Semiconductor doping

The controlled introduction of impurities to change a semiconductor's charge carrier populations. Donor impurities and acceptor impurities can produce an n-type semiconductor or a p-type semiconductor, respectively; compensation and temperature also matter.

The number of dopant sites per volume. For a roughly homogeneous three-dimensional distribution, the characteristic site separation is of order $n^{-1/3}$. This counts introduced sites rather than automatically counting free conduction band electrons: semiconductor carrier freeze-out or compensation can make the mobile carrier density much smaller.

An energy band formed when localized impurity levels hybridize. For hydrogenic donor states, a typical donor separation $n^{-1/3}$ gives an exponentially small hopping integral and hence an exponentially narrow bandwidth when $n^{-1/3}\gg a_b$. An idealized half-filled band with two spin states per donor has spin-summed density of states of order $2n/W$ per volume; its Pauli paramagnetism is $\chi \sim {\mu }_0(g{\mu }_B/2{)}^2(2n/W)$. That expression requires $k_{B}T\ll W$ and does not by itself describe localization or strong Coulomb interactions.

Before using a Maxwell–Boltzmann chemical-potential formula in a doped semiconductor, compare the carrier density to its conduction-band effective density of states. When n is much larger than $N_c$, a nondegenerate logarithmic estimate is inconsistent. Use Fermi–Dirac statistics or a controlled degenerate expansion instead. A small effective mass strongly reduces $N_c$ and raises the Fermi energy at fixed density, so even an otherwise familiar donor concentration can put the chemical potential inside the conduction band.

At sufficiently low temperature, thermal ionization of shallow dopants becomes inefficient, reducing the mobile carrier density. An acceptor-doped sample may then have many neutral acceptors and few mobile electron holes. Extracting an activation energy requires a model of dopant ionization and compensation; the mobile density need not have the same simple activation exponent in every compensation regime.

Introduction of dopants by accelerating ions into a solid. Their depth distribution, lattice damage and electrical activation depend on implantation and subsequent annealing.

Modulation doping places ionized dopants away from the region in which their supplied charge carriers move. In a semiconductor heterostructure, band alignment can transfer electrons from a doped layer into a nearby lower-energy channel.

This separation reduces scattering from nearby charged impurities while maintaining a substantial mobile-carrier population. It supports a two-dimensional electron gas and the channel of a high-electron-mobility transistor. Separation is not complete decoupling: the donors still set electrostatics, and remote impurities, interfaces and other scattering mechanisms can still limit charge carrier mobility.

A two-dimensional electron gas supplied by remote donors in aluminium gallium arsenide and confined on the gallium arsenide side of a heterojunction. An undoped spacer separates ionized donors from mobile carriers, reducing impurity scattering. A conduction-band band offset and self-consistent band bending create the confinement well.

A compensated semiconductor contains both donor impurities and acceptor impurities. In a nondegenerate equilibrium model with both species fully ionized, charge neutrality gives $n-p=\Delta =N_D-N_A$, while the semiconductor semiconductor mass-action law gives $np=n_i^2$. Solving yieldsFor $\Delta >0$ and $n_i\ll \Delta$, $n\simeq \Delta$ and $p\simeq n_i^2/\Delta$. Numerically the minority concentration is better obtained as $n_i^2/n$ than by subtracting nearly equal numbers. If dopants are only partly ionized, replace $N_D,N_A$ in charge neutrality by their ionized concentrations and determine the chemical potential self-consistently.

An amphoteric dopant can act as a donor impurity or an acceptor impurity according to its incorporated lattice site or defect configuration. Silicon in gallium arsenide is the standard example: substitution on the gallium sublattice gives a shallow donor, while substitution on the arsenic sublattice gives a shallow acceptor. Its electrical effect therefore depends on growth conditions as well as total silicon concentration.^[ref]

For a nondegenerate semiconductor with donor density $N_D$, no acceptors, and donor energy $E_D=E_c-E_b$, an occupied donor has two electron spin states while an empty donor has one. Its ionized fraction is therefore $N_D^{+}/N_D=[1+2e^{(E_F-E_D)/(k_{B}T)}{]}^{-1}$. The conduction-band relation $n=N_c{e}^{(E_F-E_c)/(k_{B}T)}$ gives $N_D^{+}=N_D/[1+(2n/N_c)e^{E_b/(k_{B}T)}]$. Combining charge neutrality, $n-p=N_D^{+}$, with the semiconductor mass-action law, $np=n_i^2$, proves the displayed equation.

