Sze, Physics of Semiconductor Devices, 1st Ed., has a good discussion of the emission wavelength vs. bandgap (wish I could find my second edition
).
As to the application of the Shockley equation to the forward voltage, Sze remarks: "The Shockley equation adequately predicts the current-voltage characteristics of germanium p-n junctions at low-current densities. For Si and GaAs p-n junctions , however, the ideal equation can only give qualitiative agreement. The departures from thje ideal are mainly due to the following effects: ......" (Sze, op. cit. pg. 102).
After a few hairy pages we get to "The experimental results in general can be represented by the following empirical form Jf ~ exp(qV/nkT), where the factor n = 2 when the recombination current dominates...and n = 1 when the diffusion current dominates...When both current are comparable, n has a value between 1 and 2."
I recall a brief mention of this in Horowitz and Hill as well.
Note that this means the temperature coefficient also varies with current. It also explains why a particular circuit I saw, that "linearized" a transistor stage using a diode worked well, whereas replacing the diode with a diode-connected transistor did not.
The bandgap enters in to the saturation current density and its temperature coefficient, and Sze shows the expression Js = T^([3 + γ]/2)*exp (-Eg/kT), and remarks that the first term is not important compared to the exponential term. Actually, you rarely see this first term at all, although it actually does get significant at very low temperatures.
When I had a brief email correspondence going with Barrie Gilbert and asked how some experimental heterojunction bipolars they were developing at Analog Devices got betas of 100,000 or more, he got impatient and told me to go and read Sze :roll: I think that's a little along the lines of telling a person interested in chemistry to go and study the Schrödinger equation, but maybe he was having a bad day :razz:
