I should probably be fired for my prior answer because things like pn-junction diodes can definitely have an exponential-like dependence on temperature.
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!
Intrinsic semiconductors or the semiconductors that are lightly doped with impurities have temperature ranges where the concentration of the free charge carriers increases exponentially with increasing temperature, like in the NTC (negative temperature coefficient) thermistors.
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.
Here the resistance decreases in the opposite direction, so it cannot be explained by changes in the concentration of free carriers, like in semiconductors.
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.
