Diode Ideality Factor
The ideality factor adjusts the original Shockley diode equation to fit non-Germanium diodes more closely. There's also a series resistance term. 

Equation 3.13 from the original 1950 paper from Bell System Technical Journal:



The factor was added a few years later in a fascinating paper on radioactive p-n junction power sources. 



With silicon diodes the ideality factor is closer to 2 than 1. It can be higher than 2. A diode-connected silicon transistor operated at reasonable currents has an ideality factor close to 1. 

I don't think it is ever less than 1 for a real diode, but that may not be correct.