Non-volatile memory devices and arrays are described that utilize band engineered gate-stacks and multiple charge trapping layers allowing a multiple trapping site gate-insulator stack memory cell that utilizes a band engineered direct tunneling or crested barrier tunnel layer and charge blocking layer for high speed programming/erasure. Charge retention is enhanced by utilization of nano-crystals and/or bulk trapping materials in a composite non-conductive trapping layer and a high K dielectric insulating layers. The band-gap engineered gate-stack with asymmetric direct tunneling or crested barrier tunnel layers of the non-volatile memory cells of embodiments of the present invention allow for low voltage high speed tunneling programming and erase with electrons and holes, while maintaining high charge blocking barriers and deep carrier trapping sites for good charge retention. Memory cell embodiments of the present invention allow multiple levels of bit storage in a memory cell through multiple charge centroids and/or multiple threshold voltage levels.