Some shallow-impurity atoms capture free carriers by emitting acoustical phonons. This process requires a ladder of closely-spaced energy levels connecting an energy-level associated with this impurity to a free-carrier band. The rate of cascade capture decreases with sample temperature. So the lifetimes of free carriers captured by this impurity increases with sample temperature. The acoustical phonons carry away energy and momentum that was originally in the free carrier.

Some studies where an increase of free-carrier lifetime with temperature have rejected the possibility of cascade capture because the temperature is too high. The direct capture of free carriers is considered more likely in these studies. I don't understand this. Direct capture involves tunnelling. The energy may be carried away by local vibrational states of the impurity atom. The theory is a bit unclear at this point. However, acoustical phonons are not involved.

Even if the rate decreases with temperature, there should be a non-zero rate of cascade capture even at very high temperatures. Furthermore, the same argument should apply to direct capture. The rate of direct capture also decreases with sample temperature. Therefore, there should be no direct capture at high temperatures.

A related issue is the following. Cascade capture seems to be always associated with traps rather than recombination centres. It seems to me that cascade capture of free carriers should sometimes be observed in recombination centres. So why aren't recombination centres considered which emit acoustical phonons?