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I find a difficult time understanding why does only the fermi level shifts in a pn junction under an external electric field? Why are the conduction and valence bands not affected by it?

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On applying external electric field, not only the fermi level but also the band edges shift in pn junctions. The barriers in band edges actually changes with bias. See the diagrams below where the initial barrier was the built-in potential (\$V_{bi}\$) at zero bias and it changes with the applied bias.

enter image description here

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