Electrical conduction

In the classical model, a body is insulating if it does not contain mobile electrons. In a conductor, electrons are loosely bound to the nuclei and can move in the crystal lattice.

If n is the density of free electrons, v their average speed, in a bar of length L, of section S with a voltage V between the ends, the current density J = I/S is equal to J = n.e.v.

The speed of the electrons is proportional to the force to which they are subjected, therefore to the electric field E = V/L.

If µ designates mobility, we have: v = µ.E

J = n.e.µ.E = σ.E = E/ρ

The classical model has been replaced by the quantum model of energy bands. In the isolated atom the electrons occupy discrete energy levels. In a crystal, as a result of interactions between atoms, these discrete levels expand and the electrons occupy permitted energy bands separated by forbidden bands. The distribution of electrons in the levels obeys the laws of statistical thermodynamics. At absolute zero, only the lowest energy levels are populated.

Complément

In insulators, the lowest energy bands are completely full. The height of the bandgap is large (≈ 5 eV). There are no accessible energy levels and no conduction. For example, the resistivity of diamond is ρ = 1.1012 Ω.m

In the conductors, the last occupied band is partially filled: there are many levels available and the conduction is high. For metals that are good conductors, we obtain:

ρ Ag = 1.6.10–8 Ω.m; ρ Cu = 1.7.10–8 Ω.m; ρ Al = 2.8.10–8 Ω.m

For semiconductors, the filling rate of the last occupied band is either very low or very high. The height of the band gap is low (≈ 1 eV). Conduction is weak and varies greatly with temperature. For silicon and germanium, we measure at 300 K:

ρ Si = 2400 Ω.m ; ρ Ge = 0,5 Ω.m