= Solution
Donor doping supplies conduction <electrons> to <n-type semiconductor>; acceptor doping supplies <holes> to <p-type semiconductor>. In the nondegenerate, fully ionized limit, majority densities are approximately $n_n=N_D$ and $p_p=N_A$, while minority densities follow $np=n_i^2$. On contact, majority carriers diffuse across the junction, exposing positive donor ions on the n side and negative acceptor ions on the p side. The depletion-region <electric field> points from n to p and opposes further <diffusion>. At <thermal equilibrium> the <Fermi level> is uniform and drift balances <diffusion>, with no net current.
For an abrupt junction with uniform doping and permittivity $\varepsilon_s$, charge neutrality gives $N_Dx_n=N_Ax_p$. <Poisson's equation> then gives
$$
V_{\rm bi}=\frac{k_BT}{q}\log\frac{N_AN_D}{n_i^2},\qquad
\boxed{W=x_n+x_p=
\sqrt{\frac{2\varepsilon_s}{q}\left(\frac1{N_A}+\frac1{N_D}\right)(V_{\rm bi}-V)}}.
$$
Positive applied $V$ means <forward bias>: it lowers the barrier and narrows the <depletion region>. Most depletion width lies in the more lightly doped side. The formula assumes the <depletion approximation> and fails once its operating assumptions break down.
<forward bias> increases minority densities at the depletion edges by the factor $e^{qV/k_BT}$. In the neutral material, the steady <diffusion>–<recombination> equation gives exponentially decaying excess minorities, with lengths $L_n=\sqrt{D_n\tau_n}$ and $L_p=\sqrt{D_p\tau_p}$. Taking the carrier gradients at the junction yields
$$
\boxed{I=I_S\left(e^{qV/k_BT}-1\right),\qquad
I_S=qA_jn_i^2\left(\frac{D_n}{L_nN_A}+\frac{D_p}{L_pN_D}\right)}.
$$
This ideal <diode> law assumes low injection, nondegenerate statistics, negligible depletion-region <recombination> and series resistance, and sufficiently long neutral regions. Its exponential comes from the bias-dependent injection barrier; there is no universal abrupt “turn-on” voltage. <reverse bias> suppresses injection, giving an approximate $-I_S$ current until additional leakage or breakdown matters.<The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors by William Shockley>[Bell System Technical Journal 28, pp. 435–489, minority-carrier injection and rectifying-junction theory]{ref}
Rectification uses the asymmetric <current-voltage characteristic>; controlled reverse breakdown enables voltage references, with current limiting. Bias-dependent depletion capacitance enables varactors. <photodiodes> and <solar cells> collect optically generated carriers through the junction field and diffusion toward it; efficient <light-emitting diodes> use radiative <electron>–<hole> <recombination>, preferably in a direct-gap material. Series resistance, high injection, trap-assisted <recombination>, heating and finite carrier storage explain departures from the ideal steady-state law.
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