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A semiconductor whose mobile charge carrier population requires Fermi-Dirac statistics rather than a dilute Maxwell-Boltzmann distribution. In a strongly degenerate n-type semiconductor, the Fermi level lies inside the conduction band; in a strongly degenerate p-type semiconductor, it lies inside the valence band. The occupied and empty quantum states on opposite sides of a tunnel diode can then overlap in energy. Heavy semiconductor doping both changes these occupations and narrows the depletion region; a narrow barrier and accessible final quantum states serve different roles.

Thermal generation of electron-hole pairs within a reverse-biased depletion region can supply a leakage electric current in addition to neutral-region diffusion. For a simple midgap trap with equal electron and hole lifetime parameters, its scale is $e\mathcal{A}W{n}_i/(2{\tau }_g)$, where $W$ is depletion width and ${\tau }_g$ the lifetime parameter. The electric field separates generated carriers. Unlike ideal diode minority-carrier diffusion current, this contribution scales approximately with $n_i$ rather than $n_i^2$ and can vary with reverse voltage through $W$. Consequently a measured activation energy need not equal the full band gap.

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.

Integrating the electric field across a linearly graded p-n junction gives $V_b-V=e\alpha w^3/(12{\epsilon }_s)$, where positive $V$ means forward bias. Thus the depletion region width depends on the cube root of the remaining barrier voltage. Both the space-charge density and integration range grow with width; this differs from the square-root width law of an abrupt uniformly doped junction.

At equilibrium, electrons initially diffuse from the n side and holes from the p side. Recombination leaves ionized donors and acceptors in a depletion region, producing an internal electric field and band bending. The Fermi level is constant across the equilibrium p-n junction. With p material on the left, both band edges lie higher on the p side, as in the left panel.

Forward bias, with the p side positive relative to n, reduces the injection barrier. Electrons and holes enter the junction region and can recombine radiatively, emitting photons with energy close to the band gap. Under bias, separate electron and hole quasi-Fermi levels replace a single equilibrium Fermi level.

Both GaN polymorphs have a direct band gap, so a conduction-band electron and valence-band hole can recombine without a large crystal-momentum change. Silicon and germanium have indirect band gaps, requiring phonon assistance for near-edge radiative recombination, which makes efficient band-edge light emission much harder. A direct gap enables the process; defects, carrier confinement and competing nonradiative channels still determine device efficiency.

Absorbed photons with energy at least the band gap generate electron–hole pairs. The built-in field of the p-n junction separates carriers generated in or reaching the depletion region; minority carriers generated nearby can diffuse into that collection region. Selective contacts carry electrons and holes into an external circuit before they recombine.

Define positive electric current into the positive-voltage terminal. In the ideal illuminated-diode model,The dark curve has $I_{\mathrm{ph}}=0$; illumination shifts it downward. At short circuit $I\simeq -I_{\mathrm{ph}}$; at open circuit $I=0$ and $V=V_{\mathrm{oc}}$. Between them the illuminated cell lies in quadrant IV and supplies power $P_{\mathrm{out}}=-VI>0$. The maximum power point maximizes the area of the rectangle with sides $V$ and $-I$:It is neither the open-circuit point nor the short-circuit point, both of which supply zero power.

forward bias opposes the built-in electric field and lowers the junction barrier from approximately $V_{\mathrm{bi}}$ to $V_{\mathrm{bi}}-V$. In the abrupt-junction depletion approximation,Thus the depletion region narrows and carriers cross the reduced barrier more readily. electron injection into the p side and hole injection into the n side increase strongly, allowing radiative recombination in an LED.

The injected minority-carrier concentrations, and therefore the ideal diffusion electric current, vary approximately as $\exp (qV/k_{B}T)$. This produces the steep forward branch of the electric current–voltage curve. One should not extrapolate the depletion-width formula to claim that the junction must lose its entire depletion region before conducting; appreciable electric current flows while a barrier remains, and high injection and series resistance eventually invalidate the simple formulas.

