I would like to know if anyone knows of a material where the conductivity drops when the current increases? Usually, it's the opposite.
This also occurs in traditional (filament) light bulbs.
The filament resides in an evacuated glass envelope, backfilled with a percentage of inert gas, to prevent oxidization and immediate failure. The filament is constructed of a fine tungsten (likely alloy) wire, wrapped into a microscopically-tight spiral, then this wrapped again into a slightly larger spiral. The double-wrapping greatly increases the effective length, and thus the total electrical resistance. The length is specifically chosen to be compatible with various standard supply voltages (120v, 240v etc.) The diameter of the filament is also responsible for the "power" of the lamp, with larger diameters making more light (and using more energy.)
The moment a light switch is thrown, the bulb appears as a low resistance (high conductance) as the filament is cold. This causes a large current to flow for a very short amount of time, which quickly heats the filament to red-hot. As the temperature increases, the filament becomes more resistive (less conductive) and the temperature-increase-rate slows. Eventually (perhaps 0.1s) later, this rate drops to zero, and the bulb is glowing white-hot in an equilibrium state.
Addition: I think most (if not all) pure metals exhibit this behavior (decreasing conductance / increasing resistance with increasing temperature.) Such is termed a "PTC" or positive temperature coefficient of resistance. As the atoms heat up, they are forced to expand, increasing the inter-atom distance, which makes them less electrically conductive. Tungsten is used for lamps due to it's extremely high melting point, which allows a wider range of operation, up to the white-hot point which few other materials can survive.
Semi-conductor materials (such as atomically doped silicon used in practically all computer systems today) tend to have NTC or negative temperature coefficients, meaning they become more conductive at higher temperatures. Thus "heat sinking" (keeping them cool) is needed for dense integrated circuits today.
This occurs in superconductors. In the simplest version (called type-I BCS superconductors) there is a resistanceless state at low temperature, magnetic field, and current density. When the current density is raised over a critical value, a transition into a state with finite resistance occurs. This was discovered by Kamerlingh Onnes in the 1910s in mercury. In other types of superconductors, the details may be more complex, but the central point of a resistanceless state disappearing at high current density remains.