Predicting the spin-lattice order of frustrated systems from first principles

A novel general method of describing the spin-lattice interactions in magnetic solids is proposed in terms of first-principles calculations. The spin exchange and Dzyaloshinskii-Moriya interactions, as well as their derivatives with respect to atomic displacements, can be evaluated efficiently on the basis of density-functional calculations for four ordered spin states. By taking into consideration the spin-spin interactions, the phonons, and the coupling between them, we show that the ground-state structure of a representative spin-frustrated spinel, MgCr${}_{2}$O${}_{4}$, is tetragonally distorted, in agreement with experiments. However, our calculations find the lowest energy for the collinear spin ground state, in contrast to previously suggested noncollinear models.