# Doping (semiconductor) **Doping** is the deliberate introduction of impurity atoms into a [[Semiconductor|semiconductor]] in order to fix how many mobile charge carriers it has and what sign they carry. A pure [[Silicon|silicon]] crystal at room temperature is a poor conductor, with about ten billion carriers per cubic centimetre against a metal's ten thousand billion billion; one [[Phosphorus|phosphorus]] atom per million silicon atoms raises that by six orders of magnitude.[^derived-dop][^openstax-v3-96] Because the added carriers are placed by the maker rather than by chance, doping is what makes a semiconductor an engineering material. In the microsim below the reader sets a donor density N_d and a [[Temperature|temperature]] and watches the gap opened by the [[Electronic_band_structure|band-structure]] sim fill with carriers. Two equations answer. The intrinsic density is `n_i = sqrt(N_c·N_v)·exp(-E_g/(2·k·T))`, about 10¹⁰ per cubic centimetre for silicon at 300 K; once donors are added, charge neutrality and the law of mass action give `n ~ N_d` and `p = n_i^2/N_d`, and the [[Fermi_level|Fermi level]] is pulled from mid-gap toward the conduction-band edge.[^spec-m38][^likharev-sm6] The conductivity readout is `sigma = e·(n·mu_n + p·mu_p)` against temperature, so the three regimes appear in order: freeze-out, a broad extrinsic plateau where every dopant is ionized, and an intrinsic runaway where thermal pairs swamp the dopants.[^spec-m38] On the [[Materials_science]] flagship this article serves the *Semiconductors and doping* section of Part VII — Research, a sibling of the shared C31 set: it takes the gap that [[Electronic_band_structure|electronic band structure]] opens and [[Band_gap|band gap]] measures, and fills it. ## History Doping is older than its theory. Early crystal rectifiers — a metal whisker pressed onto galena or silicon — worked erratically, and their behaviour depended on which part of the crystal the whisker touched. The explanation, that trace impurities were setting the local carrier type and density, could not be acted on until two things existed: a band theory explaining why a few foreign atoms in a million change conduction by a factor of a million, and a purification technology that made "a few" mean a few.[^openstax-v3-96] Zone refining and the [[Czochralski_method|Czochralski]] pulling of single [[Crystal_growth|crystals]] supplied the second, reducing unintended impurities below the intended ones and making the dopant concentration a number the maker chose. The point-contact [[Transistor|transistor]] reported by John Bardeen and Walter Brattain in 1948 was the result, and the junction devices that followed turned doping into the central operation of an industry.[^bardeen1948] Every later development — the planar process, ion implantation, shallow junctions, [[Semiconductor_device_fabrication|device fabrication]] at nanometre scale — has been a way of placing dopant atoms more precisely, in smaller volumes, with sharper boundaries. ## Carrier concentration In an intrinsic semiconductor every conduction electron leaves a hole behind, so n = p = n_i: N_c and N_v set the prefactor, and the [[Band_gap|band gap]] E_g over twice the thermal energy kT sets the exponent.[^spec-m38][^likharev-sm6] The exponent makes the numbers extreme. For silicon, E_g/2 = 0.56 eV against kT = 25.9 meV at 300 K, a ratio of 21.7, so the exponential alone is 4×10⁻¹⁰.[^derived-dop] Doping replaces the exponential with arithmetic. A donor contributes an electron to the conduction band without leaving a hole, so charge neutrality gives n ≈ N_d whenever N_d exceeds n_i, while the product np = n_i² is fixed by generation and recombination, so the minority density collapses to p = n_i²/N_d. At N_d = 10¹⁶ per cubic centimetre that is n = 10¹⁶ and p = 10⁴, a ratio of 10¹² between the two carrier populations in one crystal.[^derived-dop] The Portal Book's table gives pure silicon a resistivity of 2.3×10³ Ω·m; raising n by the factor 10⁶ lowers it to about 2×10⁻³ Ω·m, and the crystal has crossed six of the twenty-five decades separating [[Copper|copper]] from PTFE.[^openstax-93][^derived-dop] An acceptor does the mirror image, pinning p ≈ N_a. *Try:* hold N_d at 10¹⁶ and sweep temperature. Below about 50 K the curve bends down as donors recapture their electrons; to several hundred kelvin it is almost flat, n pinned at N_d; then it turns up as n_i overtakes N_d and the crystal forgets it was doped. ## Effect on band structure A dopant does not change the bands; it adds a level inside the gap. A donor is a substitutional atom with one more valence electron than the host, and that extra [[Electron|electron]] is bound to the now positively charged core by a screened Coulomb attraction — a hydrogen atom built inside the crystal, with the free-space mass replaced by the conduction-band effective mass and the vacuum permittivity by the host's. The hydrogenic estimate E_d = 13.6 eV·(m*/m)/ε_r², with silicon's effective mass of 0.26 electron masses and a relative permittivity near 12, gives about 25 meV: a level just below the conduction-band edge, of the same order as kT at room temperature.