Why is the Fermi level a constant in thermal equilibrium?

[wantommy]

Why is the Fermi level a constant in thermal equilibrium?

just like PN diode (P=material 1 ; N=material 2)
the fermi levels are the same (Ef1=Ef2) in thermal equilibrium

i confused that
why there is no energy transfer
so that can judge each energy E will obey

rate from 1 to 2 ~ N1(E)f1(E)*N2(E)[1-f2(E)] ...(*)

rate from 2 to 1 ~ N2(E)f2(E)*N1(E)[1-f1(E)] ...(**)


does the electron hop from 1(Ea) to 2(Eb)
and another electron fell from 2(Ec) to 1(Ed)?

where Eb-Ea=Ec-Ed (←the energy differences are still the same, obey energy conservation)

so it is possible for electron to hop from material1's Ea to material2's Eb
it won't just transfer in the same energy

Why the book say "each energy" will obey eqs.(*) & (**)?

I think it should be

integral E1 and E2 {N1(E1)f1(E1)*N2(E2)[1-f2(E2)]}dE1 dE2 = integral {N2(E1)f2(E1)*N1(E2)[1-f1(E2)]}dE1 dE2

(it's possible to hop anywhere, just obey "rate from 1 to 2=rate from 2 to 1")
but i calculated it
i can't derive that Ef1=Ef2?

Is anything wrong?@@

Thanks for helping!:shy:

 

[eljose79]

The reason why the Fermi level is a constant in thermal equilibrium is due to the laws of thermodynamics and the principle of detailed balance. In thermal equilibrium, there is no net transfer of energy between the two materials, meaning that the rates of electron hopping from one material to the other and vice versa are equal. This is what is represented by equations (*) and (**).

In thermal equilibrium, the Fermi levels are equal because if they were not, there would be a net flow of electrons from the material with the higher Fermi level to the one with the lower Fermi level, violating the principle of detailed balance. This would result in a non-equilibrium state, which is not possible in thermal equilibrium.

As for your calculations, it is important to note that the Fermi level is not a single energy level, but rather a range of energies where the probability of finding an electron is 50%. Therefore, in your calculations, you should integrate over a range of energies rather than just considering a single energy level. This will give you the correct result of Ef1=Ef2.

I hope this helps clarify the concept of the Fermi level and its constancy in thermal equilibrium. Let me know if you have any further questions.