Electrical conduction is the current (movement of charged particles) through a material in response to an electric field. The underlying mechanism for this movement depends on the material.

Conduction is well-described by Ohm's Law, which states that the current is proportional to the applied electric field. The ease with which current density (current per area) j appears in a material is measured by the conductivity σ, defined as:

j = σ E

or its reciprocal resistivity ρ:

j = E / ρ

In anisotropic materials, σ and ρ are tensors.

Solids (including insulating solids)

In crystalline solids, atoms interact with their neighbors, and the energy levels of the electrons in isolated atoms turn into bands. Whether a material conducts or not is determined by its band structure. Electrons, being fermions, follow the Pauli exclusion principle, meaning that two electrons cannot occupy the same state. Thus electrons in a solid fill up the energy bands up to a certain level, called the Fermi energy. Bands which are completely full of electrons cannot conduct electricity, because there is no state of nearby energy to which the electrons can jump. Materials in which all bands are full (i.e. the Fermi energy is between two bands) are insulators.

Metals

Metals are good conductors because they have unfilled space in the valence energy band. In the absence of an electric field, there exist electrons travelling in all directions and many different velocities up to the Fermi velocity (the velocity of electrons at the Fermi energy). When an electric field is applied, a slight imbalance develops and mobile electrons flow. Electrons in this band can be accelerated by the field because there are plenty of nearby unfilled states in the band.

Resistance comes about in a metal because of scattering of the electrons from defects in the lattice or by phonons. A crude theory of conduction in simple metals is the Drude model, in which scattering is characterized by a relaxation time τ. The conductivity is then given by the formula

where n is the density of conduction electrons, e is the electron charge, and m is the electron mass. A better model is the so-called semiclassical theory, in which the effect of the periodic potential of the lattice on the electrons gives them an effective mass.

Semiconductors

A solid with filled bands is an insulator, but at finite temperature, electrons can be thermally excited from the valence band to the next highest, the conduction band. The fraction of electrons excited in this way depends on the temperature and the band gap, the energy difference between the two bands. Exciting these electrons into the conduction band leaves behind positively charged holes in the valence band, which can also conduct electricity. See semiconductor for more details.

In semiconductors, impurities greatly affect the concentration and type of charge carriers. Donor (n-type) impurities have extra valence electrons with energies very close to the conduction band which can be easily thermally excited to the conduction band. Acceptor (p-type) impurities capture electrons from the valence band, allowing the easy formation of holes. If an insulator is doped with enough impurities, a Mott transition can occur, and the insulator turns into a conductor.

Superconductors

In metals and certain other materials, a transition occurs at low temperature to the superconducting state. By an interaction mediated by some other part of the system (in metals, phonons), the electrons pair up into Cooper pairs. The bosonic Cooper pairs form a superfluid which has zero resistance.

Electrolytes

Electric currents in electrolytes are flows of electrically charged atoms (ions). For example, if an electric field is placed on a solution of Na⁺ and Cl^–, the sodium ions will move towards the negative electrode (cathode), and the chlorine ions will move towards the positive electrode (anode). If the conditions are right, redox reactions will take place at the electrode surfaces, releasing electrons from the chlorine, and allow electrons to be absorbed into the sodium.

Water ice and certain solid electrolytes called proton conductors contain positive hydrogen ions which are free to move. In these materials, currents of electricity are composed of moving protons.

Gases and plasmas

In neutral gases, electrical conductivity is very low. They act as a dielectric or insulator, up until the electric field reaches a breakdown value, freeing the electrons from the atoms in an avalanche process thus forming a plasma. This plasma provides mobile electrons and positive ions, acting as a conductor which supports electric currents and forms a spark, arc or lightning. In ordinary air below the breakdown field, the dominant source of electrical conduction is via mobile ions produced by radioactive gases and cosmic rays.

Plasma is the state of matter where some of the electrons in a gas are stripped or "ionized" from their molecules or atoms. A plasma can be formed by high temperature, or by application of an electric field as noted above. Electrical conduction in a plasma is due to the motion of the electrons and the negatively- or positively-charged ions.