[^likharev-qm-meff][^derived-dop] That coincidence is the whole design. The level is shallow enough that thermal energy empties it at 300 K, so every donor delivers its electron, and deep enough that at 50 K, where kT is 4.3 meV, most do not.[^derived-dop] Heavy doping changes the picture. When dopant atoms are close enough that their hydrogenic orbits overlap, the discrete level broadens into an impurity band, and at high enough density that band merges with the host band and the edge itself shifts — band-gap narrowing. A semiconductor doped past that point conducts at any temperature and is called degenerate: its [[Fermi_level|Fermi level]] has moved inside a band, and it behaves like a poor [[Metallic_bonding|metal]]. ### Relationship to carrier concentration (low doping) At low doping the Fermi level's position follows directly from the carrier density. Writing the intrinsic level E_i for mid-gap, the displacement is `E_F − E_i = k·T·ln(n/n_i)`, so every decade of doping moves the Fermi level by kT·ln 10, which is 60 meV at 300 K.[^derived-dop] Doping silicon to 10¹⁴ per cubic centimetre moves it 238 meV above mid-gap; 10¹⁶ moves it 357 meV, to within about 200 meV of the conduction-band edge; 10¹⁸ moves it 476 meV and has nearly reached the edge, which is where the low-doping formula stops being valid and degeneracy begins.[^derived-dop] The sim draws this as a line sliding up the gap while the donor slider moves, and the same displacement is what sets the built-in voltage of a [[P–n_junction|p–n junction]] when doped regions of opposite type are joined: the two Fermi levels must align, and the band edges bend by the difference the two dopings imply. ## Techniques of doping and synthesis Dopants are introduced either while the crystal is being formed or afterwards, and the choice sets how sharply the doped region can be bounded. Growth doping reaches the whole volume but cannot be patterned; post-growth doping can be patterned to a mask but reaches only as deep as diffusion or an ion beam carries it. Every method is judged on the same four counts: the concentration it reaches, the uniformity it holds across a wafer, the sharpness of the boundary it leaves, and the lattice damage it does on the way in. ### Doping during crystal growth The simplest route adds the dopant to the melt, so the growing [[Crystal_structure|crystal]] incorporates it as it solidifies. The difficulty is segregation: a dopant is more soluble in the liquid than in the solid, so the first material to freeze is poorer in dopant and the melt is steadily enriched, leaving a boule whose doping varies along its length. The same effect run deliberately and repeatedly is zone refining, which sweeps impurities to one end of an ingot and is how device-grade silicon is purified in the first place. Epitaxial growth avoids the problem by adding dopant to the gas stream, so concentration changes abruptly between layers. ### Post-growth doping Two methods dominate after the crystal exists. In [[Diffusion|diffusion]] doping the wafer is held in a dopant-rich atmosphere at high temperature and the impurity diffuses in from the surface, giving a smooth profile whose depth grows as the square root of time and whose shape cannot be chosen apart from its depth. In ion implantation dopant ions are accelerated to tens or hundreds of kiloelectronvolts and fired into the wafer; dose and energy set concentration and depth separately, and masking is exact. The cost is damage: implantation displaces host atoms, and an [[Annealing_(materials_science)|anneal]] is needed both to repair the lattice and to move dopants onto substitutional sites, where alone they are electrically active. ### Spin-on glass A dopant-bearing silicate solution is spun onto the wafer as a thin film, baked to a glass and driven in by a high-temperature step, after which the glass is etched away. The method needs no implanter and no dopant gas handling, and the surface concentration is set by the chemistry of the film rather than a beam current, which suits large-area work such as [[Solar_cell|solar cells]]. ### Neutron transmutation doping The only method that dopes uniformly through a thick crystal turns host atoms into dopants in place. A silicon ingot is irradiated in a [[Nuclear_reactor|nuclear reactor]]; the [[Isotope|isotope]] silicon-30 captures a [[Neutron|neutron]] to become silicon-31, which beta-decays to [[Phosphorus|phosphorus]]-31 — the beta emission raising the atomic number by one while the mass number stays put.