Structure effect on intrinsic piezoelectricity in septuple-atomic-layer $mathrm{MSi_2N_4}$ (M=Mo and W)

The recently experimentally synthesized monolayer $mathrm{MoSi_2N_4}$ and $mathrm{WSi_2N_4}$ (textcolor[rgb]{0.00,0.00,1.00}{Science 369, 670-674 (2020})) lack inversion symmetry, which allows them to become piezoelectric. In this work, based on ab initio calculations, we report structure effect on intrinsic piezoelectricity in septuple-atomic-layer $mathrm{MSi_2N_4}$ (M=Mo and W), and six structures ($alpha_i$ ($i$=1 to 6)) are considered with the same space group.It is found that $mathrm{MSi_2N_4}$ (M=Mo and W) with $alpha_i$ ($i$=1 to 6) all are indirect band gap semiconductors. Calculated results show that $mathrm{MoSi_2N_4}$ and $mathrm{WSi_2N_4}$ monolayers have the same structural dependence on piezoelectric strain and stress coefficients ($d_{11}$ and $e_{11}$), together with the ionic and electronic contributions to $e_{11}$.Finally, we investigate the intrinsic piezoelectricity of monolayer $mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) with $alpha_1$ and $alpha_2$ phases expect $mathrm{CrGe_2N_4}$, because they all are semiconductors and their enthalpies of formation between $alpha_1$ and $alpha_2$ phases are very close. The most important result is that monolayer $mathrm{MA_2Z_4}$ containing P atom have more stronger piezoelectric polarization than one including N atom. The largest $d_{11}$ among $mathrm{MA_2N_4}$ materials is 1.85 pm/V, which is close to the smallest $d_{11}$ of 1.65 pm/V in $mathrm{MA_2P_4}$ monolayers. For $mathrm{MA_2P_4}$, the largest $d_{11}$ is up to 6.12 pm/V. Among the 22 monolayers, $alpha_1$-$mathrm{CrSi_2P_4}$, $alpha_1$-$mathrm{MoSi_2P_4}$, $alpha_1$-$mathrm{CrGe_2P_4}$, $alpha_1$-$mathrm{MoGe_2P_4}$ and $alpha_2$-$mathrm{CrGe_2P_4}$ have large $d_{11}$, which are greater than or close to 5 pm/V, a typical value for bulk piezoelectric materials.