Bias stabilization of a transistor
The diode current has the formula

Idiode = Is *  e ^ [Q * Vdiode / (K * T)]

WIth the T being temperature, and a common voltage at 1milliAmp being about 0.6 volts,
we can simply divide that 0.6 volts by room temperature of 290 degree K,
and we have about 2.1 milliVolts change per degree. This however requires the WRONG POLARITY of voltage change.

Digging further, wikipedia says I am wrong, and that the change of Is will be the far greater (and opposite polarity) effect. Check out Wikipedia "diode equation".

examining "reverse saturation current" on Wikipedia, we see the useful statement "a rule of thumb is that the current doubles for every 10 &deg; C".
We can use that, along with 18 milliVolts being adequate to cause Idiode to double, and see this predicts 18/10 == 1.8 milliVolts decrease.