[^ball-nuclear] Because thermal neutrons pass through silicon with little attenuation and the isotope is distributed uniformly by nature, the phosphorus distribution is uniform to a degree no diffusion or implantation can match. The method is slow and leaves the ingot radioactive until short-lived activity decays, so it is kept for high-power devices, where non-uniform doping would make one part of a large area carry the current. ## Dopant elements A dopant must sit substitutionally on a host site, be electrically active there, and have a level shallow enough to ionize at the operating temperature. Those conditions narrow the periodic table sharply: for any host only a handful of elements qualify, and the ones in routine use are chosen as much for how they behave during processing — solubility, diffusivity, whether they stay put through a later anneal — as for the electrical result, which is nearly the same whichever is used. ### Group IV semiconductors For [[Silicon|silicon]] and [[Germanium|germanium]], each host atom contributes four valence electrons to four bonds, so the neighbouring columns supply the dopants: group V atoms have one electron too many and act as donors, group III atoms one too few and act as acceptors. The size of the effect hardly depends on which element within the column is chosen, because the bound state is hydrogenic and cares about the host's effective mass and permittivity rather than the impurity's identity; what differs is solubility, diffusivity and the exact depth of the level. ### Silicon dopants The donors in routine use are [[Phosphorus|phosphorus]], [[Arsenic|arsenic]] and [[Antimony|antimony]]; the acceptor is almost always [[Boron|boron]]. The choice among donors is made on transport, not electrical behaviour: phosphorus diffuses quickly and suits deep, lightly doped regions; arsenic diffuses slowly and is preferred where a junction must stay shallow through later heat treatment. [[Gold|Gold]] and other deep-level impurities are sometimes added on purpose, not to supply carriers but to remove them, giving recombination centres that shorten carrier lifetime where fast switching matters. ### Other semiconductors In a compound semiconductor there are two sublattices, and a dopant's behaviour depends on which it occupies. A group IV atom on the group III site of a III–V compound is a donor and on the group V site an acceptor, so one element can do either job — amphoteric doping, controlled by the ratio of the two species during growth. Wide-gap materials bring a harder problem: it is often easy to dope them one way and nearly impossible the other, because native [[Vacancy_defect|vacancies]] and antisites form spontaneously and compensate the intended dopant as fast as it is added. ## Compensation A crystal usually contains both donors and acceptors at once, and only the difference between them counts. Electrons from donors fall into acceptor levels, so the free-carrier density is |N_d − N_a| while the scattering follows the total N_d + N_a: a compensated crystal has the carrier density of a lightly doped one and the [[Electron_mobility|mobility]] of a heavily doped one. Compensation is a nuisance when accidental, as in the wide-gap materials above, and a tool when deliberate — counter-doping converts an n-type region to p-type without removing the original dopant, which is how successive implants build a device from one wafer. It also sets a practical floor on how lightly a region can be doped, since the intended concentration must exceed the unintended one — which is why the starting wafer's background doping, not the implanter, fixes the lightest region a process can make. ## Doping in conductive polymers The word carries over to [[Polymer|polymers]] with a change of mechanism. A conjugated chain is doped not by substitution but by oxidation or reduction: an electron is removed from, or added to, the chain, and a counter-ion moves in to preserve neutrality. The charge is not a free carrier in the crystalline sense but a local distortion of the chain carrying charge with it, moving along a chain readily and between chains with difficulty. Doping levels are enormous by inorganic standards — a charge every few repeat units rather than one per million atoms. ## Doping in organic molecular semiconductors In a molecular solid the dopant is a second molecule chosen for its electron affinity or ionization energy, and doping is a charge transfer between the two species where they touch. Because the molecules are held by weak interactions rather than covalent bonds, the dopant can diffuse and aggregate, and stability rather than carrier density is the limiting problem. Efficiency is low: only a fraction of the added molecules transfer charge, and much of what is transferred stays bound to the ionized dopant. ## Magnetic doping A dopant carrying a magnetic moment adds spin to the problem. Substituting a fraction of the cations with a transition-metal ion gives a dilute magnetic semiconductor, in which the localized moments couple through the carriers the doping itself supplies, so the ordering temperature depends on carrier density and can be altered electrically. The attraction is one crystal carrying both charge and [[Spin_(physics)|spin]] information; the obstacle is that the transition temperatures reached are well below room temperature, and that the magnetic ions tend to precipitate as a separate [[Ferromagnetism|ferromagnetic]] phase whose signal is easily mistaken for the intended effect. ## Single dopants in semiconductors As devices shrink, the dopant atoms in the active volume become few enough to count, and their statistical fluctuation becomes a design constraint: two nominally identical transistors differ because one received a few more dopants. Turned around, the same limit is an opportunity. A single donor in silicon is a hydrogenic system with a well-defined ground state, addressable by an [[Electric_field|electric field]] and readable