Its left side increases strictly for $n>0$ and its right side decreases, so there is one physical solution, with $n\ge n_i$. In the freeze-out regime, holes are negligible and solving the quadratic gives $n=2N_D/[1+\sqrt{1+(8N_D/N_c)e^{E_b/(k_{B}T)}}]$. Strong dopant freeze-out reduces this to $n\simeq \sqrt{N_D{N}_c/2}{e}^{-E_b/(2k_{B}T)}$. An extrinsic plateau requires $(2N_D/N_c)e^{E_b/(k_{B}T)}\ll 1$ and $n_i\ll N_D$, not simply $k_{B}T\gg E_b$.

The comparison $E_B\ll k_{B}T$ alone does not establish full ionization for arbitrary donor concentration. Localized-level occupation depends on the chemical potential through charge neutrality. If donor orbit radii greatly exceed their separation, an independent isolated-hydrogenic-level model itself fails and overlapping impurity states or band reconstruction are needed. A stipulated extended-carrier density can still define a conductivity calculation, but it must not simultaneously be interpreted as an independently verified dilute-donor equilibrium result.

Substitutional semiconductor doping replaces selected host atoms with impurity atoms on ordinary lattice sites. A shallow donor impurity can supply a mobile conduction electron and remain positively charged; a shallow acceptor impurity can capture an electron and leave a mobile valence hole, remaining negatively charged. In nondegenerate, substantially ionized material, donor-dominated doping raises the chemical potential toward the conduction edge and acceptor-dominated doping lowers it toward the valence edge. Dopants do not create net bulk charge in a neutral sample: fixed impurity charge compensates the mobile carrier charge. Freeze-out, compensation and degeneracy qualify the simplest relation between dopant and carrier densities.

An intrinsic semiconductor has electron and hole densities $n=p=n_i$, so ${\sigma }_i=e{n}_i({\mu }_e+{\mu }_h)$. In a fully ionised, uncompensated n-type sample with $N_D\gg n_i$, $n\simeq N_D$, $p\simeq n_i^2/N_D$, andThus a carrier-density ratio alone is not generally the electrical conductivity ratio. If electron mobility is unchanged and electron transport dominates the intrinsic conductivity, it reduces to $N_D/n_i$; equal electron and hole mobilities instead give $N_D/(2n_i)$. Ionised-impurity scattering can change the doped mobility.

At sufficiently low temperature, donor or acceptor ionization becomes energetically unfavourable. Carriers remain bound to dopants rather than contributing to conduction, reducing the mobile carrier density. This dopant freeze-out is distinct from the high-temperature intrinsic regime, where thermal excitation across the band gap overwhelms the net dopant contribution.

An impurity level is an electronic energy level associated with a dopant or defect in a semiconductor. A shallow donor impurity introduces a level just below the conduction band, and a shallow acceptor impurity introduces one just above the valence band. Under electrostatic band bending, these levels shift with the local band edges in the same material; their occupation is determined relative to the equilibrium chemical potential.

A sequence of bound-to-bound optical transitions of a hydrogenic donor state or hydrogenic acceptor state, converging on a photoionization threshold $R^{\ast }$. For an initially occupied $1s$ state, ordinary electric dipole transitions reach $np$ states; above the threshold, the final carrier can enter a band continuum.

An n-type semiconductor has more mobile electrons than electron holes. The letter describes the predominant mobile charge carriers, not a net negative electric charge of the whole material.

A p-type semiconductor has more mobile electron holes than electrons; it can nevertheless be electrically neutral overall.

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.