Non-volatile memory devices and arrays are described that facilitate the use of band-gap engineered gate stacks with asymmetric tunnel barriers in floating gate memory cells in NOR or NAND memory architectures that allow for direct tunneling programming and erase with electrons and holes, while maintaining high charge blocking barriers and deep carrier trapping sites for good charge retention. The direct tunneling program and erase capability reduces damage to the gate stack and the crystal lattice from high energy carriers, reducing write fatigue and leakage issues and enhancing device lifespan. Memory cells of the present invention also allow multiple bit storage in a single memory cell, and allow for programming and erase with reduced voltages. A positive voltage erase process via hole tunneling is also provided.