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For a planar abrupt p-n junction with ionized densities $N_A,N_D$, p-side width $w_p$ and n-side width $w_n$, zero external field requires $N_A{w}_p=N_D{w}_n$. Integrating Poisson equation gives $V_{\mathrm{bi}}-V=e(N_A{w}_p^2+N_D{w}_n^2)/(2{\epsilon }_s)$, with positive $V$ denoting forward bias. Therefore $W=w_n+w_p$ has the displayed value, $w_n=N_{A}W/(N_A+N_D)$ and $w_p=N_{D}W/(N_A+N_D)$. The positive depletion capacitance per area is $C_A={\epsilon }_s/W$. It equals $d{Q}_p/dV=-d{Q}_n/dV$ when $Q_p=-e{N}_A{w}_p$ and $Q_n=e{N}_D{w}_n$ are signed space charges per area. Differentiating a positive charge magnitude with respect to forward voltage instead gives a negative derivative. The square-root law assumes partially depleted regions, negligible mobile charge and no substantial injected-carrier storage.

A bipolar junction transistor has two p-n junctions sharing a thin base region, arranged n-p-n or p-n-p. In its forward amplification regime, the emitter junction injects minority carriers; many cross the base before electron-hole recombination and reach the reverse-biased collector junction.^[ref]

The built-in potential of a p-n junction is the electrostatic potential difference established when carrier diffusion exposes ionized dopants until the opposing drift and diffusion currents balance. The equilibrium electrochemical potential is constant even though the electric potential and band edges vary. In a nondegenerate homojunction with neutral-edge majority densities $p_p,n_n$, $e{V}_b=k_{B}T\log (p_p{n}_n/n_i^2)$. This is band bending within the same material, not necessarily a discontinuous offset of different materials' band edges.

For a zero-bias abrupt p-n junction, choose the p side at $x<0$ and the n side at $x>0$. Ionized acceptors and donors remain fixed while mobile electrons and electron holes diffuse and establish a built-in electric field. In the depletion approximation, mobile carriers are neglected on $-x_p<x<x_n$ and the net charge density isCharge neutrality gives $N_A{x}_p=N_D{x}_n$. The quasi-neutral bulk can have a large mobile carrier density while its net charge density is nearly zero, because fixed dopant charge compensates it. With complete ionization and nondegenerate carriers, equilibrium also gives $np=n_i^2$; real carrier profiles vary smoothly rather than forming exact step functions at depletion edges.

The experimental distinction is concrete. A 1962 gallium arsenide p-n junction laser used top and bottom contacts, parallel polished faces, and current pulses with the device in liquid nitrogen. An infrared image converter recorded the radiation pattern; a spectrometer measured narrowing above laser threshold. Comparing the same device below and above laser threshold supplied a control. Pulsed, cooled operation did not demonstrate continuous operation at room temperature.^[ref]

Zhores I. Alferov's 1970 team photolithographically patterned stripe double heterostructure lasers on silver-covered copper plates. He reports sustained laser emission at room temperature; Fig. 4 shows the device structure. Independently, Izuo Hayashi, Morton B. Panish and collaborators demonstrated continuous double heterostructure laser operation with diamond heat sinks.^[ref]

The below-threshold comparison tests the onset of stimulated emission; continuous operation additionally requires sustained removal of waste heat. Neither observation alone measures the separate band offsets, optical overlap, or every loss mechanism. The schematic above therefore explains a design principle rather than reconstructing the historical apparatus.

The depletion approximation replaces the junction's transition region by a region without mobile carriers, leaving only fixed ionized dopants. For a uniformly doped abrupt p-n junction, $\rho =-e{N}_A$ on the p side, $\rho =+e{N}_D$ on the n side and $\rho \simeq 0$ outside. Charge neutrality gives $N_A{x}_p=N_D{x}_n$. Integrating Poisson equation gives the electric field and the electrostatic band bending.

The region around a p-n junction with reduced mobile charge carrier density.

