The p-n Junction in Depth

The p-n junction is a single crystal doped p-type on one side and n-type on the other. The survey lesson sketched its equilibrium and its rectifying diode curve; this lesson derives them. The transport equations supply the tools — Poisson's equation for the field, minority-carrier diffusion for the current — and the result is the Shockley diode equation, the foundation of every semiconductor device.

Equilibrium and the built-in potential

When the two regions are joined, electrons pour from the electron-rich n side into the p side and holes the other way. Each departing electron leaves behind a fixed, ionized donor $N_{d}^{+}$ ; each departing hole leaves a fixed, ionized acceptor $N_{a}^{-}$ . A space-charge double layer builds up — negative on the p side, positive on the n side — and its field opposes further diffusion. Equilibrium is reached when the Fermi level is flat across the whole crystal, the thermodynamic condition for no net particle flow.

Flattening the Fermi level forces the bands to bend. Far into the n side the conduction band lies close below $E_{F}$ ; far into the p side it lies well above. The total bend is the built-in potential $V_{bi}$ . Its value follows from the carrier statistics: in the neutral n region $n = N_{d} = n_{i} e^{(E_{F} - E_{i}) / k_{B} T}$ , and in the neutral p region $p = N_{a} = n_{i} e^{(E_{i} - E_{F}) / k_{B} T}$ . The difference in the intrinsic level between the two sides is $e V_{bi}$ , giving

V_{bi} = \frac{k _{B} T}{e} ln \frac{N _{a} N _{d}}{n _{i}^{2}} .

The depletion approximation

To find the field and potential, model the space-charge region with the depletion approximation: the transition region $- x_{p} < x < x_{n}$ is swept entirely clear of mobile carriers, leaving only the fixed dopant charge, while the neutral regions beyond are perfectly screened. The charge density is then a step,

ρ (x) = {- e N_{a}, + e N_{d}, - x_{p} < x < 0, 0 < x < x_{n},

and zero elsewhere. Overall neutrality of the depletion region requires equal total charge on the two sides,

N_{a} x_{p} = N_{d} x_{n},

so the depletion layer extends further into the more lightly doped side. Poisson's equation $d E / d x = ρ / ε$ (with $ε$ the semiconductor permittivity) integrates the step charge into a triangular field that peaks at the metallurgical junction $x = 0$ and falls linearly to zero at each depletion edge:

E_{max} = \frac{e N _{d} x _{n}}{ε} = \frac{e N _{a} x _{p}}{ε} .

Integrating once more, $d φ / d x = - E$ , gives a potential that varies quadratically through the depletion region, its total drop equal to the area of the field triangle:

V_{bi} = \frac{1}{2} E_{max} (x_{p} + x_{n}) = \frac{e}{2 ε} (N_{a} x_{p}^{2} + N_{d} x_{n}^{2}) .

Solving with the neutrality condition for the total depletion width $W = x_{p} + x_{n}$ ,

W = \frac{2 ε V _{bi}}{e} (\frac{1}{N _{a}} + \frac{1}{N _{d}}) .

For the symmetric silicon junction above, $W \approx 0.3 μ m$ — far narrower than the tens of microns of minority-carrier diffusion length, confirming that recombination happens in the neutral regions, not the depletion layer.

The depletion region in three stacked profiles. The fixed charge is a step (negative acceptors on the p side, positive donors on the n side); its integral is a triangular field peaking at the junction; the field's integral is the smooth potential step of height V_bi.

Band bending under bias

An external voltage $V$ drops almost entirely across the high-resistance depletion region, adding to or subtracting from the built-in step. Every equilibrium formula carries over with $V_{bi} \to V_{bi} - V$ (taking $V > 0$ for forward bias, p side positive):

W (V) = \frac{2 ε ( V _{bi} - V )}{e} (\frac{1}{N _{a}} + \frac{1}{N _{d}}) .

Forward bias ( $V > 0$ ) lowers the barrier to $V_{bi} - V$ , narrows the depletion layer, and lets carriers diffuse across in large numbers. Reverse bias ( $V < 0$ ) raises the barrier, widens the depletion layer, and chokes off diffusion. The junction conducts one way — the rectifying action.

Band bending in three bias conditions. At equilibrium the flat Fermi level forces a barrier eV_bi. Forward bias raises the p-side bands, shrinking the barrier and flooding the junction; reverse bias lowers them, widening the barrier and blocking diffusion.

