# P–n junction **The p–n junction** is the boundary inside a single crystal of [[Semiconductor|semiconductor]] where a region doped to conduct by positive carriers, the p-type side, meets a region doped to conduct by [[Electron|electrons]], the n-type side. It is not a mechanical contact between two pieces of material: the lattice runs continuously through it, and only the [[Doping_(semiconductor)|doping]] changes.[^sze-ch2] Mobile carriers diffuse across the boundary and leave behind the fixed charges of the ionised dopant atoms, so a thin layer emptied of mobile carriers — the [[Depletion_region|depletion region]] — forms there, and with it a built-in [[Electric_field|electric field]] that drives carriers back. The result is a structure that passes current freely one way and almost not at all the other: the [[Diode|diode]]. The same boundary, repeated and shaped, is the working part of the [[Transistor|transistor]], the [[Solar_cell|solar cell]] and the [[Light-emitting_diode|light-emitting diode]].[^up3-devices] In the microsim below the reader drags one control, the applied bias `V`, from −1 V to +0.8 V, and a three-dimensional band diagram bends in response. Two equations answer. The depletion width `W = sqrt(2·eps·(V_bi - V)·(N_a + N_d)/(q·N_a·N_d))` narrows under forward bias and widens under reverse bias; the Shockley equation `I = I_s·(exp(q·V/(n·k·T)) - 1)` turns the current on exponentially as `V` approaches the built-in voltage `V_bi = (k·T/q)·ln(N_a·N_d/n_i^2)`, which for [[Silicon|silicon]] doped 10¹⁶ cm⁻³ on both sides at 300 K reads 0.72 V. The sim shows `V_bi`, `W` and `I` together, so the reader can see that the barrier the field makes and the current the diode passes are two readings of one quantity. On the [[Materials_science]] flagship this page serves Part VIII, Industry, in the section *Semiconductors*, where the crystal-growing and patterning pages — [[Czochralski_method|Czochralski growth]] and [[Photolithography|photolithography]] — meet the electrical behaviour they exist to produce. An [[Integrated_circuit|integrated circuit]] is billions of these boundaries, drawn by [[Semiconductor_device_fabrication|fabrication]] rather than assembled. ## Properties A pure semiconductor at room temperature is a poor conductor, because almost all its valence [[Electron|electrons]] are bound and the [[Band_gap|band gap]] — 1.12 eV in silicon, 0.66 eV in [[Germanium|germanium]] — is many times the thermal energy `k·T` of 0.026 eV.[^up3-ch9] Doping changes that. Replacing a few silicon atoms per million with a group-V element such as phosphorus leaves one electron per impurity too many for the bonds, and it sits in a shallow donor level a few tens of millielectronvolts below the conduction band, from which thermal energy frees it: the material becomes n-type and conducts by electrons. A group-III element such as boron leaves one electron too few, creating a mobile vacancy in the bonding — a hole — that behaves as a positive carrier: the material becomes p-type.[^up3-ch9][^likharev-sm6] In each case one carrier is in large majority and the other, thermally generated, is in small minority; their concentrations satisfy `n·p = n_i^2` at equilibrium, so doping one up drives the other down.[^sze-ch1] The junction is the place where these two doped regions meet within one crystal. Russell Ohl at Bell Telephone Laboratories found the effect in 1940 in a silicon rod that happened to contain an impurity boundary: illuminated, the rod produced a voltage across that boundary, and it conducted far better one way than the other.[^crystalfire] What the boundary does depends entirely on the voltage across it, and the pair's three subsections below are the three cases the microsim's single slider sweeps: no applied voltage, forward bias, reverse bias. ### Equilibrium (zero bias) With no external connection, electrons on the n-side face a steep concentration drop across the boundary and [[Diffusion|diffuse]] into the p-side; holes diffuse the other way. Each departing carrier leaves behind an ionised dopant atom locked into the lattice — a negative acceptor ion on the p-side, a positive donor ion on the n-side. Those fixed charges are not neutralised, so a dipole layer of space charge builds up, negative on the p-side and positive on the n-side, and its [[Electric_field|electric field]] points from n to p, opposing the diffusion that created it.