Phonon-induced renormalization of exchange interactions in metallic two-dimensional magnets

The presence of spin-polarized charge carriers in metallic magnets provides a mechanism for spin-lattice interactions mediated by electron-phonon coupling. Here, we present a theory of this mechanism used to estimate its effect on the exchange interactions in two-dimensional (2D) magnets. Starting from a square lattice model at half filling, we show that the presence of electron-phonon coupling with equilibrium phonon distribution leads to a notable suppression of exchange interactions with temperature. We then apply our approach to the prototypical 2D metallic ferromagnet, ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$, with moderate electron-phonon coupling. We find that the exchange interactions undergo a renormalization, leading to a softening of the magnon modes, and suppression of the Curie temperature by $\sim 10$%. We expect that this effect can be further enhanced in systems with strong electron-phonon coupling, as well as for the nonequilibrium distribution of phonons induced by strong laser fields or charge currents.