An illustrative abrupt-junction depletion approximation assumes uniform, fully ionized donor impurities and acceptor impurities, negligible mobile space charge in the depleted layer, and permittivity ${\epsilon }_s$. For area $A$, doping number densities $N_A,N_D$ and built-in electric potential difference $V_{\mathrm{bi}}$, it givesHere $q>0$ is the elementary charge and $V$ is positive for forward bias. Increasing both doping densities reduces $W$ in this estimate. That explains the preparation strategy for a tunnel diode; it does not make this sharp-depletion approximation a precise microscopic model of a degenerate semiconductor.

For a long, nondegenerate, ideal p-n junction under low injection, the diode saturation current is set by minority carriers supplied from the neutral regions. Their equilibrium densities are $n_{p0}=n_i^2/N_A$ and $p_{n0}=n_i^2/N_D$. Solving the steady diffusion and recombination equations gives $I_S=e\mathcal{A}(D_n{n}_{p0}/L_n+D_p{p}_{n0}/L_p)$, where $\mathcal{A}$ is junction area, $D_n,D_p$ are diffusion coefficients, and $L_n,L_p$ are diffusion lengths. Reverse bias suppresses opposite majority-carrier injection exponentially while collecting this minority supply. Thus $I_S$ is approximately bias independent before breakdown, but depends strongly on temperature through $n_i^2$ and on transport, doping, and geometry.

A drift current is an electric current caused by the response of mobile charge carriers to an electric field. It is distinct from the diffusion contribution due to concentration gradients. At equilibrium in a p-n junction, the two contributions cancel although each can be nonzero.

Leo Esaki's 1958 report describes alloy-fabricated germanium p-n junctions, with acceptor impurity concentration $1.6\times 10^{19}{\mathrm{cm}}^{-3}$ and donor impurity concentration approximately $10^{19}{\mathrm{cm}}^{-3}$. Its readout was the current-voltage characteristic: a forward peak occurred at $0.035\pm 0.005\mathrm{V}$. The report says specimens could be reproduced; above $0.3\mathrm{V}$ the forward branch approximately followed the ordinary Shockley diode equation.^[ref]

In his 1973 lecture, Leo Esaki recalls an earlier junction whose strong room-temperature reverse conduction preceded a negative-slope forward segment revealed by cooling. Its estimated zero-bias width was about $200$ ångströms ($20\mathrm{nm}$), supported by capacitance measurements. Increasing semiconductor doping further made the negative slope visible across the measured temperatures, including room temperature. Thus preparation and temperature variation were experimental controls, not merely changes to a theoretical diagram. The low onset voltage argued against avalanche production of electrons and electron holes; the energy band explanation connected the forward decline to loss of available final states.^[ref]

The historical readouts were junction electric current versus applied electric potential difference and a separate capacitance check; preparation and temperature were varied. The cited descriptions do not identify a meter model or give a complete wiring diagram.

For a modern repetition, measure junction voltage directly while stepping the supply and recording electric current, repeat both sweep directions, and compare temperatures and differently doped specimens. These are proposed controls, not a reconstruction of the historical apparatus. Lead electrical resistance, Joule heating, contact quality and voltage resolution can obscure the result. The tunnel-diode load-line stability calculation also predicts that an external resistor can make a negative-slope operating point inaccessible in a slow sweep. A jump in terminal readings then need not mean the underlying current-voltage characteristic is discontinuous.

In a diffusion-dominated p-n junction, $I_s=C(T)n_i^2$ with a transport prefactor determined by minority-carrier diffusion and dopant densities. Combining this with the intrinsic carrier concentration gives, for fixed gap and effective masses,The slope against $1/T$ is $-E_g/k_B$. A measured diode ideality factor helps identify the transport regime: generation or recombination current can have a different dependence on $n_i$, so its activation slope cannot automatically be interpreted with the diffusion formula.

For a p-n junction, biasing with the p-type semiconductor at positive electric potential difference relative to the n-type semiconductor.

Let each side of a symmetric abrupt p-n junction have thickness $d$, area $\mathcal{A}$ and doping density $N$. Partial depletion gives $w^2={\epsilon }_s(V_{\mathrm{bi}}-V)/(eN)$. Both sides become fully depleted at $V_{\mathrm{fd}}=V_{\mathrm{bi}}-eN{d}^2/{\epsilon }_s$. Beyond that point, the ionized dopant charges are fixed, but an incremental terminal voltage changes the electric field uniformly across the full thickness $2d$. Gauss's law then gives $d{Q}_{\mathrm{terminal}}={\epsilon }_s\mathcal{A}dV/(2d)$, up to terminal orientation. The measured positive capacitance approaches $C_{\mathrm{fd}}$ rather than vanishing. This geometrical result assumes ideal contacts, negligible mobile carriers, no breakdown and no other series capacitances.