[^sze-ch2] Equilibrium is reached when the field's drift current exactly cancels the diffusion current, separately for electrons and for holes, at every point. The net current is zero, but neither component is: the junction is in a balance of two large opposed flows, not at rest. The thermodynamic statement of that balance is that the [[Fermi_level|Fermi level]] is flat across the whole crystal; the bands must therefore bend to accommodate it, and the amount of bending is the built-in potential `V_bi = (k·T/q)·ln(N_a·N_d/n_i^2)` where `N_a` and `N_d` are the acceptor and donor concentrations, `n_i` the intrinsic carrier concentration and `k·T/q` the thermal voltage, 25.85 mV at 300 K.[^sze-ch2][^si-brochure] For silicon with `N_a = N_d = 10¹⁶ cm⁻³` and `n_i = 9.65×10⁹ cm⁻³`, the logarithm is ln(1.07×10¹²) = 27.7 and `V_bi` = 0.72 V (derived).[^sproul-green] The depletion region that carries this potential is 0.43 µm wide, and the peak field at the boundary, at the apex of a triangular field profile, is about 33 kV/cm (derived).[^sze-ch2] `V_bi` cannot be measured by touching a voltmeter to the two ends of the crystal. The metal contacts form their own junctions with the semiconductor, and around the closed loop the contact potentials sum to exactly cancel `V_bi` — as they must, since a current drawn from an isolated junction in the dark would be free energy and would lower the system's [[Entropy|entropy]] for nothing.[^sze-ch2] ### Forward bias Connecting the positive terminal of a supply to the p-side and the negative to the n-side is forward bias. The applied voltage falls almost entirely across the depletion region, because that is the only part of the crystal with no mobile carriers and hence a high [[Electrical_resistivity_and_conductivity|resistivity]]; it opposes the built-in field, so the barrier the carriers face drops from `q·V_bi` to `q·(V_bi − V)`. Less space charge is needed to support the smaller potential step, so the depletion region narrows: at V = 0.5 V the silicon junction above shrinks from 0.43 µm to 0.24 µm (derived).[^sze-ch2] With the barrier lowered, far more majority carriers have enough thermal energy to cross. Electrons are injected into the p-side and holes into the n-side, where each is a minority carrier in unfamiliar territory; they diffuse away from the junction and recombine within a diffusion length, and the current is supplied by the supply at the contacts. Because the fraction of carriers with energy enough to surmount a barrier of height `q·(V_bi − V)` is a Boltzmann factor, the current rises exponentially with `V`: at 300 K, and for an ideal junction, every extra 59.5 mV multiplies it by ten (derived).[^sze-ch2][^si-brochure] That steepness is why a diode appears to have a "turn-on voltage". Nothing switches; the current simply climbs so fast that it passes from microamperes to amperes over a few tenths of a volt, and the apparent threshold — near 0.7 V for silicon, 0.3 V for germanium — is set by the [[Band_gap|band gap]], which fixes `n_i` and hence `I_s`.[^up3-ch9] In a [[Light-emitting_diode|light-emitting diode]] built from a [[Direct_and_indirect_band_gaps|direct-gap]] material, the injected minority carriers recombine by emitting a [[Photon|photon]] of roughly the gap energy rather than by heating the lattice through [[Phonon|phonons]], and the same forward current that warms a silicon diode makes light.[^up3-devices] ### Reverse bias Reversing the supply, positive to the n-side, adds to the built-in field. The barrier rises to `q·(V_bi + |V|)`, more space charge is required, and the depletion region widens: at −5 V the silicon junction reaches 1.2 µm (derived).