It seems like all the explanations for diode mechanics simply state that if N-type semiconductor is connected to V+ of the battery, the depletion region grows so current cannot pass.
It seems, however, that the few free electrons (minority carriers) on the P-type semiconductor can move across the layer and vice versa for holes on the N-type semiconductor. This creates the reverse saturation current. This is logical since the electrons move to high potential and holes move to low potential. However, why can't we produce a proper current from this? Electrons emitted from the battery can flow through the P-type semiconductor and across the depletion layer.
Is this main problem that there just aren't enough free electrons on the P-type side to make this happen?