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Why is Fermi level always constant in an equilibrium problem?

I find it difficult to visualize (or concretely understand) the fact that Fermi level should universally stay constant in an equilibrium between different materials. I have seen this so many times that it is now a natural part of my analysis when looking at semiconductors from a energy-band perspective, however, when I think of it, it occurs to me that why on earth do we have to line up the electrochemical potential even in materials that have different work functions such as N+ Poly – Oxide – N-type semiconductors (MOS Capacitors)?

Why do we have to start analysing by equating the Fermi levels so that we can justify that we need a flat-band voltage to flatten the bands?

Thank you for the replies.

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    Mark Lundstrom

    Think of the Fermi level like the water level of a lake. It tells us how the electronic states are filled up with electrons. If two different parts of the lake have different water level, water will flow until the wat level of the lake is uniform. It’s the same with electrons and the Fermi level.

    Another way of looking at this is to recall that the drift-diffusion current equation can equivalently be written as Jn = n mu_n (dFn/dx) where n is the electron density, mu_n is the mobility, and Fn is the quasi-Fermi level (like the Fermi level but defined out of equilibrium. In equilibrium, there is no current flow, so the quasi-Fermi level (which is the Fermi level in equilibrium) has to be spatially uniform. (In the previous analogy, we would say that the flow of water is driven by gradients in the Fermi level. In equilibrium there is no flow, so the Fermi level is constant across the lake.

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