Piezoelectricity Enhancement and Band Structure Modification of Single Atomic Shift in MoS$_{\mathrm{2}}$ Supercell Monolayer

A monolayer of transition metal dichalcolgenides (TMDCs, TM: Mo, W, DC: S, Se, Te) has second-order nonlinearity including piezoelectricity responding to an external field due to spatial inversion asymmetry. The intrinsic piezoelectric coefficient (e$_{\mathrm{11}})$ of MoS$_{\mathrm{2}}$ without any atomic shift has \textasciitilde 298 pC/m, where e$_{\mathrm{11}}$ indicates the sum of ionic and electronic polarizations along the armchair direction responding to the uniaxial atomic shift along the armchair direction. The piezoelectric coefficients (e$_{\mathrm{11}})$ of MoS$_{\mathrm{2}}$ supercell with a single atomic shift of Mo- and S-ion positively (20{\%}) along the armchair direction were increased to \textasciitilde 350 pC/m and \textasciitilde 305 pC/m, respectively. Meanwhile, the piezoelectric coefficients (e$_{\mathrm{11}})$ of MoS$_{\mathrm{2}}$ supercell with a single atomic shift of Mo- and S-ion positively (20{\%}) along the zigzag direction have \textasciitilde 330 pC/m. The bandgap energy at the K point in the first Brillion zone of a single atomic shift either Mo- and S-ions positively (20{\%}) along the armchair direction in the MoS$_{\mathrm{2}}$ atomic cell is largely reduced to \textasciitilde 0.06 eV compared to the intrinsic bandgap (1.96 eV) of MoS$_{\mathrm{2}}$ without atomic stain. The large piezoelectricity enhancement and bandgap modification due to a single atomic shift in TMDCs may open astonishing scientific research and applications including quantum information processing and optomechanics in the pico-scale atomic layer.