Fully auxetic and multifunctional of two-dimensional δ -GeS and δ -GeSe

Auxetic materials, with negative Poisson's ratio (NPR), exhibit enhanced fracture toughness and shear resistance, due to their unusual mechanical behavior. Although a considerable number of two-dimensional (2D) NPR materials have been predicted theoretically, most of these materials have only in-plane or out-of-plane NPR. In this paper, using first-principles calculations, we found that monolayer $\ensuremath{\delta}$-GeS and $\ensuremath{\delta}$-GeSe show fully auxetic behaviors with both in-plane and out-of-plane NPRs at the same time. Meanwhile, monolayer $\ensuremath{\delta}$-GeSe has a direct bandgap and strong absorption in the blue and violet ranges. Owing to the large spontaneous polarization and strong dipole interaction, they can maintain the ferrielectric properties even at room temperature. Further, they have a considerable piezoelectric coefficient about one order of magnitude larger than recently studied 2D $\mathrm{MoS}_{2}$ and $h$-BN, due to small elastic stiffness constants. The rare fully auxeticity combined with various properties of monolayer $\ensuremath{\delta}$-GeS and $\ensuremath{\delta}$-GeSe could lead to novel multifunctional applications.