Temperature impact on laser emission

[Q.B.]

Hi everyone,

I'm trying to understand why laser diodes need a stronger input current to start lasing when their temperature increases.

If I tried to add thermal transitions to the rate equations governing the evolutions of an atom quantum level populations (let's say $n_{up}$ and $n_{down}$ for a two-level problem) in a laser amplification medium, I would write:
$$\frac{dn_{up}}{dt}= partwithstimulatedemission, absorption, andpumping + (somefactor)(n_{down}e^{(E_{down}-E_{up})/kT} - n_{up}e^{(E_{up}-E_{down})/kT})$$
$$\frac{dn_{down}}{dt}= partwithstimulatedemission, absorption, andpumping - (somefactor)(n_{down}e^{(E_{down}-E_{up})/kT} - n_{up}e^{(E_{up}-E_{down})/kT})$$
Which seems to make population inversion easier when temperature increases.

However, non-radiative transitions (with phonons for instance) might also be supported by the temperature increase. Are these non-radiative processes in the end taking over the thermal fluctuations which help population inversion, and explain the phenomenon observed in diodes?

Thanks in advance!

 

[Henryk]

Q.B.
The Boltzmann factor $exp( - \frac {E}{k_bT} )$ applies only for a system at equilibrium. At equilibrium, the probability of population of the higher energy level will always be lower than that for a lower energy level. To get laser operating you need to create a population inversion, that is more electrons in higher energy levels than lower energy levels and the Boltzmann factor is simply not applicable in a non-equilibrium case.

In laser diodes, the population inversion is achieved by injection using a p-n junction. The number of electrons in the upper state depends on the injection current minus recombination. At higher temperature, the recombination rate increases and that's why you need higher current to achieve the population inversion.