Phenomenological Theory of Magnetoelastic Modes in Hexagonal Ferromagnets of Easy-Plane Type

Abstract

The magnetoelastic-mode theory of easy-plane hexagonal ferromagnets is approached from general principles of small oscillations around the equilibrium configuration. A general formula for normal-mode energies is obtained by diagonalizing the Hamiltonian, properly retaining terms quadratic in magnon-magnon, phonon-phonon, and magon-phonon operators. It is found that magnons and phonons are strongly mixed in the region of small $k$ for transverse waves propagating in certain directions parallel to the hexagonal plane when an external field is applied along the hexagonal hard direction and is close to $\frac{36K_h}{M}$ (hexagonal anisotropic field). On the other hand, the spin waves propagating along the $c$ axis are well approximated by the Turov-Shavrov frozen-lattice model because the extremely strong axial anisotropy makes the dynamical interaction (quadratic magnon-phonon terms) diminished. The dipolar effect is also discussed; and it is shown that, in the hard-direction case, magnons and phonons are mixed in relatively small amounts for the transverse waves propagating perpendicular to the magnetization, owing to the volume dipolar effect. Application of our theory to Tb is made with the use of numerical values of the temperature-dependent anisotropic and magneto-elastic coupling parameters. The lower-branch magnetoelastic modes may be responsible for the absorption of microwaves in the low-frequency-resonance experiments.