I was fiddling with the Shockley equation and discovered that you can calculate the temperature coefficient of a diode from measurements taken at room temperature.
Assuming you only need to deal with currents well above the saturation current, which is the case in practical circuits, you can characterize the diode's behavior with an equation of the form ln I = a Vf + b, where I is the diode's current and Vf is the voltage across it. b is a negative number.
Let T(m) represent the temperature, in Kelvins, at which you take measurements to establish parameters a and b. The diode's temperature coefficient = (ln I - b) / (T(m) x a)
Using digital vom's I took data points for a 1N4003, 1N4148, 1N5818 schottky and MBR3035PT schottky, determined parameters a and b for each diode, and calculated the temperature coefficients. For a wide range of currents, the temperature coefficients of the 1N4003 and
1N4148 calculate near the 2.2 mV/K one sees cited so often. The calculations show lower temperature coefficients for the schottkys.Also interesting is the fact that schottkys have an emission coefficient near unity, and follow the 60 mV per current decade rule of thumb. Standard silicon diodes have emissions coefficients closer to 2 and do not follow the 60 mV "rule," but I suppose that rule may apply to diode-connected transistors, or to the relationship of transistor collector current to Vbe. I'll have to look into that.