Thermal dependence of PL measurements:quasi-fermi levels

[rukichi09]

Hello!Happy New Year!

I am currently working on the thermal dependence of photoluminescence measurements of Zinc oxide. however, I am investigating numerically. At thermal equilibrium the generation of carriers is equal to the recombination rate, away from thermal equilibrium- away from the respective band edges by kBT, the distribution of carriers is expressed by Boltzmann distribution and this is where quasi-fermi levels enter the picture. I would like to ask for help in understanding quasi-fermi levels and their relationship with temperature.

Also, quasi-Fermi levels are related to the electron and hole densities n and p,is it possible to express n and p as a function of T?

 

[snapback]

hello

Actually this concept is not my favourite one in physics, but I hope I can nevertheless help you a little to visualize this concept. I have mainly encountered the notion of quasi Fermi levels (or imrefs) , when dealing with semiconductor junction devices (pn-diode, schottky diode).

Imagine you have thermal equilibrium and an p-type semiconductor at a certain temperature where all acceptors are ionized (exhaustion). The concentrations of free electrons n0, free holes p0 and concentrations of ionized acceptors N_A⁺ adjust in a way that the condition of neutrality is satisfied. In this state we have n0 << p0 $$\approx$$N_A⁺ and the concetrations of electrons in CB or holes in VB are specified by a unique Fermi level Ef.

Now, assume that you disturb the equilibrium by exciting electrons from valence band and thus creating electrons and holes and that these additional electrons and holes can live for a quite a long time.

If the pair concentration is low compared to the hole (majority carrier) concentration p0, you see that the hole concentration is almost unchanged but the concentration of electrons changes very significantly.

Now you can approximate:

you say, oh well, the hole concentration has not changed, so the Fermi level for holes is like in thermal equilibrium but since I have many more free electrons than before, their concentration is governed by a different Fermi level (which is closer to conduction band than before). But ...then you have a different "Fermi level" for electrons and for holes... to remind that we are no longer dealing with thermal equilibrium the term "Quasi-Fermi-level" was introduced for this situation.

I wouldn't dare to speculate anything about temperature dependence of quasi-fermi levels, since, when you disturb thermal equilibrium, you need (stricly speaking ) to abandon the concept of a temperature.

If you library hat this book: https://www.amazon.com/dp/0780334795/?tag=pfamazon01-20, you should check chapter 9. This is a very nice book, especially if you want to calculate various things in semiconductors, despite it is devoted mainly to transport.

Good luck


PS: I'm curious what do you intent to calculate ? the dependence of PL spectrum with temperature ?

 

[rukichi09]

Yes, I calculated the PL spectra at thermal equilibrium using the Roosbroeck-Shockley relation.
Now, I'm trying to calculate PL at higher temperatures and observe what happens.

 

[snapback]

this calculation is really a formidable task.

please be aware that a lot of calculations (or formulas) which use the concept of quasi Fermi levels (an example formula for the non-degenerated case would be n=n_i*exp(($$\phi$$_n-$$\phi$$_i)/kT) (where n_i is the intrinsic electron concentration, $$\phi$$_n is the quasi Fermi level for electrons and $$\phi$$_i is the intrinsic Fermi level)) implicitly assume that the energetical distribution of electrons in the conduction band is Boltzmann-like. This might not necessarily be the case, when the electrons are excited optically like in PL measurements, especially if you have a semiconductor with short lifetimes. Be cautious when using them !

The subtleties of the concept of "quasi Fermi levels" are deeply discussed in section 4.1.2 of "Semiconductor statistics" by Blakemore, republished by Dover Publications Inc in 2002.

Good luck