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fluidistic

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In the textbook "Electronic transport in mesoscopic systems" by Datta, page 37 reads

but then completely diverges and never actually show why this statement holds. In fact, 1 or 2 pages below, it is shown that the difference in energy of the conduction electrons (those which produce the current) is worth ##2e|\vec E|L_m## where ##L_m## is the mean free path. In other words, the energy width around the Fermi level is a quantity that does not have any ##k_BT## term, and is proportional to the electric field's strength (which makes entirely sense to me). So what's going on here?! "It is very easy to see X" and then he went to prove "Y"? Am I missing something?Datta said:It is easy to see why the current flows entirely within a few ##k_BT## of the quasi-Fermi energy.

Now using intuition and the free electron model, I can understand why in the case of a thermal perturbation of order ##k_BT##, only electrons that aren't below ##E_F-k_BT## can interact and modify their energy with the field. It is due to Pauli exclusion principle which prevents electrons much below the surface to gain an energy of ##k_BT## because they would gain an energy leaving them into already occupied states, which is forbidden.

When I apply the exact same reasoning but instead of using a perturbation of order ##k_BT## (usually between ##10^{-5}## eV and ##10^{-3}## eV), I use a voltage of 1 V across a 1 cm long sample, so ##2e|\vec E|L_m## is about ##10^{-6}## eV, which is extremely small compared to ##E_F## (order of 1 eV), I get a width around ##E_F## that has a magnitude like Datta's calculations, namely ##e|\vec E|L_m## which has nothing to do with ##k_BT## and is, in this case, about 40 times smaller than raising the temperature from absolute 0 to 1 K.

I do not see where I go wrong. How does one make ##k_BT## appear regarding the electrons that create a current? And where do I go wrong?