Magnetic-induced phonon anisotropy in ZnCr$_2$O$_4$ from first principles

We have studied the influence of magnetic order on the optical phonons of the geometrically frustrated spinel ZnCr$_2$O$_4$ from first-principles. By mapping the first-principles phonon calculations onto a Heisenberg-like model, we developed a method to calculate exchange derivatives and subsequently the spin-phonon couping parameter from first-principles. All calculations were performed within LSDA+U.