A non-volatile memory device with a programmable leakage can be formed employing a trench capacitor. After formation of a deep trench, a metal-insulator-metal stack is formed on surfaces of the deep trench employing a dielectric material that develops leakage path filaments upon application of a programming bias voltage. A set of programming transistors and a leakage readout device can be formed to program, and to read, the state of the leakage level. The non-volatile memory device can be formed concurrently with formation of a dynamic random access memory (DRAM) device by forming a plurality of deep trenches, depositing a stack of an outer metal layer and a node dielectric layer, patterning the node dielectric layer to provide a first node dielectric for each non-volatile memory device that is thinner than a second node dielectric for each DRAM device, and forming an inner metal layer.