Coexistence of intrinsic piezoelectricity and ferromagnetism induced by small biaxial strain in septuple-atomic-layer $mathrm{VSi_2P_4}$

The septuple-atomic-layer $mathrm{VSi_2P_4}$ with the same structure of experimentally synthesized $mathrm{MoSi_2N_4}$ is predicted to be a spin-gapless semiconductor (SGS). In this work, the biaxial strain is applied to tune electronic properties of $mathrm{VSi_2P_4}$, and it spans a wide range of properties upon the increasing strain from ferromagnetic metal (FMM) to SGS to ferromagnetic semiconductor (FMS) to SGS to ferromagnetic half-metal (FMHM). Due to broken inversion symmetry, the coexistence of ferromagnetism and piezoelectricity can be achieved in FMS $mathrm{VSi_2P_4}$ with strain range of 0% to 4%. The calculated piezoelectric strain coefficients $d_{11}$ for 1%, 2% and 3% strains are 4.61 pm/V, 4.94 pm/V and 5.27 pm/V, respectively, which are greater than or close to a typical value of 5 pm/V for bulk piezoelectric materials. Finally, similar to $mathrm{VSi_2P_4}$, the coexistence of piezoelectricity and ferromagnetism can be realized by strain in the $mathrm{VSi_2N_4}$ monolayer. Our works show that $mathrm{VSi_2P_4}$ in FMS phase with intrinsic piezoelectric properties can have potential applications in spin electronic devices.