Density of States
The density of states g(E) is the number of single-particle quantum states per unit energy interval (and usually per unit volume) available at energy E. It is fixed by a system's dispersion relation and dimensionality; for free electrons in three dimensions g(E) ∝ √E. Combined with an occupation function such as the Fermi–Dirac distribution, it determines carrier densities, electronic heat capacity, magnetic susceptibility and optical absorption.
Curious
Electrons fill available states from the lowest energy upward until they run out, reaching the Fermi energy. How quickly they fill up depends on how many states exist at each energy. In a metal, only the electrons near the Fermi energy can respond to heat or an electric field, because the states just below are already full. So the number of states right at the Fermi energy controls how the metal conducts and how much heat its electrons can absorb.
Dimension matters: electrons squeezed into thin layers (quantum wells) or wires have very different densities of states from those in bulk material, a fact engineered in modern lasers and transistors (see Particle in a Box).
Exploring
| Dimension | Free-electron g(E) | Example system |
|---|---|---|
| 3D | ∝ E1/2 | Bulk metal |
| 2D | constant (per subband) | Quantum well (2D electron gas) |
| 1D | ∝ E−1/2 | Quantum wire, carbon nanotube |
| 0D | discrete delta functions | Quantum dot |
Deep Dive
Derivation. With periodic boundary conditions in a box of volume V, allowed wavevectors form a grid with one point per (2π)³/V of k-space. Counting points inside a sphere of radius k, including a factor 2 for spin, and using E = ℏ²k²/2m gives N(E) = (V/3π²)(2mE/ℏ²)3/2, so g(E) = dN/dE = (V/2π²)(2m/ℏ²)3/2E1/2. For a general band, g(E) = (2/(2π)³) ∫ dS/|∇kE| over the constant-energy surface, which diverges at van Hove singularities where the band is flat.
Role in semiconductors. Near band edges, the same formula applies with the effective mass (see Effective Mass), giving effective densities of states Nc and Nv that, with the band gap, fix the intrinsic carrier concentration ni = √(NcNv) exp(−Eg/2kBT) used in p-n junction calculations.
Measurement. Photoemission spectroscopy, tunnelling spectroscopy (whose conductance is proportional to the local density of states), and low-temperature heat capacity probe g(E). In superconductors, the density of states develops a gap at the Fermi level, observed directly by tunnelling and consistent with BCS theory.
Concept Relationship Map
Requires: Brillouin zone, Fermi energy
Enables: heat capacity, carrier statistics, tunnelling spectroscopy
Commonly confused with: the occupied-state distribution g(E)f(E)
Sources
- Sommerfeld, A. (1928). Zeitschrift für Physik, 47, 1–32.
- 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.