[^sze-ch2] Majority-carrier diffusion across the barrier is now suppressed by the same exponential factor that amplified it in forward bias, and within a few hundred millivolts of reverse bias it is negligible. What is left is the drift of minority carriers, and this does not depend on the barrier at all. Any electron thermally generated within a diffusion length of the depletion region on the p-side will, if it wanders into the field, be swept across it; the field cannot make more of them. The reverse current therefore saturates at a small, almost voltage-independent value `I_s`, the same quantity that sets the scale of the forward current. Its magnitude follows `n_i^2`, which varies as exp(−E_g/k·T), so it climbs steeply with [[Temperature|temperature]] — a leakage that limits how hot a silicon device may run.[^sze-ch1] Two further behaviours belong to reverse bias. The space-charge layer is an insulator between two conductors, so the junction has a capacitance per unit area `C/A = eps/W`; since `W` grows as the square root of the reverse voltage, the capacitance falls with it, and a junction used this way is a voltage-tunable capacitor. For the silicon junction above, `C/A` is 24 nF/cm² at zero bias (derived).[^sze-ch2] And if the field reaches the material's critical value — about 3×10⁵ V/cm in lightly doped silicon — the junction breaks down, either by avalanche multiplication, in which a carrier accelerated by the field ionises lattice atoms and starts a cascade, or, in heavily doped narrow junctions, by direct tunnelling.[^sze-ch1] A reverse-biased junction under illumination is a photodiode: light absorbed in the depletion region creates pairs that the field separates at once, which is also how a [[Solar_cell|solar cell]] works, the same diode operated as a generator instead of a load.[^up3-devices] ## Governing equations Both of the microsim's readouts come from a single idealisation, the depletion approximation: treat the space-charge layer as completely emptied of mobile carriers and the material outside it as perfectly neutral, so that the charge density is a step function, `−q·N_a` on the p-side of the boundary and `+q·N_d` on the n-side, and zero everywhere else.[^sze-ch2] Charge neutrality of the crystal as a whole then requires that the two blocks of space charge carry equal and opposite charge, `N_a·x_p = N_d·x_n`, where `x_p` and `x_n` are the distances the depletion region extends into each side and `W = x_p + x_n`. The more heavily a side is doped, the less the layer intrudes into it: a junction with one side doped a hundred times harder than the other is depleted almost entirely on the light side, which is why practical devices set their depletion width by the doping of a single lightly doped region. The approximation is good because the transition from neutral to depleted happens over a few Debye lengths, tens of nanometres in moderately doped silicon, much less than `W` itself. Its two consequences, worked below, are the two quantities the sim tracks against the reader's bias slider. ### Size of depletion region Inside the space-charge layer Poisson's equation reads `dE/dx = rho/eps`, with `eps` the permittivity of the semiconductor (11.7 times that of free space for silicon).[^sze-ch1] A constant charge density therefore makes the field vary linearly with position, rising from zero at the outer edge of the p-side layer to a maximum at the metallurgical boundary and falling linearly back to zero at the outer edge of the n-side layer. The profile is a triangle, and its peak is `E_max = q·N_a·x_p/eps = q·N_d·x_n/eps` the two expressions being equal by the neutrality condition. The potential difference across the layer is the area under that triangle, `(1/2)·E_max·W`, and it must equal the total potential step `V_bi − V`, where `V` is the applied bias, positive for forward.[^sze-ch2] Eliminating `E_max`, `x_p` and `x_n` between these relations gives the depletion width the sim displays: `W = sqrt(2·eps·(V_bi - V)·(N_a + N_d)/(q·N_a·N_d))` For the symmetric silicon junction of the lead — `N_a = N_d = 10¹⁶ cm⁻³`, `V_bi` = 0.72 V, `eps` = 11.7 ε₀ — this evaluates to 0.43 µm at zero bias, 0.24 µm at +0.5 V, and 1.2 µm at −5 V (derived).