Condensed matter physics
Free-electron gas and the Fermi level
The free-electron model treats conduction electrons as a fermion gas in a large volume governed by Fermi–Dirac statistics. Filling states at T=0 determines the Fermi surface, Fermi energy, and degeneracy pressure.
In the simplest model, ionic lattice potentials, electron interactions, and scattering are neglected, and conduction electrons are treated as free particles in a box. Despite its simplicity, it estimates Fermi scales and explains the small electronic heat capacity of metals.
Definition: Fermi energy and wave number
n is the free-electron density and m the electron mass (or an appropriate effective mass). kF is the radius of the spherical Fermi surface in isotropic 3D k-space; EF is its energy at T=0.
Counting states
In volume V with two spin states, the number of states inside the sphere of radius kF is N=VkF³/(3π²). Setting n=N/V gives the formula above. The spin degeneracy of two is included; omitting it causes a substantial density error.
For a 3D parabolic dispersion, the density of states grows as the square root of energy. Only a thin shell of order kBT around EF changes occupation as temperature rises, since states deep below EF are already filled.
Example: Worked example
For density n, how does EF change if n increases by a factor of 8?
Solution
Since EF ∝ n^(2/3), it increases by 8^(2/3)=4.
Quick check
The Fermi sphere results from filling wave vectors out to kF, counting both spin states. At T=0 the quantum pressure is not thermal: Pauli exclusion forces compression to raise kF and the total energy. For a three-dimensional free-electron gas the energy density is 3nEF/5 and the pressure is 2nEF/5. This degeneracy pressure follows from state filling; important Coulomb interactions or lattice effects require a more complete model.
In real metals the Fermi surface is usually not spherical: the periodic potential warps bands and can create multiple electron or hole pockets. de Haas–van Alphen oscillations in magnetic field reveal extremal cross-sections of that surface. The free-electron gas is therefore a useful starting point for intuition, while band structure and effective mass determine quantitative transport properties.
If electron density rises eightfold, by what factor does the free-electron Fermi energy rise?
Which sign is in the denominator of the corresponding distribution?
References
- Landau, Lifshitz (1980). Statistical Physics
- Charles Kittel, Herbert Kroemer (1980). Thermal Physics