A diode is a two-port electronic component that allows current to flow more easily in one direction than the other. The most common type of diode today is the p-n diode, made out of a semiconductor junction. Also common is the Schottky diode, made out of a metal-semiconductor junction. The behavior of both these types is described by the Shockley equation, in which the current is exponential with forward voltage. This exponential relationship also describes the current flowing through a bipolar junction transistor, although in that context the relation is called the Ebers-Moll model of the transistor.
Our first derivation is the temperature coefficient of the semiconductor diode's forward voltage drop. This temperature coefficient is negative, meaning that the voltage across a diode or a base-emitter junction for a particular current decreases as we heat up the device. The negative temperature coefficient is the basis of thermal runaway, in which a diode driven by a fixed voltage gets hotter and hotter and passes more and more current until it suffers damage.
If we differentiate the Shockley equation with respect to temperature, and assume the saturation current, IS, is constant with temperature, we find that the temperature coefficient of the forward voltage drop must be positive, not negative. The source of the negative temperature coefficient turns out to be the strong behavior of IS with temperature. We take an approximate relation between saturation current and temperature, and combine it with the ideal Shockley equation to obtain an estimate of the change in a semiconductor diode's forward voltage drop with temperature.
In the above derivation, we assume the Shockley equation describes our diode accurately. That is to say: the current increases exponentially with forward voltage, and the constant in the exponential is q/kT. For Shottky diodes and bipolar transistors, the exponential constant is indeed very close to the ideal value. But for silicon PN diodes like the BAV99, the exponential constant is closer to one half the ideal value. The ideality factor of a diode is the number by which we divide q/kT to get the actual constant that governs the diode current's exponential dependence upon voltage. When we measure current versus voltage in the lab, we get an ideality factor of around 2.0 for PN diodes and 1.0 for bipolar transistor base-emitter junctions.
Assuming a ideality factor 2.0 for the BAV99, we can repeat the above derivation and obtain IS = 4 nA. This compares well with the 10-nA reverse current of the BAV99 at 25°C. We include the factor 2.0 in the exponential term for B and get 2 μA/K3, which is a hundred thousand times smaller than our ideal calculation. Nevertheless, the dramatic changes in IS and B do not affect the derivative of V with respect to T. The Ln(B) term is still small compared to the 3Ln(T) term. But the ideality factor does divide the temperature dependence by a factor of two, so we arrive at −1.2 mV/°C instead of −2.4 mV/°C. Looking at the data sheet, we see that for a current of 10 mA, the voltage drop decreases by around 250 mV for a 190°C increase in temperature, which is around −1.3 mV/°C. In this circuit, we use a chain of seven BAV99DW diodes to generate a voltage for a logic power supply. We expect the voltage across this chain to drop by around 8 mV/°C.