Non-radiative recombination vs. radiative recombination It seems as a common knowledge, but I cannot understand:
Why does non-radiative decay (electron-hole recombination) in direct-gap semiconductors occur faster than radiative decay?
 A: I think there is some problem in understanding. In direct band gap materials most of the radiation is through radiative recombination. 
The process of radiative recombination is slow in the indirect band gap materials compared to the radiative recombination in direct band gap materials. 
Also in the case of indirect band gap materials the radiative recombination is slower than non radiative recombination. In indirect band gap materials the de-excitation is mainly through non radiative process. 
The non radiative decay is a result of the de-excitation of electron from conduction band to valance band in the presence of a phonon. The excessive energy and momentum is then carried out by the phonon. 
As you know that when the electron is excited from the valance band to the conduction band in indirect band gap materials, first the electrons tend to come to ground state of the conduction band. The conduction band minima is not matching with the top state of the valance band in $\omega-k$ space. Since all the vacancies in the valance band are in the top state, the transition from the conduction band to valance band becomes difficult/forbidden as momentum can not remain conserved i.e. the lifetime of the excited state increases quite a bit. Same can be seen in this Wikipedia link.
Involvement of the phonon helps in conserving the momentum and results in faster non-radiative transition in indirect band gap materials.
I hope this will help
