Energy bands: conductor, semiconductor and insulator, band gap, doping and LED colour

What it does

In a single atom an electron can only have certain energies. In a crystal, billions of atoms sit close together and those sharp levels spread out into bands. Two bands matter: the valence band, full of the electrons that hold the atoms together, and the conduction band above it, where an electron is free to wander and carry a current. The energy step between them is the band gap, Eg.

• Material. Copper has no gap: its bands overlap, so it always conducts. Germanium (0.66 eV), silicon (1.12 eV) and gallium arsenide (1.42 eV) are semiconductors. Diamond (about 5.5 eV) is an insulator.
• Temperature. Heat lifts a few electrons across the gap. Each one leaves a hole behind in the valence band, so a pure semiconductor has equal numbers of free electrons and holes, nᵢ. Slide the temperature and watch the dots: the number grows very fast as it warms.
• Doping. Add phosphorus (five outer electrons) and you get n-type silicon, with a donor level just under the conduction band. Add boron (three outer electrons) and you get p-type, with an acceptor level just above the valence band. The page shows the majority and minority carriers and checks that n × p = nᵢ².
• LED colour. When an electron drops back across the gap it can give out light with energy about Eg. Slide Eg and see the wavelength, λ = 1240 / Eg (λ in nm, Eg in eV), and the colour.

This fits Class 12 Physics: semiconductor electronics (energy bands, intrinsic and extrinsic semiconductors, n-type and p-type) and links back to the energy levels of atoms. Next, join p-type and n-type silicon in the PN junction; for the full physics of bands see band structure explorer, and for real carrier numbers use the carrier concentration calculator.

How to use it

1. Pick a material. The diagram shows the two bands and the gap, drawn to scale.
2. Move T. Filled dots are free electrons, open circles are holes. The picture is on a log scale: every extra pair of dots means about ten times more carriers.
3. For germanium, silicon or GaAs, choose n-type or p-type and set the number of dopant atoms N per cubic centimetre.
4. In the second card, move Eg to change the LED.
5. Try the Check yourself questions; New questions gives a fresh set.

Worked example

Silicon at 300 K: the page shows nᵢ = 9.78 × 10⁹ cm⁻³ (the textbook value is about 1 to 1.5 × 10¹⁰; this page uses one rough model for every material). Warm it to 400 K and there are about 346 times more carriers. That is why a semiconductor’s resistance falls when it is heated, while a metal’s rises.

Now choose n-type with N = 1.00 × 10¹⁶ cm⁻³. Every phosphorus atom gives one electron, so nₑ ≈ 1.00 × 10¹⁶ cm⁻³, and the holes fall to nₕ = nᵢ² / nₑ = (9.78 × 10⁹)² / 10¹⁶ ≈ 9.57 × 10³ cm⁻³.

LED: with Eg = 1.90 eV, λ = 1240 / 1.90 = 653 nm, which is red. A band gap of about 2.7 eV gives blue (459 nm); silicon’s 1.12 eV gives 1107 nm, which is infrared.

Tips

• Doping is tiny: 10¹⁶ atoms per cm³ is about one phosphorus atom for every five million silicon atoms, yet it raises the electrons a million times.
• An n-type crystal is still electrically neutral: each free electron is balanced by a positive phosphorus ion.
• Silicon is a poor light emitter; LEDs use compounds such as gallium arsenide phosphide or indium gallium nitride.

FAQ

Why does the gap decide the colour? The light comes from an electron falling across the gap, so the photon’s energy is close to Eg, and E = hc/λ.

Are the numbers exact? No. The band gaps are rounded and the carrier formula uses one rough constant for every material, so the carrier numbers are “about”. The trends (huge rise with temperature, n × p constant) are right.

Is anything saved? No. Everything runs in your browser and nothing is stored or sent.

Is this an official textbook tool? No. It is an independent practice page that follows the same school topics.

Last updated 2026-11-09.