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I understand that the Fermi level at the p-n junction must be the same between the materials forming the junction. To achieve this, the band energies of the n- and p-type materials are bent.

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However, far away from the p-n junction, the band diagram (incl. the Fermi level) should be the same as for isolated materials. This would imply that the Fermi level is higher (lower) away from the junction than at the junction. Does that mean that the Fermi level can vary within the same material?

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No Fermi level variation inside a piece of N-type or P-type material. The contact creates a voltage difference between the joined N- and P-blocks and that voltage affects to both pieces by moving the Fermi levels to the same equilibrium value. Essentially the difference of the Fermi levels just causes the contact phenomenons and the balance is reached when the Fermi levels are equal in both pieces.

Think it as you had a 3 V battery. Let the minus pole of it be connected to the ground. Connect a conductive wire to the plus pole. Do not let the wire touch anything else. The whole wire would have +3 volts when compared to the ground, no voltage reduction at the distant end of the wire - assuming the possible transient just after connecting the wire has died away and there's no voltage drop caused by a continuous DC current and resistance.

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