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We propose non-volatile memory (NVM) designs based on Piezoelectric Strain FET (PeFET) utilizing a piezoelectric/ferroelectric (PE/FE such as PZT) coupled with 2D Transition Metal Dichalcogenide (2D-TMD such as MoS2) transistor. The proposed NVMs store bit information in the form of polarization (P) of the FE/PE, use electric-field driven P-switching for write and employ piezoelectricity induced dynamic bandgap modulation of 2D-TMD channel for bit sensing. We analyze PeFET with COMSOL based 3D modeling showing that the circuit-driven optimization of PeFET geometry is essential to achieve effective hammer-and-nail effect and adequate bandgap modulation for NVM read. Our results show that distinguishability of binary states to up to 11X is achieved in PeFETs.We propose various flavors of PeFET NVMs, namely (a) high density (HD) NVM featuring a compact access-transistor-less bit-cell, (b) 1T-1PeFET NVM with segmented architecture, targeted for optimized write energy and latency and (c) cross-coupled (CC) NVM offering a trade-off between area and latency.PeFET NVMs offer up to 7X smaller cell area, 66% lower write energy, 87% lower read energy and 44% faster read compared to 2D-FET SRAM. This comes at the cost of high write latency in PeFET NVMs, which can be minimized by virtue of optimized PE geometry.