Condensed matter physics
Fermi–Dirac statistics
The Fermi–Dirac distribution describes equilibrium fermion occupations subject to Pauli exclusion. At absolute zero states fill up to the Fermi energy; finite temperature smears the occupation edge.
Fermions such as electrons, protons, and neutrons obey Pauli exclusion: each single-particle quantum state can hold at most one fermion per spin state. Their occupation probability follows Fermi–Dirac statistics.
Definition: Fermi–Dirac occupation
f(ε) is the probability that a state of energy ε is occupied; μ is the chemical potential. At nonzero temperature f(μ)=1/2. At T=0 it becomes a step at the Fermi energy.
Fermi surface and degenerate matter
In a metal, conduction electrons fill states up to the Fermi level. Only electrons within an energy range of order kBT around μ are readily excited, so many thermal and electronic properties depend on structure near the Fermi surface.
The Fermi temperature T_F=E_F/kB is usually far above ordinary metal temperatures, so conduction electrons form a degenerate quantum gas. The classical Maxwell–Boltzmann approximation fails when quantum occupation matters.
Example: Worked example
At T>0, a state has ε−μ=0. What is its occupation probability?
Solution
Substitution gives f=1/(e⁰+1)=1/2.
Quick check
At low temperature, the Sommerfeld expansion estimates thermodynamic quantities. For a smooth function g(ε), the occupied-state integral is its T=0 value plus the leading correction π²(kBT)²g′(μ)/6. Thus only electrons within an energy window of order kBT around the Fermi level can change their thermal occupation; in ordinary metals the electronic heat capacity is proportional to T. The expansion assumes that the density of states varies slowly over the scale kBT.
When kBT is small compared with EF, occupation resembles a step: states below μ are nearly full and those above it nearly empty. Heating smears only a narrow energy layer; because particle number stays fixed, μ(T) shifts slightly. For comparison with measurements, integrate g(ε)f(ε) rather than reading f at one energy. This explains why the electronic thermal response of a metal is far smaller than a classical gas prediction.
What is the Fermi–Dirac occupation at ε=μ?
Which sign is in the denominator of the corresponding distribution?
References
- Landau, Lifshitz (1980). Statistical Physics
- Charles Kittel, Herbert Kroemer (1980). Thermal Physics