Diode Equation needs a Knee Transplant

Adjust N, which is a constant that depends upon the manufacturing process and semiconductor material. A value somewhere in the neighborhood of 0.5 will probably do the trick.

There is no band gap voltage in the "low voltage" diode equation. As the applied voltage exceeds the band gap voltage about 0.6 for Si and

-19,

if the equation is not at equilibrium, a potential barrier wont develope across the P-N junction so its not that important.

Here's one more for all the contradictory advise:

Increase Io. Dramatically.

Perhaps take some measurements of the diode voltage at several different current values (i.e. 1uA if you can get that low, 10uA, 100uA, 1mA -- don't exceed the diode ratings). Then do a curve fit on semilog paper

-- figure that you are going to see Io e^(eV/NkT) only, the "- 1" part will drop right out.

Yes, go the empirical route. You can use ordinary graph paper if you want, by graphing log I versus V, or by graphing I versus exp(V). The -1 drops out to insignificance, as Tim says. You will find that the data points lie on a very straight line, until self-heating kicks in, whereupon the graph will start to curve away from the straight line established at low currents (constant temperature) where you see the purely exponential side of the relationship. For small diodes I agree with Tim's suggested current range.

You will have to take more than three measurements if you want to see just where the curve diverges from the ideal, as a result of self-heating effects. Try it. I was pretty impressed at how precisely nature mirrors the math.