Tunnel diode VI equation
Try looking for eg


  tunnel diode transfer function


You are unlikely to get a single tidy expression due to the non linear nature of the beast and its "discontinuous" stable transfer function - which is what it's all about.

Some equations here near 1st diagrams.

The "magic" part is the negative resistance region - shown in red below. 

  

Below is the classic Esaki diode / Tunnel diode transfer function and you can try to fit formulae to that if you think it useful. Unlike most other devices you cxan have the same current at 3 different voltages - with two being stable, and the intermediate voltage being a transient one due to it being in a region of negative resistance. 



You could model this as something like y=x^3 with some offsets and shape adjustments but that would be missing the main point of the device. 



Related:

Useful comment in [oscillator applications](http://www.powerguru.org/crystal-oscillator-
design-and-negative-resistance/)



Here is a completely made up example.
The formula is shown below the graph.
This was made "by eye" solely on the basis of approximating an Esaki diode V/I curve. By adjusting parameters you could probably get a "close enough" fit to a real diode.
I doubt if this equation is especially useful but it does allow you to get a rough model of a real device. Axis units are semi arbitrary - adjust parameters to suit.
If you increase voltage above the maximum seen here you'll get a nasty shock due to the (5-Volts) term - no suggestion that this occurs in real life :-)





Off the Wall:  

Plot an energy versus velocity curve for a mass approaching and exceeding light speed.
For mass use standard
M_relativistic = Rest_mass/(1- V^2/C^2)
Use Energy = 0.5 m x v^2.
Ignore (for now) complex-imaginary aspect for V>C.
eg assume for now that jE = E.
Plot energy against V.
Wow!
Ponder.
-> You get a tunneling type curve with two velocities > C that have the same energy as a point for V &lt; C for values of V only above a certain % of light speed. E goes to infinity at V=C as expected. BUT the curve is suggestive of a tunnel effect across C with two velocities > C with the same energy. There's more, but that's enough OTW already :-).

Implications of the tachyonic jE are, of course, unknown.