The Shockley diode equation

The current follows from minority-carrier injection. Lowering the barrier by $e V$ raises the minority concentration at each depletion edge by the Boltzmann factor $e^{e V / k_{B} T}$ — the law of the junction. On the n side the hole concentration at the edge becomes $p_{n} (x_{n}) = p_{n 0} e^{e V / k_{B} T}$ , an excess

δ p_{n} = p_{n 0} (e^{e V / k_{B} T} - 1), p_{n 0} = \frac{n _{i}^{2}}{N _{d}} .

These excess holes diffuse into the neutral n region and recombine over a diffusion length $L_{p}$ , exactly the injection profile derived from the continuity equation. The diffusion current at the edge is $J_{p} = e D_{p} δ p_{n} / L_{p}$ . Adding the symmetric electron injection into the p side gives the total current density,

J = e (\frac{D _{p} p _{n 0}}{L _{p}} + \frac{D _{n} n _{p 0}}{L _{n}}) (e^{e V / k_{B} T} - 1) = J_{0} (e^{e V / k_{B} T} - 1) .

The saturation current is the diode's Achilles heel and its thermometer: because $I_{0} \propto n_{i}^{2} \propto e^{- E_{g} / k_{B} T}$ , it roughly doubles every $10 K$ , which is why germanium diodes ( $E_{g} = 0.67 eV$ ) leak far more than silicon ( $1.12 eV$ ) and why high-temperature electronics turns to wide-gap materials.

The ideal-diode characteristic I = I_0(exp(eV/k_BT) - 1). Forward bias rises exponentially past the turn-on near V_bi; reverse bias saturates at the tiny current I_0, magnified in the inset scale, giving one-way conduction.

Junction and diffusion capacitance

A junction stores charge two ways, and each is a capacitance. In reverse and small forward bias the stored charge is the fixed dopant charge in the depletion layer, $Q = e A N_{d} x_{n}$ . Differentiating with the depletion width gives the junction (depletion) capacitance

C_{j} = \frac{d Q}{d V} = \frac{ε A}{W ( V )} = A \frac{e ε}{2 ( V _{bi} - V )} \frac{N _{a} N _{d}}{N _{a} + N _{d}},

the same form as a parallel-plate capacitor of gap $W$ , but with $W$ tunable by the reverse bias. A junction used this way is a voltage-controlled capacitor, the varactor that tunes radio-frequency oscillators; a plot of $C_{j}^{- 2}$ versus $V$ is a straight line whose slope gives the doping.

Under strong forward bias a second, larger capacitance dominates: the stored excess minority charge $Q = I τ$ changes with the current, giving the diffusion capacitance $C_{d} = d Q / d V = (e / k_{B} T) I τ$ , proportional to the forward current and to the lifetime. Diffusion capacitance is what slows a diode's switching from conducting to blocking — the stored charge must be removed first.

Breakdown

Beyond a critical reverse voltage the tiny saturation current turns sharply into a large reverse current — breakdown. Two distinct mechanisms produce it.

Avalanche breakdown. In the strong field of a wide depletion region a carrier gains enough kinetic energy between collisions to knock a bound electron across the gap by impact ionization, creating a new electron-hole pair; the offspring are accelerated in turn, and the multiplication runs away. Avalanche dominates in lightly doped junctions, has a breakdown voltage that rises with temperature (hotter lattice scatters carriers before they reach ionizing energy), and sets the reverse rating of power and high-voltage diodes.
Zener breakdown. In a heavily doped, narrow junction the depletion layer is so thin that valence electrons on the p side tunnel directly into empty conduction states on the n side once the bands overlap under reverse bias. Zener (tunneling) breakdown dominates below about $5 V$ in silicon and has a breakdown voltage that falls with temperature. Zener diodes exploit the sharp, stable breakdown edge as a voltage reference.

The reverse breakdown region. Below the breakdown voltage only the saturation current flows; at V_BR the current turns down steeply as avalanche multiplication or Zener tunneling sets in, clamping the voltage across the diode.

The junction physics assembled here — the built-in field, the exponential injection current, the stored charge, and the breakdown edge — is the vocabulary of every device. Cascading two junctions makes a transistor, and running a junction as a light source or a photon collector makes the optoelectronic devices of the final lesson.