Built-In Potential

When a p-type semiconductor (rich in positive 'holes') is joined directly to an n-type semiconductor (rich in free electrons), something remarkable happens at the interface even before any external voltage is applied. Electrons from the n-side diffuse across the junction and recombine with holes on the p-side, while holes drift in the opposite direction. This mutual annihilation of charge carriers near the boundary creates a thin region — called the depletion region — that is stripped of mobile charges and is left with fixed, immobile ions carrying net charge. Those fixed charges set up a built-in electric field pointing from the n-side (positive donor ions) toward the p-side (negative acceptor ions). The work done against this field on a unit positive charge as it crosses the junction is called the built-in potential, V₀ (also written V_bi or contact potential). Typical values range from about 0.3 V for germanium to 0.7 V for silicon and up to 1.4 V for gallium arsenide. This internal voltage is not something you can measure with a voltmeter placed across the terminals — it is self-consistently cancelled by contact potentials at the metal leads — but it is absolutely real inside the device and governs almost every aspect of diode and transistor behaviour. The built-in potential acts as a potential-energy barrier that stops further diffusion once equilibrium is reached: the drift current due to the electric field exactly balances the diffusion current due to the carrier concentration gradient. Understanding V₀ is therefore the foundation for understanding how a diode conducts in forward bias, blocks current in reverse bias, emits light in an LED, or generates electricity in a solar cell.

Key takeaways
  • Carrier Concentrations Before Contact
  • Diffusion Across the Junction
  • The Depletion Region and Its Electric Field
  • Equilibrium and the Built-In Potential
  • Depletion Width and Its Dependence on Doping

Carrier Concentrations Before Contact

In an n-type semiconductor, the donor doping density Nₐ sets the electron concentration n₀ ≈ Nₐ and the hole concentration p₀ = nᵢ²/Nₐ, where nᵢ is the intrinsic carrier concentration (~1.5×10¹⁰ cm⁻³ in Si at 300 K). The p-side is the mirror image. Both sides are individually electrically neutral before contact.

Diffusion Across the Junction

When the two semiconductors are brought into contact, the huge concentration gradient drives electrons from n to p and holes from p to n — exactly as a drop of ink spreads through water. Near the interface, electrons and holes recombine and disappear, leaving behind positively charged donor ions (⁺) on the n-side and negatively charged acceptor ions (⁻) on the p-side.

The Depletion Region and Its Electric Field

The exposed fixed ions form a space-charge region. By Gauss's law, the positive charge sheet on the n-side and the negative charge sheet on the p-side create an electric field E directed from n to p. This field opposes further diffusion (it pushes electrons back to the n-side and holes back to the p-side), so the depletion region widens only until the drift current equals the diffusion current.

Equilibrium and the Built-In Potential

At thermal equilibrium no net current flows. The potential difference across the depletion region is V₀ = (kT/q) ln(Nₐ Nₐ / nᵢ²), derived by setting the electrochemical potentials (Fermi levels) equal on both sides. Here kT/q ≈ 0.026 V at 300 K is the thermal voltage Vₜ. The Fermi level is flat throughout the entire junction at equilibrium — a powerful way to visualise that no current is flowing.

Depletion Width and Its Dependence on Doping

The total depletion width W = xₙ + xₚ, where charge neutrality requires Nₐ xₙ = Nₐ xₚ (using Nₐ for donor and Nₐ for acceptor densities). Solving Poisson's equation gives W = √(2ε V₀/q · (1/Nₐ + 1/Nₐ)). Heavier doping → smaller depletion width but also higher V₀, so the two effects partially compete. This relationship is exploited in tunnel diodes and avalanche photodiodes.

The Key Equation

V0=kTqln⁡ ⁣(ND NAni 2)V_0 = \frac{kT}{q}\ln\!\left(\frac{N_D\, N_A}{n_i^{\,2}}\right)
  • V_0 — Built-in potential (V): Electrostatic potential difference across the depletion region at thermal equilibrium.
  • k — Boltzmann constant (J K⁻¹): 1.381 × 10⁻²³ J K⁻¹; relates thermal energy to temperature.
  • T — Absolute temperature (K): Device temperature; at room temperature T = 300 K.
  • q — Elementary charge (C): 1.602 × 10⁻¹⁹ C; charge of a single proton.
  • N_D — Donor concentration (cm⁻³): Number density of ionised donor atoms on the n-side (e.g., phosphorus in silicon).
  • N_A — Acceptor concentration (cm⁻³): Number density of ionised acceptor atoms on the p-side (e.g., boron in silicon).
  • n_i — Intrinsic carrier concentration (cm⁻³): Carrier density in undoped semiconductor; ~1.5 × 10¹⁰ cm⁻³ for Si at 300 K.

Worked Example

Problem. A silicon p–n junction at 300 K has a donor concentration Nₐ = 10¹⁶ cm⁻³ on the n-side and an acceptor concentration Nₐ = 10¹⁷ cm⁻³ on the p-side. The intrinsic carrier concentration of silicon is nᵢ = 1.5 × 10¹⁰ cm⁻³. Calculate the built-in potential V₀.

V0≈0.75 VV_0 \approx 0.75\text{ V}

Answer. The built-in potential is approximately 0.75 V, consistent with the well-known ~0.7 V barrier in silicon diodes.

Figure
Diagram
n-typep-typeDepletionRegione⁻ e⁻e⁻ e⁻⊕ ⊕⊕ ⊕⊕⊕⊕⊕⊕⊕⊕⊖⊖⊖⊖⊖⊖E (electric field)VV₀ ≈ 0.7 V (Si)Potentialxₙxₚ
Origins · 1 minWilliam Shockley and how this was discovered
WS
William Shockley
1910–1989 · American

Shockley co-invented the bipolar junction transistor at Bell Labs in 1947 (with Bardeen and Brattain) and developed the theoretical framework for p–n junctions, including the diode equation that explicitly incorporates the built-in potential. His 1950 textbook 'Electrons and Holes in Semiconductors' became the definitive reference for a generation of solid-state physicists. He shared the 1956 Nobel Prize in Physics for this work. Shockley was so competitive that when Bardeen and Brattain invented the first working transistor while he was away, he rushed to derive a superior design — the junction transistor — in a single inspired burst over Christmas 1947, writing the key equations in a hotel room.

  • Depletion Region — The depletion region is the spatial zone in which the built-in potential physically resides; understanding its width and charge density is inseparable from calculating V₀.
  • Forward and Reverse Bias — Applying an external voltage either reduces (forward bias) or increases (reverse bias) the built-in potential barrier, directly controlling whether current flows through the diode.
  • Fermi Level — At equilibrium the Fermi level is constant across the junction; the built-in potential equals the difference in Fermi levels of the isolated n- and p-type materials before contact.
  • Doping in Semiconductors — The donor (Nₐ) and acceptor (Nₐ) concentrations appear directly in the V₀ formula; higher doping increases the built-in potential logarithmically.