Junction Formation
A p–n junction is the fundamental building block of nearly every modern electronic device — from the LED in your phone screen to the solar cell on a rooftop. It is formed when a piece of p-type semiconductor (doped with electron-accepting impurities, leaving behind positive 'holes') is brought into intimate contact with a piece of n-type semiconductor (doped with electron-donating impurities, creating free electrons). What makes this interface remarkable is not the materials themselves, but the physics that spontaneously unfolds the instant they meet. At the moment of junction formation, a steep concentration gradient exists: electrons are abundant on the n-side and scarce on the p-side, while holes are abundant on the p-side and scarce on the n-side. This gradient drives diffusion — the same process that causes a drop of ink to spread through water. Electrons diffuse from n to p; holes diffuse from p to n. As charged carriers cross the junction, they leave behind ionised donor and acceptor atoms that cannot move. These fixed charges create an electric field pointing from n to p that opposes further diffusion. The region swept clear of mobile carriers is called the depletion region, and the voltage it produces is the built-in potential V₀. Equilibrium is reached when the electric drift force exactly balances the diffusion tendency, so net current across the junction is zero. This balance between drift and diffusion — and the resulting depletion region — gives the p–n junction its extraordinary ability to rectify current, emit light, detect photons, and amplify signals. Understanding junction formation is therefore the gateway to understanding all of semiconductor electronics and modern photonics.
- Starting Conditions: Two Doped Semiconductors
- Diffusion: Carriers Cross the Junction
- Ionised Atoms and the Depletion Region
- The Built-in Electric Field
- Equilibrium: Drift Balances Diffusion
Starting Conditions: Two Doped Semiconductors
In n-type silicon, each phosphorus atom donates one free electron while remaining as a fixed positive ion (P⁺). The semiconductor is electrically neutral overall because mobile electrons balance the fixed donor ions. In p-type silicon, each boron atom accepts one electron (creating a hole) and becomes a fixed negative ion (B⁻). Again, the material is neutral because mobile holes balance fixed acceptor ions. Carrier concentrations obey n·p = nᵢ², where nᵢ ≈ 1.5 × 10¹⁰ cm⁻³ for silicon at 300 K.
Diffusion: Carriers Cross the Junction
When the two materials are joined, electrons near the junction see a region of very low electron concentration on the p-side and diffuse across, driven by Fick's law: J_diff = -D (dn/dx), where D is the diffusion coefficient. Similarly, holes diffuse from p to n. This is like perfume molecules spreading from a concentrated region into fresh air — no external force is needed, only the concentration gradient.
Ionised Atoms and the Depletion Region
As electrons leave the n-side near the junction, they expose the fixed positive donor ions (P⁺) they used to neutralise. On the p-side, holes leave behind fixed negative acceptor ions (B⁻). A region develops on each side that is depleted of mobile carriers — the depletion region (also called the space-charge region). Its width W depends on the doping concentrations Nₐ and Nₐ and the built-in potential V₀. A typical depletion width in silicon is 0.1–1 μm.
The Built-in Electric Field
The exposed fixed charges (positive on the n-side, negative on the p-side) create an electric field E pointing from n to p. By Gauss's law, the field strength peaks at the metallurgical junction. This field exerts a force on any remaining mobile carriers: electrons are pushed back to the n-side (drift opposes diffusion), and holes are pushed back to the p-side. Drift current density is J_drift = σE = nqμE, where μ is carrier mobility.
Equilibrium: Drift Balances Diffusion
Equilibrium is established when drift current exactly cancels diffusion current for both electrons and holes separately. The resulting potential energy difference across the junction is the built-in potential V₀. For silicon doped with Nₐ (acceptors) and Nₐ (donors), V₀ = (kT/q) ln(Nₐ·Nₐ/nᵢ²). At room temperature kT/q ≈ 0.026 V, and typical doping gives V₀ ≈ 0.6–0.7 V for silicon. No external work is needed to maintain this — it is a thermodynamic equilibrium.
The Key Equation
- V_0 — Built-in (contact) potential (V): The electrostatic potential difference across the depletion region at thermal equilibrium; it cannot be measured directly with a voltmeter.
- k — Boltzmann constant (J K⁻¹): Relates temperature to thermal energy; k = 1.381 × 10⁻²³ J K⁻¹.
- T — Absolute temperature (K): Temperature in kelvin; at room temperature T ≈ 300 K.
- q — Elementary charge (C): Magnitude of electron charge; q = 1.602 × 10⁻¹⁹ C.
- N_a — Acceptor concentration (cm⁻³): Number density of acceptor (e.g. boron) atoms in the p-type region.
- N_d — Donor concentration (cm⁻³): Number density of donor (e.g. phosphorus) atoms in the n-type region.
- n_i — Intrinsic carrier concentration (cm⁻³): Carrier concentration in undoped silicon; nᵢ ≈ 1.5 × 10¹⁰ cm⁻³ at 300 K.
Worked Example
Problem. A silicon p–n junction is doped with Nₐ = 10¹⁶ cm⁻³ acceptors and Nₐ = 10¹⁵ cm⁻³ donors. Calculate the built-in potential V₀ at T = 300 K. (nᵢ = 1.5 × 10¹⁰ cm⁻³, kT/q = 0.02585 V)
Answer. The built-in potential is approximately 0.634 V, consistent with the typical 0.6–0.7 V range for silicon junctions.
Origins · 1 minWilliam Shockley and how this was discovered
Shockley, together with John Bardeen and Walter Brattain, invented the transistor at Bell Labs in 1947 and developed the theoretical framework describing p–n junctions, diffusion currents, and the diode equation. Their work earned the 1956 Nobel Prize in Physics and launched the semiconductor age. Shockley was so competitive that after winning the Nobel Prize he left Bell Labs to start his own company in California — and the engineers who quit his firm went on to found Fairchild Semiconductor and, later, Intel, effectively creating Silicon Valley.
Related Concepts
- Diode I–V Characteristics — Once junction formation is understood, the Shockley diode equation I = I₀(e^(qV/kT)−1) describes how the depletion region widens under reverse bias and collapses under forward bias, producing the characteristic non-linear I–V curve.
- Semiconductor Doping — Doping determines Nₐ and Nₐ, which directly set the built-in potential and depletion width — without understanding doping, the concentrations in the junction equations have no physical basis.
- Photovoltaic Effect in Solar Cells — A solar cell is a p–n junction illuminated by light; photons generate electron-hole pairs that are separated by the built-in electric field of the depletion region, converting light directly to electrical energy.
- Drift and Diffusion of Charge Carriers — The two competing transport mechanisms — drift driven by electric field and diffusion driven by concentration gradient — are the core physics of junction formation and apply throughout all semiconductor devices.