which comprise piezoelectric and magnetostrictive layers [3–7]. Applying a voltage to the piezoelectric layer(s) leads to
the formation of strain in the compound. This strain introduces an effective magnetic
anisotropy field in the magnetostrictive ferromagnetic component, leading to an coupling between voltage (electric field) and magnetization. The coupling is bidirectional
since rotating the magnetization of a magnetostrictive layer also induces strain in the
compound and consequently a polarization in the piezoelectric. Hence, such magnetoelectric schemes provide indirect coupling between electricity and magnetism
mediated by elastodynamics. Since generating large electric fields in the piezoelectric layers can be energy efficiency, capacitive magnetoelectric transducers promise
a much higher energy efficiency than their current-based counterparts.
The coupling scheme of a magnetoelectric transducer can be split into two parts,
(i) piezoelectric coupling between electric and elastic domains, and (ii) magnetostrictive coupling between elastic and magnetic domains. Here, we investigate the second
part and focus especially on the behavior at GHz frequencies that are relevant for fast
electronic devices. In this frequency range, elastic waves (hypersound) interact with
magnetic waves (spin waves), forming hybrid magnetoelastic waves under resonant
conditions. The physics of the magnetoelastic resonance and the resulting magnetoelastic waves have been described in bulk and infinitesimally thin films decades
ago [8–17]. However, modern applications of magnetoelastic waves in magnetoelectric and spintronic devices are based on nm-thick films. The finite thickness of
these magnetic films alters the dynamic dipolar field with respect to infinitesimally
thin films. Consequently, also the magnetoelastic coupling and the behavior of the
magnetoelastic waves change due to the finite film thickness. In this chapter, the
magnetoelastic theory and equations are extended to describe magnetoelastic waves
in thin films of finite thickness.
The chapter begins by introducing basic magnetic interactions and by reviewing the properties of spin waves in bulk and thin film ferromagnets. The waves are
described by a general formalism to calculate the eigensystem. The effect of the
finite film thickness is then incorporated in this formalism. In the second part, linear
elasticity and elastic waves in thin films are discussed. In the third part, the magnetoelastic interactions together with the combination of the magnetodynamic and