Why is the Fermi level a constant in thermal equilibrium?

  1. 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 wont 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 cant derive that Ef1=Ef2?

    Is anything wrong?@@

    Thanks for helping!!:shy:
     
  2. jcsd
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