What's a typical electron-hole recombination time of semiconductors? Have some trouble finding a useful reference to answer the following questions:


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*What's a typical electron-hole recombination time of semiconductors? And how does it depend on the temperature and the band gap? If there's no formula, examples will do.

*What's the probability of an electron being kicked into the conduction band merely by thermal motion? Is it $\exp(-\Delta E/T)$, where $\Delta E$ is the band gap and $T$ the temperature?
That thermal excitation should lead to noise in the system, is that correct?
 A: Some observations:


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*in diodes, the recombination time is of order a nanosecond. It's as fast for similar materials with a high carrier density. The time will depend on the density but also other things and I am not aware of a simple universal formula. It's useful to know that the intra-band collisions that bring the bands into their well-known distributions are much faster and take as little as 0.1 picosecond. The inter-band processes' characteristic time may go to many microseconds or longer.

*yes and no, the exponential determines the ratio of carriers that want to sit on the energetically disfavored side divided by those on the dominant side. However, if you talk about the probability that a particular process occurs, you have to specify "in what time interval" you want this process to occur (and how many electrons you watch etc.) - because the result will surely depend on $dt$. It's about the "rate of the process". Moreover, in the equilibrium, recombination and generation are balanced.
More generally, it is not true that the recombination appears primarily due to thermal noise that kicks the carrier through the whole band. Quite on the contrary, recombination - and the opposite process of generation - of the carriers in the real world is dominated by the impurities which locally produce new states for the carriers in the middle of the gap, making their jump easier (because it may be divided).
A: The highest value of electron-hole recombination time I know of is around 1 millisecond, in very high quality single-crystal silicon. This material has very little defect-based (SRH) recombination (because of the high quality), and very little radiative recombination (because it has an indirect gap), and very little Auger recombination (if the doping is not too large). 1 millisecond is extremely unusual, a triumph of engineering. Nanoseconds would be more usual. Sometimes in high-speed devices, the semiconductor is purposefully engineered to have unusually low lifetime.
Recombination tends to occur faster at higher temperature, although I don't think it's a vast difference in most cases. It depends on the dominant recombination mechanism. Certainly, phonon-mediated recombination would occur appreciably faster at high temperatures. Likewise, the effect of changing the band gap depends on what the dominant recombination mechanism is, including factors like whether the band gap is direct or indirect, and where the impurity levels sit relative to the band edges, and how the band gap change alters carrier densities (in the case of Auger recombination).
If you attach a semiconductor to leads, it will have Johnson–Nyquist noise related to its resistance and temperature, just like anything else. The fact that the electron thermal agitation may involve jumping from the valence to the conduction band does not make any special difference, except insofar as it affects the overall resistance.