[^sze-ch2] The square-root law is the shape to watch in the sim: doubling the reverse voltage does not double the width, and pushing the forward bias towards `V_bi` collapses it towards zero, at which point the depletion approximation itself fails and the formula must be abandoned. When one side is much more heavily doped, `(N_a + N_d)/(N_a·N_d)` reduces to `1/N_light` and the expression becomes the familiar one-sided form `W ≈ sqrt(2·eps·(V_bi − V)/(q·N_light))`.[^sze-ch2] ### Current across depletion region The forward current follows from what the lowered barrier does to the minority-carrier populations at the edges of the depletion region. Because the majority carriers on each side remain in quasi-equilibrium with the barrier, the minority concentration injected at the far edge is raised above its equilibrium value by exactly the Boltzmann factor of the applied bias — the "law of the junction", `p_n(x_n) = p_n0·exp(q·V/(k·T))`.[^sze-ch2] Each injected carrier then diffuses into the neutral region and recombines, and solving the diffusion equation with recombination gives an exponential decay over the diffusion length `L = sqrt(D·tau)`, where `D` is the diffusion coefficient and `tau` the minority-carrier lifetime. The current is the gradient at the edge, and summing the electron and hole contributions gives the Shockley ideal-diode equation: `I = I_s·(exp(q·V/(n·k·T)) - 1)` with `I_s = q·A·(D_p·p_n0/L_p + D_n·n_p0/L_n)`, derived by William Shockley in 1949 in the paper that also set out the junction transistor; he shared the 1956 Nobel Prize in Physics with John Bardeen and Walter Brattain.[^shockley1949][^nobel1956] The equation reads differently in the two directions. For `V` more than a few `k·T/q` positive, the −1 is negligible and `I ≈ I_s·exp(q·V/(n·k·T))`, a straight line of slope 59.5 mV per decade on a semi-logarithmic plot at 300 K (derived, for `n` = 1). For `V` more than a few `k·T/q` negative, the exponential is negligible and `I ≈ −I_s`, the saturation current of the previous section. The ideality factor `n` lies between 1, when the current is carried by diffusion in the neutral regions, and 2, when recombination inside the depletion region dominates — as it does at low forward bias in silicon.[^sze-ch2] ILLUSTRATIVE: the band-bending drawn by the sim is the depletion-approximation profile, a quadratic potential on each side of the boundary, and the current curve is the ideal equation with a single fitted `I_s`. Real diodes depart from it at both ends — at high current the voltage dropped across the neutral bulk resistance and high-level injection bend the curve away from the exponential, and at high reverse voltage breakdown ends the saturation.[^sze-ch2] The sim reports `V_bi`, `W` and `I` for the doping the reader has chosen, not for any particular manufactured device. ## See also - [[Silicon]] — the crystal in which almost all junctions are made - [[Diode]] - [[Depletion_region]] - [[Semiconductor]] - [[Transistor]] - [[Czochralski_method]] - [[Photolithography]] - [[Integrated_circuit]] - [[Semiconductor_device_fabrication]] - [[Band_gap]] - [[Doping_(semiconductor)]] - [[Solar_cell]] ## References [^sze-ch1]: Sze, S. M.; Ng, K. K. *Physics of Semiconductor Devices*, 3rd ed. (Wiley, 2007), Ch. 1 "Physics and Properties of Semiconductors — A Review" (page to pin): the mass-action law `n·p = n_i²`, the static relative permittivity of silicon (11.7), the temperature dependence of `n_i`, and the critical field for avalanche breakdown in lightly doped silicon (≈3×10⁵ V/cm). [^sze-ch2]: Sze, S. M.; Ng, K. K. *Physics of Semiconductor Devices*, 3rd ed. (Wiley, 2007), Ch. 2 "p-n Junctions" (page to pin): the depletion approximation, `V_bi`, the triangular field profile and `W`, the law of the junction, the Shockley equation, the ideality factor, depletion capacitance and the cancellation of `V_bi` around a contacted loop. [^up3-ch9]: OpenStax (Sanny, J.; Ling, S. J.). *University Physics Volume 3* (2016), Ch. 9 Condensed