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We establish the sign of the linear magnetoelectric (ME) coefficient, $\alpha$, in chromia, Cr$_2$O$_3$. Cr$_2$O$_3$ is the prototypical linear ME material, in which an electric (magnetic) field induces a linearly proportional magnetization (polarization), and a single magnetic domain can be selected by annealing in combined magnetic (H) and electric (E) fields. Opposite antiferromagnetic domains have opposite ME responses, and which antiferromagnetic domain corresponds to which sign of response has previously been unclear. We use density functional theory (DFT) to calculate the magnetic response of a single antiferromagnetic domain of Cr$_2$O$_3$ to an applied in-plane electric field at 0 K. We find that the domain with nearest neighbor magnetic moments oriented away from (towards) each other has a negative (positive) in-plane ME coefficient, $\alpha_{\perp}$, at 0 K. We show that this sign is consistent with all other DFT calculations in the literature that specified the domain orientation, independent of the choice of DFT code or functional, the method used to apply the field, and whether the direct (magnetic field) or inverse (electric field) ME response was calculated. Next, we reanalyze our previously published spherical neutron polarimetry data to determine the antiferromagnetic domain produced by annealing in combined E and H fields oriented along the crystallographic symmetry axis at room temperature. We find that the antiferromagnetic domain with nearest-neighbor magnetic moments oriented away from (towards) each other is produced by annealing in (anti-)parallel E and H fields, corresponding to a positive (negative) axial ME coefficient, $\alpha_{\parallel}$, at room temperature. Since $\alpha_{\perp}$ at 0 K and $\alpha_{\parallel}$ at room temperature are known to be of opposite sign, our computational and experimental results are consistent., Comment: 11 pages, 5 figures

Multiferroics are a unique class of materials where magnetic and ferroelectric orders coexist. The research on multiferroics contributes significantly to the fundamental understanding of the strong correlations between different material degrees of freedom and provides an energy-efficient route toward the electrical control of magnetism. While multiple ABO3 oxide perovskites have been identified as being multiferroic, their magnetoelectric coupling strength is often weak, necessitating the material search in different compounds. Here, we report the observation of room-temperature multiferroic orders in multi-anion SrNbO3-xNx thin films. In these samples, the multi-anion state enables the room-temperature ferromagnetic ordering of the Nb d-electrons. Simultaneously, we find ferroelectric responses that originate from the structural symmetry breaking associated with both the off-center displacements of Nb and the geometric displacements of Sr, depending on the relative O-N arrangements within the Nb-centered octahedra. Our findings not only diversify the available multiferroic material pool but also demonstrate a new multiferroism design strategy via multi-anion engineering.

In this work we present a new method for the calculation of the electrostrictive properties of materials using density functional theory. The method relies on the thermodynamical equivalence, in a dielectric, of the quadratic mechanical responses (stress or strain) to applied electric stimulus (electric or polarisation fields) to the strain or stress dependence of its dielectric susceptibility or stiffness tensors. Comparing with current finite-field methodologies for the calculation of electrostriction, we demonstrate that our presented methodology offers significant advantages of efficiency, robustness, and ease of use. These advantages render tractable the highthroughput theoretical investigation into the largely unknown electrostrictive properties of materials.