Treat the $p^{+}$ strips as much more strongly doped than the $n$ bulk, so the depletion layer lies predominantly in the latter. Fully ionized donors leave uniform space charge $\rho =e{N}_D$. In one dimension,At the depletion boundary $x=d$, the minimum-bias solution has $E(d)=0$. Thus $E(x)=e{N}_D(x-d)/{ϵ}_{\mathrm{Si}}$, and the magnitude of the potential drop isFor $N_D=10^{18}{\mathrm{m}}^{-3}$, $d=300\mu \mathrm{m}$, and ${ϵ}_r=11.6$, this isThe applied reverse bias is approximately this value; more precisely the built-in junction potential contributes to the total depletion voltage and is subtracted from the required external bias. It is small on the scale of seventy volts and is not specified. Here the stated free-carrier density is used as the effective donor density of the neutral bulk before depletion.

Joining p-type and n-type regions initially allows majority carriers to diffuse across and recombine. Exposed negative acceptor charge on the p side and positive donor charge on the n side form a depletion region. Its electric field points from n to p and balances diffusion at equilibrium. The Fermi level is then spatially constant, while the conduction and valence band edges bend across the built-in potential; this equilibrium barrier cannot supply continuous power to a load.

forward bias makes p positive relative to n, reducing the barrier from $q{V}_{bi}$ to $q(V_{bi}-V)$. electrons enter p material and holes enter n material, becoming minority carriers whose diffusion and recombination sustain the current. reverse bias enlarges the barrier and leaves a much smaller minority-carrier current until breakdown. The ideal diffusion model gives $I=I_0[e^{qV/k_{B}T}-1]$, the Shockley diode equation; depletion-region recombination, series resistance and breakdown limit this approximation. Minority-carrier injection on both sides is essential to the junction theory.^[ref]

Absorption of photons with $h\nu \gtrsim E_g$ creates electron-hole pairs, the upward band-diagram arrow. In the depletion region the built-in field drives electrons towards n material and holes towards p material; diffusion also delivers carriers generated within a diffusion length outside that region. Selective contacts collect the two signs. Illumination maintains a nonequilibrium quasi-Fermi-level splitting, allowing a voltage and useful current through an external load; it is the light, rather than the equilibrium barrier, that supplies the energy.

Taking forward diode current as positive, an ideal illuminated junction haspower is delivered where $VI<0$ in this convention. Reflection, below-gap transmission, carrier thermalization, recombination and resistive losses limit real conversion; even an ideal single junction has radiative detailed-balance limits.^[ref]

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 ${\epsilon }_s$, charge neutrality gives $N_D{x}_n=N_A{x}_p$. Poisson's equation then givesPositive 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 $e^{qV/k_{B}T}$. 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 yieldsThis 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.^[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.

A p-n junction brings p-type and n-type regions into contact. Majority-carrier diffusion initially leaves exposed ionized acceptors and donors, producing a depletion region and a built-in potential. At thermal equilibrium the electrochemical potential is constant: drift and diffusion currents cancel. The internal barrier is not a freely extractable battery voltage.

For a nondegenerate abrupt homojunction,Here $V>0$ denotes forward bias and the depletion approximation is assumed. forward bias lowers the barrier and injects minority carriers; reverse bias widens the depleted region and suppresses ordinary majority injection. In the ideal diffusion regime,Recombination can change the ideality factor; series resistance, high-level injection, leakage and breakdown limit this simple Shockley diode equation. The original junction theory explicitly treats both electron and hole currents, which is essential to carrier injection and transistor action.^[ref]

As a rectifying diode, the junction conducts strongly in one bias direction and weakly in the other before breakdown. This permits power rectification, signal detection and electronic switching. Stored minority charge affects turn-off time, so a static exponential current-voltage relation alone does not specify switching speed. A junction capacitance $C_j\simeq {ϵ}_{s}A/W$ varies with reverse bias and can be used in a voltage-controlled capacitor; heavily doped or high-field junctions can instead exploit controlled tunnelling or avalanche breakdown.

In a photodiode, absorbed photons generate electron-hole pairs and the depleted-region field assists collection. reverse bias widens the collecting region and can improve speed, but adds dark-current noise and breakdown constraints. In a solar cell, the same carrier generation supplies current against an external voltage. An ideal illuminated characteristic is $I=I_s(e^{qV/(k_{B}T)}-1)-I_{\mathrm{ph}}$: at open circuit the forward diode current balances photocurrent, while useful power is taken at an intermediate operating point. Neither short circuit nor open circuit delivers power to a load.

