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Superconducting qubit circuits require high quality low critical current density Josephson tunnel junctions. We have developed high quality epitaxial NbN/AlN/NbN tunnel junctions with critical current density, Jc, ranging from as low as a few A/cm2 to a few tens of kA/cm2. In this work, we measured current-voltage characteristics for junctions with a Jc ~ 32 A/cm2 in the temperature range from 17 mK to above 6.7 K. All the junctions are in good quality and have a high gap voltage (>; 5.5 mV). We found that the junctions' sub-gap resistances were temperature dependent above 2.5 K and saturated at temperatures below 2.5 K. A junction sub-gap leakage current Ir of about 23 pA was measured. The voltage noise power spectral density for 3 junctions with different sizes has been measured at the base temperature of 17 mK with the junctions biased above the gap voltage. The spectral densities clearly show 1/f behavior at low frequency and are proportional to the junction's linear dimension.