through the current of a nearby channel — the endpoint of a technology whose whole history has been the placing of impurity atoms more precisely. ## Modulation doping Doping supplies carriers and, in the same act, the ionized impurities that scatter them, which puts a ceiling on [[Electron_mobility|mobility]]. Modulation doping separates the two. The dopants sit in a wide-gap layer and the carriers they release fall into an adjacent narrow-gap layer, where a thin undoped spacer keeps them apart from the ions they came from. The result is a two-dimensional electron gas with carrier densities set by doping and mobilities far above what that density allows in bulk, which is what makes high-electron-mobility transistors possible. ## See also - [[Intrinsic_semiconductor]] — the n = p = n_i starting point - [[Extrinsic_semiconductor]] — the doped case - [[Electron_mobility]] — the μ in the conductivity readout - [[Fermi_level]] — what the donor slider moves - [[Semiconductor]] - [[Electronic_band_structure]] — the parent sim, which opens the gap - [[Band_gap]] — the E_g in the exponential - [[P–n_junction]] ## References [^openstax-v3-96]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, "Condensed Matter Physics" (pp. 393–440), §9.6 Semiconductors and Doping: donors and acceptors, impurity levels in the gap, and the n-type/p-type classification (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^likharev-sm6]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 6, "Elements of kinetics" (pp. 187–225), §6.4, carriers in semiconductors: the intrinsic density, the law of mass action np = n_i², and the drift conductivity of a two-carrier system (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^openstax-93]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 9, "Current and Resistance" (pp. 373–416), §9.3, Table 9.1 Resistivities and Temperature Coefficients: pure silicon 2.3×10³ Ω·m at 20 °C (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^likharev-qm-meff]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, effective mass `1/m_ef = (1/hbar^2)*d2E/dq2`; silicon 0.26 mₑ in the conduction band and 0.39 mₑ in the valence band, InSb down to 0.0145 mₑ (pp. 87–89). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^bardeen1948]: Bardeen, John; Brattain, Walter H. (1948). "The Transistor, A Semi-Conductor Triode." *Physical Review* 74. Pages and DOI to pin. [^ball-nuclear]: Ball, David (2011). *Introductory Chemistry*. Chapter 15, "Nuclear Chemistry" (pp. 720–764): neutron capture, beta decay and the change in atomic number it produces (pp. 729–739; page to pin). https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^spec-m38]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M38: new sibling of `Electronic_band_structure`, the gap of row M37 filled with carriers — `n_i = sqrt(N_c·N_v)·exp(-E_g/(2·k·T))` with silicon at 10¹⁰ cm⁻³ and 300 K, a donor density N_d as the control pinning `n ~ N_d` and `p = n_i^2/N_d` while E_F is pulled toward the band edge, and `sigma = e·(n·mu_n + p·mu_p)` read out over temperature through the freeze-out, extrinsic and intrinsic regimes. [^derived-dop]: Computed for this article from the equations on the page, with k = 8.617×10⁻⁵ eV/K, E_g = 1.12 eV and n_i = 10¹⁰ cm⁻³ for silicon at 300 K as given by [^spec-m38]. kT = 25.9 meV at 300 K, 8.6 meV at 100 K, 4.3 meV at 50 K; E_g/(2kT) = 21.7 at 300 K, so the exponential factor is 4×10⁻¹⁰. At N_d = 10¹⁶ cm⁻³, p = n_i²/N_d = 10⁴ cm⁻³ and n/p = 10¹². Fermi-level displacement kT·ln(n/n_i): 238 meV at 10¹⁴, 357 meV at 10¹⁶ and 476 meV at 10¹⁸ cm⁻³, against a half-gap of 560 meV; one decade of doping is kT·ln 10 = 60 meV. Resistivity: 2.3×10³ Ω·m from [^openstax-93] divided by the carrier-density factor 10⁶ gives ≈2×10⁻³ Ω·m, ignoring the electron/hole mobility split, which is a factor of order one. The intrinsic density rises by 225 between 300 K and 400 K from the exponential alone, and by about 350 once the T^(3/2) prefactor of N_c·N_v is included, matching the figure quoted on [[Electrical_resistivity_and_conductivity]]. ILLUSTRATIVE: the hydrogenic donor estimate 13.6 eV·(m*/m)/ε_r² = 25 meV uses m* = 0.26 mₑ from [^likharev-qm-meff] with a relative permittivity taken as 12, a round value not read from a Portal Book; only the order of magnitude and its comparison with kT are claimed. ## External links - The Wikipedia pair's *External links* section lists dopant tables, implantation range calculators and process references; the Portal Book chapters above are the sources of this page. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Doping_(semiconductor).json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Doping (semiconductor)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Doping_(semiconductor).html" data-title="Doping (semiconductor)"></div> *Built from `MICROSIM_GUIDE/specs/sims/Doping_(semiconductor).json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Doping_(semiconductor)) : [Wikitube](https://en.wikitube.io/wiki/Doping_(semiconductor)) · pinned revision [1369380683](https://en.wikipedia.org/w/index.php?oldid=1369380683) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M38 · sim pending (matter/Doping_(semiconductor)).*