Would semi-conductor like properties result in something like this - thermal noise pushing electrons into the band gap?
In my head I was thinking of pure elements, like bulk metals. But there definitely could be more interesting things going on in the sample, like diode formation (internally or at the contacts) or weird doping profiles. That may be part of why it's hard to measure the properties!
This causes a decrease in resistance towards higher temperatures, so it has nothing to do with any explanation for a decrease in resistance towards lower temperatures.
The diodes have increasing inverse currents towards higher temperatures, which also has nothing to do with any explanation for increasing currents towards lower temperatures.
In metals, the concentration of free carriers is constant and the resistance decreases towards lower temperatures because the mobility of the free carriers increases, i.e. there is less friction between them and the crystal lattice, because the vibrations of the latter have a smaller amplitude. However this does not lead to any exponential decrease of the resistance.
The weird current dependence on voltage and temperature that can be seen here is most likely caused by an inhomogeneous sample that is composed of many small domains with different electrical properties.
Moreover, the material may be anisotropic, so extra variability is added by the random orientations of the microcrystals that compose the ceramic sample.
Logarithmic plots are a device of the devil. - Charles Richter
I'm sorry, but this result just feels to me like people are assuming that the material must be a superconductor and are analyzing all the data under that assumption rather than asking the question "is this a superconductor?"