Spin-phonon dispersion in magnetic materials

Microscopic coupling between the electron spin and the lattice vibration is responsible for an array of exotic properties from morphic effects in simple magnets to magnetodielectric coupling in multiferroic spinels and hematites. Traditionally, a single spin-phonon coupling constant is used to characterize how effectively the lattice can affect the spin, but it is hardly enough to capture novel electromagnetic behaviors to the full extent. Here, we introduce a concept of spin-phonon dispersion to project the spin moment change along the phonon crystal momentum direction, so the entire spin change can be mapped out. Different from the phonon dispersion, the spin-phonon dispersion has both positive and negative frequency branches {even in the equilibrium ground state}, which correspond to the spin enhancement and spin reduction, respectively. Our study of bcc Fe and hcp Co reveals that the spin force matrix, that is, the second-order spatial derivative of spin moment, is similar to the vibrational force matrix, but its diagonal elements are smaller than the off-diagonal ones. This leads to the distinctive spin-phonon dispersion. The concept of spin-phonon dispersion expands the traditional Elliott-Yafet theory in nonmagnetic materials to the entire Brillouin zone in magnetic materials, thus opening the door to excited states in systems such as CoF$_2$ and NiO, where a strong spin-lattice coupling is detected in the THz regime.