Metal-Nb$_{2}$O$_{5-x}$-metal memdiodes exhibiting rectification, hysteresis, and capacitance are demonstrated for applications in neuromorphic circuitry. These devices do not require any post-fabrication treatments such as filament creation by electroforming that would impede circuit scalability. Instead these devices operate due to Poole-Frenkel defect controlled transport where the high defect density is inherent to the Nb$_{2}$O$_{5-x}$ deposition rather than post-fabrication treatments. Temperature dependent measurements reveal that the dominant trap energy is 0.22 eV suggesting it results from the oxygen deficiencies in the amorphous Nb$_{2}$O$_{5-x}$. Rectification occurs due to a transition from thermionic emission to tunneling current and is present even in thick devices (> 100 nm) due to charge trapping which controls the tunneling distance. The turn-on voltage is linearly proportional to the Schottky barrier height and, in contrast to traditional metal-insulator-metal diodes, is logarithmically proportional to the device thickness. Hysteresis in the I-V curve occurs due to the current limited filling of traps.
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