**Electrical Conductivity** Conductivity is defined by Ohm’s law:

U = R I

where:

I is the current (in Amperes) through a resistor

U is the drop in potential (in Volts) across it.

The proportionality constant R is called the “resistance”, measured in Ohms (Ω).

The reciprocal of resistance (R

^{–1}) is called conductance. The unit of conductance is the Siemens (S = Ω

^{–1}).

Resistance is proportional to the length l of the sample and inversely proportional to the sample cross-section A:

R = ρ l / A

where:

ρ is the resistivity measured in Ω cm (in SI units Ω m).

Its inverse σ = ρ

^{–1} is the conductivity. The unit of conductivity is S m

^{–1}.

Electrical conductivity results from the existence of charge carriers and of the ability of those charge carriers to move.

σ = neμ

where:

σ = the electrical conductivity

n = the number of charge carriers per unit volume

e = the electronic charge

μ = the carrier mobility

In semiconductors and electrolyte solutions, one must also add an extra term due to positive charge carriers (holes or cations).

Electrical conductivity in materials may depend on direction and be anisotropic.

In a metal there is a high density of electronic states with electrons with relatively low binding energy, and ”free electrons” move easily from atom to atom under an applied electric field.

In a metal, the orbitals of the atoms overlap to form molecular orbitals: With N interacting atomic orbitals we will have N molecular orbitals. In the metal, however, or any continuous solid-state structure, N will be a very large number

(typically 10

^{22} for a 1 cm

^{3} metal piece). With so many molecular orbitals spaced together in a given range of energies, they form an apparently continuous band of energies.

The lowest unoccupied band is called the conduction band and the highest occupied one the valence band. The conductivity of the metal is due either to only-partly-filled valence or conduction bands, or to the band gap being near zero.

An electron injected at a certain position into a conduction orbital, which is delocalised over macroscopic dimensions, could leave the same orbital instantaneously at any other point in that orbital’s space.

Such a free moving electron would feel no resistance and would be super conducting (conducting without energy loss). However a moving electron may couple to the lattice, to phonons or scatter off a phonon or off a lattice imperfection for example. These increase the resistivity.

__Source:__ The Nobel Prize in Chemistry, 2000: Conductive polymers

http://nobelprize.org/nobel_prizes/chemistry/laureates/2000/chemadv.pdf