Hiroshi Amano is an electrical engineer whose work with Isamu Akasaki helped overcome two obstacles to blue light-emitting diodes: growing high-quality gallium nitride and obtaining controllable p-type semiconductor.^[ref]

A low-temperature-deposited aluminium nitride buffer layer improved subsequent gallium nitride growth on sapphire. His lecture compares the resulting surfaces using scanning electron microscopy. The research also showed that low-energy electron irradiation of magnesium-doped gallium nitride could activate p-type semiconductor, enabling a p-n junction. The two results address different requirements: structural quality and the electrical activity of the semiconductor doping.^[ref]

A homojunction joins regions of the same semiconductor material with different semiconductor doping. Its bulk band gap is the same on both sides. This distinguishes the electrostatic band bending of a simple equilibrium p-n junction from the material-dependent band offsets possible in a heterojunction.

An integrated circuit interconnect is a conducting path joining circuit regions. In Robert Noyce's patent, deposited metal strips cross retained oxide on silicon; selected openings permit contact to the semiconductor while oxide prevents unwanted contact across p-n junctions. Figs. 3–5 combine detection, filtering and electronic amplifier functions using a transistor in one structure. The key is both connectivity and insulation: a metal path must touch its intended terminals while crossing other regions without shorting them.^[ref]

Isamu Akasaki was an electrical engineer whose research helped establish gallium nitride as a material for blue and ultraviolet light-emitting diodes. Working with Hiroshi Amano and other collaborators, he pursued improvements in crystal quality and in the control of p-type and n-type semiconductor.^[ref]

The scientific motivation follows from photon energy: shorter visible wavelengths require larger transition energies. A sufficiently wide band gap is therefore necessary for blue emission, but it does not by itself produce a useful device. The material must also support carrier injection and efficient light emission. High-quality growth and an operating p-n junction turned those physical requirements into an engineering platform for full-colour displays and lighting.^[ref]

A Josephson junction couples two regions with superconductivity through a weak link. A tunnel barrier is one implementation.

The remarkable possibility is a steady electric current across an insulating barrier at zero electric potential difference. This is the DC Josephson effect, carried by coherent transfer of Cooper pairs. It differs from single-electron superconducting tunneling, which probes the quasiparticle excitation spectrum and produces a dissipative current-voltage characteristic. Both contributions can occur in the same superconducting tunnel device. Brian D. Josephson predicted the pair contribution in 1962; his paper treats transfer that leaves the quasiparticle distribution unchanged. ^[ref]

The Josephson effect turns a difference of quantum phases into a measurable electric current, and an applied electric potential difference into a rate of phase advance. The Giaever gap experiment instead turns an electric potential difference into an excitation energy scale. These complementary measurements expose two aspects of superconductivity: its coherent order parameter and its gapped quasiparticle spectrum. The microscopic pairing calculation belongs to BCS theory and the BCS gap equation. A tunnel diode uses a different device, a semiconductor p-n junction.

Adding capacitance and an external bias makes the Gauge-invariant Josephson phase dynamical. Current-biased Josephson phase dynamics explains how its trapped energy levels and escape can be measured electrically.

The incremental capacitance associated with changes of electric charge around a p-n junction. Its depletion contribution is often estimated by $C_j\simeq {\epsilon }_{s}A/W$, so an independent junction capacitance measurement can constrain the width $W$ of the depletion region under the model assumptions. Other stored electric charge, including injected minority carriers, can also contribute to a measured terminal capacitance.

A junction field-effect transistor controls a conducting semiconductor channel using reverse-biased p-n junctions at its gate. Changing gate voltage changes the depletion region and therefore the cross-section available to mobile charge carriers. A depletion-mode n-channel device normally conducts at zero gate-source voltage and is depleted by a more negative gate. The small reverse gate electric current gives high input resistance, while a source-drain supply provides output power.