A count of how many states electrons can occupy at each energy – like a chart of how many seats each row of a stadium has. Where the count is high, many electrons can join in whatever happens at that energy; inside a it is zero. pile many states onto one energy, which is why they favour unusual behaviour, and a can measure the count at a single spot.
Going deeper
How many states there are at each energy depends on how many directions electrons can move in: it grows smoothly in a bulk metal, comes in flat steps in a thin layer, and in graphene falls to zero at a single point – which is why a gate moves graphene’s filling so far.
Counting seats
Electrons in a crystal can only sit in the states its allows, and each state holds at most one electron. The density of states counts how many of those states lie in each thin slice of energy. It matters because almost everything a material does involves the electrons near the top of the filled states – at the – and the more states there are there, the more electrons can respond: to heat, to a magnetic field, to a gate, or to each other.
In a stadium picture, the band structure says where the seats are, the density of states says how many there are in each row, and the Fermi level says how far up the stadium has filled.
Why dimension matters
The shape of the count depends on how many directions electrons can move in. In an ordinary three-dimensional metal it grows smoothly with energy. In a flat two-dimensional layer with simple bands it is constant – the same number of states in every slice above the band edge – which is why each new level in a adds a sudden step. In graphene it starts at zero at the and rises in proportion to energy, so a gate that adds only a few electrons moves graphene’s Fermi level a long way.
Where a band flattens or has a saddle point, states pile up into sharp peaks called van Hove singularities. Bringing the Fermi level onto such a peak, by or by twisting two layers, is a common route to magnetism, or .
Measuring it
A scanning tunnelling microscope measures the density of states directly and locally: at a fixed tip position, how the tunnelling current changes with voltage follows the number of states at each energy, so band edges, gaps and the sharp peaks at the edge of a superconducting gap appear as features in a curve taken at one spot. The heat capacity of a metal at low temperature measures it at the Fermi level for the whole sample, and in graphene devices the capacitance of the sheet itself reveals it, because filling states that are few in number takes a measurable extra voltage.
For specialists
The number of single-particle states per unit energy (and per unit area in 2D), g(E) = Σn ∫ δ(E − εn(k)) d2k/(2π)2. For a parabolic it is constant, m*/(2πħ2) per and ; for graphene’s Dirac cone it rises linearly from zero; van Hove singularities appear at saddle points. Its value at the Fermi level sets the electronic heat capacity, Pauli susceptibility, screening and the tendency to Stoner or superconducting instabilities; scanning tunnelling spectroscopy measures the local density of states through dI/dV.
Where this comes from
Solid State PhysicsAshcroft and Mermin · Holt, Rinehart and Winston, New York (1976)