Electron-hole gas compression

There is such a thermoelectric phenomenon: when you will heat one end of a conductor, semimetal or semiconductor, electrons will flow from a hot end to the cold end (like any gas when it expands) and presumably you will get higher density of electron gas on the cold end than it initially was.

If some material has 0 Hall coefficient and most likely the same mobility of electrons and holes then electrons and holes will move to the cold end to the same extent and therefore we will likely have larger density of electron-hole gas in comparison to its initial state. If I'm correct in my assumptions and there is such a phenomenon as electron-gas compressibility, is there any limit (in different materials) to which electron-hole gas could be compressed and which physical effects will be associated with the compression?