Consider applying an external voltage to the junction as shown below, with the positive terminal connected to the P type material and the negative terminal connected to the N type material. The negative terminal will inject electrons into the N type material. This will replenish those "lost" due to migration across the junction. Similarly the positive terminal will remove electrons from the P type material, creating more holes, to replenish those lost by recombination. If the applied voltage exceeds the barrier voltage, it will completely replenish the depletion layer. Once this occurs, electron migration across the junction will resume. However the depletion layer cannot reform, as the battery is continually replenishing the majority charge carriers, as they are lost due to electron hole recombination at the junction.
The net result, is that when the external voltage is connected this way, it reduces the barrier voltage and if the applied voltage is greater than the barrier voltage, it will produce current flow through the device. (i.e. the junction is forward biased.)
The current flow is of course due to electron flow through the circuit. The free electrons travel away from the negative terminal of the battery. They flow through the conduction band in the N type semiconductor. When they cross into the P type semiconductor, they drop down into holes in the valence band and then move through the valence band, towards the positive terminal.
However we find it more useful to think of the movement of holes rather than valence electrons in P type semiconductor, (not least because the
effective number density of mobile charge carriers in P type semiconductor, corresponds directly to the number of holes in the valence band). Therefore
a more conventional description of current flow through the PN junction is;
Electrons are injected in the N type semiconductor by the negative terminal of the battery and the free electrons flow towards the junction. The positive terminal injects holes in the P type material, which also flow towards the junction. At the junction the electrons and holes recombine.
When reviewing current flow, we highlighted the importance of number density in determining current flow through conductors and semiconductors. Imagine that the doping of the N type material is much greater than the P type, hence the density of free electrons is much greater than the density of holes. We cannot have a continuous flow, with more electrons travelling towards the junction, than there are holes. (If this were initially the case, then after recombination at the junction, we would be left with a surplus of electrons. This would quickly accumulate and build up negative charge, which would then oppose the electron flow, until it was equal to the flow of holes in the P type semiconductor).
Therefore it is the material with the lowest doping level, that sets the upper limit, for the series current through the P and N junction, when a voltage is applied.