Diode current components
See the diagram below which shows the three regions in a pn-junction: two neutral regions and a depletion region.
(fig taken from wikipedia)

The fact that the junction is abrupt does not mean that there is no depletion region or voltage drop across the junction. The diagram above actually represents that of an abrupt junction.
For simplicity, it is usually assumed that the entire voltage applied across the diode appears across the depletion region and hence the drop in the neutral region is zero. Which in turn means that the electric field is zero in neutral region. There exists a non-zero field in depletion region.

The exponential decay of carriers actually happens outside depletion region, where the electric field is zero. And hence it can be assumed that the carrier concentration decay/grow exponentially with distance inside these quasi-neutral regions. This is true for both forward and reverse bias. The derivation of current equation for a diode found in textbook usually is done for some applied bias, can be forward or reverse.