Atypical breathing driven two-dimensional valley multiferroicity

Valley multiferroicity, coupled with ferro-valleytricity and primary ferroicities in a single phase, is of fundamental significance in condensed-matter physics and materials science, as it provides a convenient route to reverse the anomalous valley Hall (AVH) effect. Current research in this field focuses mainly on ferromagnetic ferro-valleytricity, whereas ferroelectric ferro-valleytricity is seldom explored. Here, using symmetry arguments and tight-binding model analysis, we report a novel mechanism of coupling ferro-valleytricity with ferroelectricity, i.e., single-phase valley multiferroicity, in a two-dimensional magnetic lattice. This mechanism correlates to the atypical breathing nature of the magnetic lattice. Importantly, the valley physics, associated with Berry curvature, can be reversed under a ferroelectric transition, thereby guaranteeing the ferroelectrically reversible AVH effect. The underlying physics are discussed in detail. Based on first-principles calculations, we further confirm valley multiferroicity in a real 2D magnetic material of single-layer Gd₂CO₂. The explored phenomena and mechanism are not only useful for fundamental research in valley multiferroics but also enable a wide range of applications in nanodevices.