  On-chip cavity magnomechanics is an emerging field exploring acoustic and
magnonic functionalities of various ferromagnetic materials and structures
using strongly confined phonons. It is expected that such cavity magnomechanics
can be extended to multilayer ferromagnets, especially synthetic
antiferromagnets (SAFs) that exhibit zero net magnetization through interlayer
exchange coupling. However, the conventional theoretical framework for a single
ferromagnet cannot be used directly because of the antiferromagnetic
magnetization dynamics associated with the interlayer exchange coupling. In
this paper, we theoretically investigate phonon-magnon coupling with a
three-layer SAF. Our formulation of the phonon-magnon coupling constants
reveals that the acoustic (optical) magnon mode dominantly couples to the
cavity phonon when the magnetization angles in the two ferromagnetic layers are
antiparallel (orthogonal). Moreover, numerical calculations including the
effects of dipole-dipole interactions and in-plane uniaxial magnetic anisotropy
allow us to predict phonon frequency shifts and linewidth broadening that can
be detected in experiments. These theoretical insights would greatly help us to
make a strategy for bringing the system into the strong coupling regime and to
devise novel control protocols in analogy to cavity quantum electrodynamics and
cavity optomechanics.
