Vbe has kind of mystified me for a long time. I finally dove into a good textbook and picked apart the equations.
This is actually the math for a bjt (diode-connected), and it has a term related to base doping level, so it may not apply to a diode that consists of a single pn junction.
But that's just as well; it makes more sense in most applications to use a diode-connected transistor anyway.
The expression for Vbe is a linear function of the log of absolute temperature.
Vbe = Vg + (kT/q)( ln(E*Ic) - (4-n) ln T)
where Vg = band gap voltage for silicon
k/q = 86.17 microvolts per degree Kelvin (Boltzmann's constant divided by electron charge)
E is a bunch of device parameters bundled together that doesn't depend on temperature, according to the text. I got an expression for E in terms of the fundamental device parameters but it's nasty.
Ic = collector current
-n is the exponent in the power law that relates average electron mobility in the base to absolute temperature. A possible value for n, according to the text, is 0.8
Taking the derivative, dVbe/dT = (k/q) (ln(E*Ic) - (4-n) (1+ ln T))
Ok, so the temperature coefficient is still a linear function of ln T. Now the question that always interested me was, what happens to the temperature coefficient as the temperature varies across a range, say -55C to 125C? According to the equation, tempco changes 166uV. If the Vbe is 2mV per degree Kelvin, then diode tempco changes about 460 ppm per degree centigrade. It changes about 8 percent from -55C to 125C.
So for example, say you are using a diode-determined voltage or current to cancel a thermal-voltage determined source in a bandgap circuit.
The thermal voltage is perfectly linear over temperature, in fact it is directly proportional to absolute temperature.
The fact that diode voltage is not quite perfectly linear points up a systemic error in the bandgap circuit architecture, and explains why it may not be possible to get a perfectly flat voltage or current reference with respect to temperature from a bandgap circuit, even in theory. But it's still pretty darn good. And there are ways of compensating even the small curvature in a bandgap reference, apparently.
It never ends.