doping concentration,temperature are the major source of reverse current
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There are two diode ratings. One is the forward current rating, and the other is the reverse breakdown voltage rating. The forward current rating is the maximum current that the diode can conduct before failing, and the reverse breakdown voltage rating is the maximum reverse voltage that can be applied before failing. Of course, the reverse voltage rating has a different meaning in a zener diode, but this answer applies to ordinary diodes.
A zener diode, in the reverse bias condition, presents a higher voltage. The current, however, is a function of the supplying circuit.
by reverse breakdown voltage & forward current capacity
What is the cause of reverse recovery time in a pn junction diode
The IN5408 diode is an ordinary silicon diode. It has a 3 amp forward current rating, and a 1000 volt peak reverse voltage rating.