A memory array is disclosed having bipolar current-voltage (IV) resistive random access memory cells with built-in “on” state rectifying IV characteristics. In one embodiment, a bipolar switching resistive random access memory cell may have a metal/solid electrolyte/semiconductor stack that forms a Schottky diode when switched to the “on” state. In another embodiment, a bipolar switching resistive random access memory cell may have a metal/solid electrolyte/tunnel barrier/electrode stack that forms a metal-insulator-metal device when switched to the “on” state. Methods of operating the memory array are also disclosed.