A forward-biased light-emitting diode injects electrons and holes that can recombine radiatively. Efficient emission favors suitable direct-gap materials and carrier confinement; the characteristic photon energy is near the semiconductor band gap. nonradiative recombination, escape losses, series resistance and heating limit efficiency. Two coupled junctions also form a bipolar junction transistor: an emitter injects minority carriers through a thin base, and a reverse-biased collector collects them. Useful current gain depends on efficient injection and limited base recombination, not simply on joining any three arbitrary regions.

These functions are experimentally distinguished by measuring dark and illuminated current-voltage curves, bias-dependent capacitance, transient switching or optical emission, with controlled temperature and current limits. The common mechanism is bias-controlled carrier injection or collection at a depletion barrier; the device geometry and operating mode determine which function is useful.

Semiconductor doping introduces impurities with levels close to the band edges. A donor impurity can release an electron to the conduction band, leaving a fixed positive ion; an n-type semiconductor has electrons as majority carriers. An acceptor impurity can take an electron from the valence band, leaving a mobile hole and fixed negative ion; a p-type semiconductor has holes as majority carriers. The opposite species remains as a minority carrier population, and a neutral bulk sample balances the ionised dopants with mobile carriers.

At a p-n junction, concentration gradients initially drive electrons and holes across the join. Their removal near the join exposes negative acceptor ions on the p side and positive donor ions on the n side, forming a depletion region. Its electric field opposes further majority-carrier diffusion. Equilibrium has a flat Fermi level, no net current and spatially bent band edges. With p on the left and n on the right, the electron band edges fall towards the n side; the difference is the equilibrium barrier in electron energy. A flat Fermi level does not mean flat conduction and valence edges.

Doping changes the mobile carrier population by introducing controlled impurity levels. In an intrinsic semiconductor, the filled valence band is separated from the nearly empty conduction band by a band gap $E_g$. Thermal excitation produces an electron-hole pair, so the electron and hole number densities obey $n=p=n_i$. A electron hole represents a missing valence-band electron and behaves as a mobile positive charge carrier. For nondegenerate bands in thermal equilibrium,Here $N_C,N_V$ are the effective density of states, $E_C,E_V$ the band edges, and $E_F$ denotes the chemical potential, conventionally called the Fermi level. The semiconductor mass-action law holds in this nondegenerate equilibrium regime.

In a tetrahedrally bonded elemental semiconductor such as silicon, a substitutional group-V atom has an extra outer electron relative to the host. It is a donor impurity: a shallow level just below $E_C$ can ionise, releasing an electron to the conduction band and leaving a fixed positively charged donor. The resulting n-type semiconductor has electrons as majority carriers and holes as minority carriers. Conversely a group-III substitution supplies one fewer bonding electron. An acceptor impurity can take an electron from the valence band, leaving a mobile electron hole and a fixed negative acceptor; the resulting p-type semiconductor has holes as majority carriers.

The impurity state is shallow because the electron or hole responds to a screened Coulomb potential and an effective mass, rather than the bare atomic potential. A simple hydrogenic donor estimate iswhere ${ϵ}_r$ is the relative permittivity. This is a useful scaling model, with real band anisotropy and central-cell effects omitted. Dopants are fixed in the lattice; their ionisation changes mobile carrier numbers without making a uniformly doped bulk sample carry a large net electric charge.

The charge neutrality condition isWhen shallow dopants are almost fully ionised and $N_D-N_A\gg n_i$, an n-type region has $n\simeq N_D-N_A$ and $p\simeq n_i^2/n$. A p-type region similarly has $p\simeq N_A-N_D$ and $n\simeq n_i^2/p$. Thus compensation by the opposite impurity type reduces the majority-carrier density; the total dopant concentration is not generally the mobile carrier density. At sufficiently high impurity density the degenerate semiconductor regime requires the full Fermi-Dirac distribution and these exponential approximations fail.