Matter Physics, pp. 393–440 (page to pin), §9.5 Band Theory of Solids and §9.6 Semiconductors and Doping: band gaps of silicon and germanium at 300 K, donor and acceptor levels, and n- and p-type conduction. https://openstax.org/books/university-physics-volume-3 [^up3-devices]: OpenStax (Sanny, J.; Ling, S. J.). *University Physics Volume 3* (2016), Ch. 9 §9.7 Semiconductor Devices, within pp. 393–440 (page to pin): the junction diode, rectification, the light-emitting diode and the photovoltaic cell. https://openstax.org/books/university-physics-volume-3/pages/9-7-semiconductor-devices [^likharev-sm6]: Likharev, K. K. *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Ch. 6, pp. 187–225 (page to pin): carrier statistics in doped semiconductors and the drift–diffusion description of transport that the junction analysis rests on. [^shockley1949]: Shockley, W. (1949). "The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors." *Bell System Technical Journal* 28 (3): 435–489 (DOI to pin). [^nobel1956]: The Nobel Prize in Physics 1956 (William Shockley, John Bardeen, Walter Houser Brattain), "for their researches on semiconductors and their discovery of the transistor effect". NobelPrize.org. https://www.nobelprize.org/prizes/physics/1956/summary/ [^sproul-green]: Sproul, A. B.; Green, M. A. (1991). "Improved value for the silicon intrinsic carrier concentration from 275 to 375 K." *Journal of Applied Physics* 70 (2): 846–854 (DOI to pin) — the source of the 300 K value `n_i` = 9.65×10⁹ cm⁻³ used here; older textbook tables give 1.0×10¹⁰ cm⁻³, which lowers `V_bi` to 0.71 V. [^si-brochure]: Bureau International des Poids et Mesures. *The International System of Units (SI)*, 9th ed. (2019), §2.3.1 — the Boltzmann constant (1.380649×10⁻²³ J/K) and the elementary charge (1.602176634×10⁻¹⁹ C) are exact by definition, so the thermal voltage `k·T/q` at T = 300 K is 25.852 mV and one decade of ideal diode current costs `ln(10)·k·T/q` = 59.5 mV (derived). [^crystalfire]: Riordan, M.; Hoddeson, L. *Crystal Fire: The Birth of the Information Age* (W. W. Norton, 1997), the chapter on the wartime silicon work at Bell Telephone Laboratories (page to pin): Russell Ohl's 1940 silicon rod with an accidental impurity boundary, its photovoltage and its rectifying behaviour. ## Further reading - Sze, S. M.; Ng, K. K. *Physics of Semiconductor Devices*, 3rd ed. (Wiley, 2007) — Chapters 1 and 2 are the standard quantitative treatment behind every number on this page. - OpenStax, *University Physics Volume 3* (2016), Ch. 9 Condensed Matter Physics — the open-access account, at the level of a first course (Portal Book 079). - Likharev, K. K. *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Ch. 6 — carrier statistics and transport, for readers who want the junction from the statistical-mechanics side (Portal Book 075). - Riordan, M.; Hoddeson, L. *Crystal Fire: The Birth of the Information Age* (1997) — how the junction was found and what was made of it. ## External links - [*University Physics Volume 3*, Chapter 9](https://openstax.org/books/university-physics-volume-3/pages/9-7-semiconductor-devices) — OpenStax, CC BY; the semiconductor-devices section - [The Nobel Prize in Physics 1956](https://www.nobelprize.org/prizes/physics/1956/summary/) — Shockley, Bardeen and Brattain, with the presentation and lectures - Further sites are listed in the Wikipedia pair's *External links*; none is reproduced here until its URL has been checked. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/P–n_junction.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *P–n junction* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/P–n_junction.html" data-title="P–n junction"></div> *Built from `MICROSIM_GUIDE/specs/sims/P–n_junction.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/P–n_junction) : [Wikitube](https://en.wikitube.io/wiki/P–n_junction) · pinned revision [1363344649](https://en.wikipedia.org/w/index.php?oldid=1363344649) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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