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.

The realization of multiferroics in nanostructures, combined with a large electric dipole and ferromagnetic ordering, could lead to new applications, such as high-density multistate data storage. Although multiferroics have been broadly studied for decades, ferromagnetic ferroelectricity is rarely explored, especially in two-dimensional (2D) systems. Here we report the discovery of 2D ferromagnetic ferroelectricity in layered transition-metal halide systems. On the basis of first-principles calculations, we reveal that a charged ${\mathrm{CrBr}}_3$ monolayer exhibits in-plane multiferroicity, which is ensured by the combination of orbital and charge ordering as realized by the asymmetric Jahn-Teller distortions of octahedral $\mathrm{Cr}─{\mathrm{Br}}_6$ units. As an example, we further show that ${({\mathrm{CrBr}}_3)}_2\mathrm{Li}$ is a ferromagnetic ferroelectric multiferroic. The explored phenomena and mechanism of multiferroics in this 2D system not only are useful for fundamental research in multiferroics but also enable a wide range of applications in nanodevices.

At B3LYP level of theory, we predict that the half-metallicity in zigzag edge graphene nanoribbon (ZGNR) can be realized when an external electric field is applied across the ribbon. The critical electric field decreases with the increase of the ribbon width to induce the half-metallicity. Both the spin polarization and half-metallicity are removed when the edge state electrons fully transferred from one side to the other under very strong electric field. The electric field range under which ZGNR remains half-metallic increases with the ribbon width. Our study demonstrates a rich field-induced spin polarization behavior, which may lead to some important applications in spinstronics.