There are three useful temperature regimes. At low temperature, dopant freeze-out leaves many shallow levels un-ionised. In the extrinsic regime, the dopants are largely ionised and set the majority-carrier density. At high enough temperature, intrinsic pair creation dominates and $n,p$ approach $n_i$. The Fermi level moves towards the conduction band with donor doping and towards the valence band with acceptor doping. The electrical conductivity isDoping therefore controls conductivity and its carrier type, but ionised impurities also scatter carriers and can reduce charge carrier mobility. Carrier density and mobility must both be considered.

A p-n junction forms when p-type and n-type regions are joined within a semiconductor. Initially the concentration gradients drive diffusion current: electrons diffuse from the n region to the p region and holes in the opposite direction. Recombination removes mobile majority carriers near the join, exposing fixed positive donors on the n side and fixed negative acceptors on the p side. This forms a depletion region and an electric field directed from n to p. Its drift current opposes diffusion. At equilibrium the net current vanishes and the Fermi level is uniform, although the band edges vary spatially through band bending.

For an abrupt nondegenerate junction with fully ionised dopants, its built-in potential isUnder the depletion approximation, the space-charge densities are $-e{N}_A$ on a p-side width $w_p$ and $+e{N}_D$ on an n-side width $w_n$. Integrating Poisson equation and imposing zero field outside gives $N_A{w}_p=N_D{w}_n$ andMost of the depletion width therefore extends into the more lightly doped side. The schematic shows the impurity levels before contact and the resulting space charge and potential after contact; there is no externally extractable equilibrium battery voltage from the built-in field alone.

A forward bias, with p positive relative to n, reduces the barrier from $V_{\mathrm{bi}}$ towards $V_{\mathrm{bi}}-V$ and injects minority carriers across the junction. In the ideal low-injection, diffusion-controlled regime this produces the Shockley diode equation, $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias increases the barrier and the depletion region; only a small thermally generated minority-carrier current flows before breakdown. Reverse-biased depletion regions also give a voltage-dependent junction capacitance. Real devices have recombination, leakage, series resistance and breakdown mechanisms beyond the ideal law. Controlled doping creates majority and minority carriers, and their junction produces a depletion barrier that makes rectification possible.

Take an abrupt equilibrium p-n junction with the p side at $x<0$ and the n side at $x>0$. In the quasi-neutral bulk, electron holes are the majority carriers on the p side and electrons are the majority carriers on the n side. With ionized dopant densities $N_A,N_D$ and nondegenerate statistics,where $n_i$ is the intrinsic carrier concentration. The semiconductor mass-action law $np=n_i^2$ applies at equilibrium. Both mobile populations are small in the depletion region compared with the dopant densities.

The charge and carrier density of an abrupt p-n junction must distinguish mobile populations from net charge. Diffusion leaves negatively charged ionized acceptor impurities on the p side and positively charged ionized donor impurities on the n side. Under the depletion approximation,The neutral bulks have large carrier densities but nearly zero net charge density. The electric field points from the positive donor region toward the negative acceptor region, balancing diffusion at equilibrium.

The figure uses equal dopant densities only to show a simple symmetric sketch. Generally the more lightly doped side has the wider depleted interval. The carrier profiles are smooth, whereas the step-shaped net charge density is the ideal depletion approximation.

Semiconductor doping introduces controlled impurities to alter equilibrium carrier populations and electrical conductivity. In an intrinsic semiconductor, thermally excited electrons in the conduction band and electron holes in the valence band have equal densities $n=p=n_i$. A shallow donor impurity, such as phosphorus in silicon, supplies an electron from an impurity level just below the conduction-band edge. Once ionized it is positively charged, and the material is n type. An acceptor impurity, such as boron in silicon, accepts a valence electron into a level just above the valence-band edge, leaving a mobile hole and a negatively charged ionized impurity; the material is p type.

With dopant densities $N_D,N_A$, charge neutrality requiresFor fully ionized, nondegenerate dopants at equilibrium, the semiconductor mass-action law gives $np=n_i^2$. In a substantially n-type sample, $N_D-N_A\gg n_i$, soThe corresponding p-type result is $p\simeq N_A-N_D$ and $n\simeq n_i^2/(N_A-N_D)$. The majority population grows while the minority population decreases; adding donors does not simply add equal numbers of free electrons and holes.

