In a novel nonvolatile memory cell formed above a substrate, a diode is paired with a reversible resistance-switching material, preferably a metal oxide or nitride such as, for example, NixOy, NbxOy, TixOy, HFxOy, AlxOy, MgxOy, CoxOy, CrxOy, VxOy, ZnxOy, ZrxOy, BxNy, and AlxNy. In preferred embodiments, the diode is formed as a vertical pillar disposed between conductors. Multiple memory levels can be stacked to form a monolithic three dimensional memory array. In some embodiments, the diode comprises germanium or a germanium alloy, which can be deposited and crystallized at relatively low temperatures, allowing use of aluminum or copper in the conductors. The memory cell of the present invention can be used as a rewriteable memory cell or a one-time-programmable memory cell, and can store two or more data states.