Zener diode equation proof (chacteristic curve)

I'm trying to prove this equation(at least I think that's the correct one):

which produces this graph:

I can prove the one that describes the PN diode but I am not sure what I should do differenly here. Any help would be appreciated!

Note: If anyone can point me somewhere I can find this proof I would be grateful as well.

• en.wikipedia.org/wiki/Shockley_diode_equation – Sunnyskyguy EE75 Jan 4 at 17:11
• What do you mean by "prove", exactly? This is not an equation of some mathematical theory. It is derived from the physical model used to describe the workings of a PN diode and it has been verified experimentally (up to a point-it doesn't describe the breakdown region). – Lorenzo Donati Jan 4 at 17:15
• Are you aware that the equation you posted (Shockley's equation) is the equation that (almost) corresponds to the graph you posted? And that is the equation you should have derived as the "PN junction equation". The difference lies in the fact that that equation, as I already said, doesn't explain the brakdown region (i.e. the region for V<Vz, where the reverse current increases dramatically). – Lorenzo Donati Jan 4 at 17:26
• Read also the Wikipedia article on Zener diodes. – Lorenzo Donati Jan 4 at 17:38
• You can model the zener as two imperfect (with ideality != 1, and series resistance and parallel capacitance) Shockley diodes in inverse parallel, one with a voltage source in series. That will produce the graph you show. There are more complex models. – Spehro Pefhany Jan 4 at 21:53