Small strain induced large piezoelectric coefficient in $\alpha$-AsP monolayer

Strain engineering can effectively tune the electronic, topological and piezoelectric properties of materials. In this work, the small strain (-4\% to 4\%) effects on piezoelectric properties of $\alpha$-AsP monolayer are studied by density functional theory (DFT). The piezoelectric stress tensors $e_{ij}$ and elastic stiffness tensors $C_{ij}$ are reported by using density functional perturbation theory (DFPT) and finite difference method (FDM). It is found that the Young's modulus of $\alpha$-AsP monolayer shows very strong anisotropy, and the armchair direction is very softer than zigzag direction, which provides possibility for tuning easily piezoelectric coefficients along the armchair direction. In considered strain range, uniaxial compressive (tensile) strain along the armchair (zigzag) direction is found to raise observably both the $e_{22}$ and $d_{22}$ (absolute value). In fact, both compressive strain along the armchair direction and tensile strain along the zigzag direction essentially reduce the lattice constants along the armchair direction, which can enhance the piezoelectric coefficients. The $e_{ij}$ of $\beta$-AsP monolayer as a function of strain is also studied to illustrate the importance of particular puckered structure of $\alpha$-AsP in enhancing the piezoelectric coefficients. A classic SnSe monolayer with puckered structure is used to further declare that small strain along the armchair direction can effectively improve the piezoelectric coefficients. For example, the $d_{22}$ of SnSe monolayer at -3.5\% strain is up to 628.8 pm/V from unstrained 175.3 pm/V. For SnSe monolayer, a large peak is observed for $e_{22}$, which is due to a structural phase transition. For $e_{16}$ of SnSe monolayer, a large peak is also observed due to the cross of lattice constants $a$ along the zigzag direction and $b$ along the armchair direction.