Energy Bands
Energy bands are the continuous ranges of energy that electrons in a crystalline solid are allowed to occupy; they arise because the discrete levels of isolated atoms split and broaden when about 1023 atoms are brought together in a periodic lattice, and they are separated by band gaps, energy ranges in which no electron states exist. How the bands are filled determines whether a material is a metal, semiconductor or insulator.
Curious
An isolated atom has sharp energy levels (see Atomic & Molecular Physics). Bring two atoms close and each level splits into two; bring a huge number together, as in a crystal, and each level splits into so many closely spaced levels that they form a practically continuous band. Between bands there can be ranges of energy that no electron is allowed to have: band gaps.
This picture explains one of the most basic facts about materials: why copper conducts electricity and glass does not. If the highest occupied band is only partly full, electrons can easily move into nearby empty states and carry current, as in metals. If it is completely full and the next band is far above, electrons are stuck, as in insulators.
Exploring
| Material | Band gap at 300 K (eV) | Class |
|---|---|---|
| Copper | none (partly filled band) | Metal |
| Germanium | 0.66 | Semiconductor |
| Silicon | 1.12 | Semiconductor |
| Gallium arsenide | 1.42 | Semiconductor |
| Diamond | 5.47 | Insulator |
Deep Dive
Two limits. In the nearly free electron model, a weak periodic potential opens gaps of size about 2|UG| at the Brillouin zone boundaries, where electron waves are Bragg reflected (see Brillouin Zone). In the tight-binding model, atomic orbitals overlap weakly; for a one-dimensional chain with hopping energy t and spacing a, E(k) = E0 − 2t cos(ka), a band of width 4t. Real band structures are computed numerically, most commonly with density functional theory, which systematically underestimates band gaps (the “band gap problem”) unless corrected by more advanced methods.
Counting states. Each band holds 2N electrons for N unit cells (the factor 2 for spin). A crystal with an odd number of electrons per cell must therefore have a partly filled band and should be a metal in simple band theory. Some such materials are in fact insulators because strong electron–electron repulsion localises the electrons (Mott insulators), a known failure of independent-electron band theory.
Direct and indirect gaps. If the top of the valence band and the bottom of the conduction band occur at the same crystal momentum, the gap is direct (GaAs), and light is absorbed and emitted efficiently, which makes such materials suitable for LEDs and lasers. Silicon's gap is indirect, so a phonon must also take part, which is why silicon is a poor light emitter despite dominating electronics (see Semiconductors).
Concept Relationship Map
Requires: Bloch's theorem, crystal structure
Enables: effective mass, density of states, insulators
Commonly confused with: band theory (the method, not the result)
Sources
- Bloch, F. (1929). Zeitschrift für Physik, 52, 555–600.
- Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley.
- Ashcroft, N.W., Mermin, N.D. (1976). Solid State Physics. Holt, Rinehart and Winston.