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.

• Received 5 October 2011

DOI:https://doi.org/10.1103/PhysRevB.84.224429