a polaron
/ POH-luh-ron /
In a simple metal an electron zips freely past the atoms. In an ionic crystal it cannot: an extra electron is a lump of negative charge, and it drags the nearby positive ions toward itself and pushes the negative ions away, denting the lattice around it. A polaron is that package — the electron (or hole) TOGETHER WITH the little bowl of lattice distortion it carries everywhere it goes. The carrier has, in effect, dug itself a pit and now must haul the pit along.
When the distortion is strong and tightly localized on essentially one site, it is a SMALL polaron, and this is common in transition-metal oxides where the carrier lives as a change in the valence of one metal ion. In Fe(1-x)O, for instance, a hole is simply an Fe3+ sitting among Fe2+; the carrier moves when the extra charge hops to a neighbouring iron, Fe2+ + Fe3+ --> Fe3+ + Fe2+, dragging its lattice dimple with it. Because each hop needs a thermal kick to rearrange the surrounding ions, small-polaron conduction is THERMALLY ACTIVATED: the mobility, and thus the conductivity, rises with temperature (mobility proportional to exp(-E_hop/kT)) — the exact opposite of a metal, whose conductivity falls as it heats.
The polaron is the honest picture of how charge really moves in most oxide ceramics, where carriers are heavy, slow, and localized rather than free and fast. It explains the low mobilities of oxide semiconductors, the hopping conduction in magnetite and in NTC thermistors (whose resistance drops steeply on heating), and the mixed conduction in fuel-cell electrode oxides. A common oversimplification to resist: treating oxide electrons as free-band carriers like those in silicon — in most ionic ceramics they are polarons, and that changes both the size and the temperature-behaviour of the conductivity.
An NTC thermistor is polaron conduction put to work: in a spinel oxide like manganese-cobalt oxide, charge hops between Mn3+ and Mn4+ on neighbouring sites. Heating the bead speeds the hops, so its resistance falls sharply and predictably with temperature — the basis of the temperature sensor in your car and thermostat.
A polaron is a carrier plus its self-made lattice dent; it moves by hopping, and hopping speeds up when you heat it.
Small-polaron conduction going UP with temperature is not a violation of anything — it is the signature that carriers are hopping over a barrier, not flowing freely as in a metal. The two mechanisms give opposite temperature trends.