Shockley Diode Equation

William Shockley published the theory of the p-n junction in 1949, and the equation carrying his name is the foundation of every semiconductor device that followed. Its shape comes from statistics rather than geometry: carriers must climb a potential barrier to cross the junction, the number with enough thermal energy to do so is Boltzmann-distributed, and forward bias lowers the barrier. So the current rises exponentially with voltage, scaled by $kT/q$ — the thermal voltage, 25.85 mV at 300 K.

Take the exponential seriously and the "0.7 V diode drop" of introductory electronics dissolves. There is no such thing as a fixed forward voltage. The drop is whatever voltage happens to be passing the current the rest of the circuit chose, and 0.7 V is simply where a small silicon diode lands at a milliamp or so. At a microamp the same diode sits nearer 0.5 V; at an amp, nearer 0.9 V. The useful way to hold it in your head is the slope: about 60 mV per decade of current for an ideal junction at room temperature. Ten times the current costs 60 mV more, which is why a diode compresses so beautifully and why log amplifiers are built from them.

The same steepness is a hazard. Driving a diode or an LED from a voltage source without a series resistor is the classic beginner's destruction: sixty millivolts too many is ten times too much current, and the component's own heating lowers its forward voltage further, which raises the current again. Diodes are driven from current sources, or from voltage sources through a resistor that sets the current — and the resistor, not the diode, is what makes the circuit predictable.

Two corrections separate this ideal equation from a real device. The ideality factor $n$ runs from 1, where diffusion current dominates, toward 2, where recombination in the depletion region does — low currents and LEDs sit nearer 2, moderate forward bias in a good silicon diode nearer 1. And the temperature behaviour is subtler than the explicit $kT/q$ suggests: the saturation current $I_s$ is itself strongly temperature-dependent, roughly doubling every 5 to 10 °C, and that dependence dominates. The net result is that at a FIXED current the forward voltage FALLS by about 2 mV per degree, which is the opposite of what the bare $kT/q$ term would lead you to expect, and it is the effect every diode-based temperature sensor is built on.