I would expect such a material as you linked to have a momentum imbalance in the x direction, which leads to mass transport in the x direction, until a new equilibrium is achieved.

That makes sense. So it electrons will flow in one direction until there is a static electric field that create a slope in the fermi level. Perohaps https://en.wikipedia.org/wiki/Electret

It says that quartz is an electret, but I can't find the band structrure.