Derive diode current and diagnose recombination

In a nondegenerate p region, equilibrium minority electrons have density $n_{p0}=n_i^2/N_A$ for complete ionization and strong p-type doping. A forward junction voltage V sets their density at the depletion edge to $n_{p0}e^{V/V_T}$ under the ideal injection assumptions. The excess above equilibrium is therefore proportional to $e^{V/V_T}-1$. Holes injected into the n region obey the corresponding relation. Applied terminal voltage equals this junction voltage only when other voltage drops are negligible.

In a neutral region with low-level injection, the majority density stays close to its equilibrium value. A linear minority lifetime approximation then gives a steady diffusion equation for excess carriers. Its long-base solution decays exponentially away from the depletion edge, with diffusion length equal to the square root of diffusion coefficient times lifetime. Taking the derivative at the edge produces minority diffusion current. The electron and hole edge contributions add because their conventional forward currents point in the same terminal direction.

A defect level can capture an electron and a hole sequentially, enabling Shockley–Read–Hall recombination without direct band-to-band emission. The net rate vanishes in thermal equilibrium because capture and emission balance. Away from equilibrium its numerator measures the departure of the electron-hole product from the intrinsic square, while the denominator contains carrier densities, trap energy through $n_1,p_1$, and capture-lifetime parameters. It is not generally proportional to either carrier density with one universal lifetime.

For a roughly midgap trap with comparable capture times, depletion-region forward recombination is strongest near the location where electron and hole populations are comparable. Their local growth can yield a current approximately proportional to $e^{V/(2V_T)}$ over an appropriate bias range. This gives an ideality factor near two. Recombination in a neutral region can still support an ideality-one diffusion current, so the phrase recombination current alone does not determine an ideality factor. Spatial location and injection regime matter.

In the forward exponential regime, ideality one gives a voltage increment $V_T\ln10$ for a decade increase in current. A nearly constant slope near twice that value suggests a different transport regime, but does not prove a unique defect mechanism. High injection can also change the exponential scale. At large current, a series-resistance drop increases terminal voltage without the same increase in junction bias, producing an apparent ideality factor that grows with current rather than remaining a pure exponential constant.

Reverse current likewise need not equal a perfectly constant saturation value. Generation in a widening depletion region, surface leakage, tunneling, and avalanche can all change it. The elementary diode law is valuable precisely because its assumptions are explicit and its departures can be diagnosed. Temperature dependence, geometry, and independent lifetime measurements help distinguish mechanisms that a single current-voltage trace cannot identify. Current conservation still holds throughout the device even as electron current converts into hole current through recombination.