The Fermi energy shifts toward the conduction band for n-type material and toward the valence band for p-type material. In the nondegenerate approximation,where $N_C,N_V$ are effective band-edge state densities. At very low temperature, dopants may not be ionized and carrier freeze-out occurs. At intermediate temperatures, ionized dopants dominate the carrier density; at sufficiently high temperature, intrinsic excitation can dominate. Heavy doping can produce degenerate carrier statistics, invalidating these Boltzmann expressions.

The electrical conductivity is $\boxed{\sigma =e(n{\mu }_e+p{\mu }_h)}$. Doping usually raises the majority carrier density, but ionized impurity scattering can lower mobility, so conductivity need not scale indefinitely with dopant concentration. Joining p-type and n-type regions creates a p-n junction, with a depletion region and built-in electric field established by carrier diffusion and compensating drift. The fixed ionized dopants, not an absence of all charge, account for the depleted region's net charge density.

Take a p-n junction made from one semiconductor, with the p-type semiconductor on the left and the n-type semiconductor on the right. At thermal equilibrium the chemical potential $\mu =E_F$ is horizontal throughout. Before contact the chemical potential lies nearer the valence band on the p side and nearer the conduction band on the n side; after contact the band edges bend to permit one common value.

In this orientation both the conduction band edge $E_C$ and the valence band edge $E_V$ decrease from p to n by $e{V}_{\mathrm{bi}}$. For a homojunction their separation $E_g=E_C-E_V$ is constant. The acceptor impurity level $E_A$ lies just above $E_V$ where acceptors are present, and the donor impurity level $E_D$ lies just below $E_C$ where donors are present. These impurity levels follow the local band bending; they do not form one additional horizontal band across the device.

The shaded depletion region has few mobile electrons or electron holes and contains fixed negative ionized acceptor impurities on the p side and positive ionized donor impurities on the n side. Thus its electric field points from n to p. Its neutral outer regions have negligible electric field, so the band edges become flat there. In the depletion approximation, charge neutrality requires $N_A{x}_p=N_D{x}_n$; the symmetric drawing illustrates $N_A=N_D$ rather than assuming that equality is universal. Diffusion and field-driven drift current cancel at equilibrium: no applied voltage does not mean zero internal electric field or zero built-in electric potential difference.

In a p-n junction, diffusion and electron-hole recombination near the contact leave a depletion region of ionized dopants. Its built-in electric field opposes further majority-carrier diffusion. At thermodynamic equilibrium drift and diffusion currents cancel. Forward bias lowers the electrostatic barrier and injects electrons and electron holes across the junction; reverse bias raises the barrier and suppresses that majority-carrier injection. For an ideal diode below breakdown,Thus a modest forward voltage gives a rapidly increasing current, whereas reverse current remains approximately $-I_s$. This asymmetric current-voltage relation makes the junction a rectifier. Real leakage, electron-hole recombination and breakdown limit the ideal expression.

A p-n junction joins p-type and n-type semiconductors. Initial carrier diffusion exposes ionized acceptors and donors, creating a depletion region and an internal electric field. At equilibrium the Fermi level is uniform and drift balances diffusion, so the built-in potential produces no net terminal current. For nondegenerate abrupt doping, its magnitude is $V_{\mathrm{bi}}=(k_{B}T/e)\ln (N_A{N}_D/n_i^2)$.

Forward bias lowers the barrier and injects minority carriers across the junction. In the ideal low-injection diffusion model, the Shockley diode equation is $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias broadens the depletion region and leaves a small leakage current until breakdown mechanisms become important.

The asymmetric current-voltage curve enables rectification. Illumination produces electron-hole pairs that the junction field can separate, giving a photodiode or solar cell. Forward injection and radiative recombination in a suitable direct-gap material give a light-emitting diode. Measuring dark and illuminated current-voltage curves distinguishes electrical injection from photogenerated current; an ideal junction need not have zero reverse leakage or identical behavior in every material.