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We say that under equilibrium, no further movement of majority carriers takes place due to the immobile ions or the electric field or built in potential stops it.

But in many cases and videos I seen, they say that diffusion current balances the drift current, resulting in zero current. Now according to me, the drift current in a p-n junction is due to minority carriers which are less, and then we should get a current because of it.

But many texts says the drift current direction is opposite to the diffusion current, but when majority charge carriers are not allowed to move further then how can one argue this that diffusion counters drift when diffusion in general is not happening only? There must be some minority current or am I missing something?

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You mixed up the terminologies in the bulk region with those in the junction. The terms majority carrier and minority carrier refer to the bulk region (far away from the junction region), for p-type bulk, the majority is hole and minority is electron, and the majority is electron minority is hole for n-type bulk. In the bulk region, the majority carrier is blocked by the building potential barrier in the depletion retion.

depletion region

In referring to the junction region, there are depletion regions, ideally no carriers at all. But in real situation (finite temperature case) the carriers are gradually changed from majority to minority across the junction as shown in the figure below.

Junction region

across the junction, there exists diffusion current due to the inhomogeneous distribution of carrier concentrations. Also, there is the drift current due to the built-in potential across the depletion regions. These two currents cancel each other, leaving the carriers in both bulk regions undisturbed.

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