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In most situations, 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 flows in a material is measured by the conductivity σ, defined as:
or its reciprocal resistivity ρ:
In anisotropic materials, σ and ρ are tensors.
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.
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 modelThe Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials. The Drude model is the application of kinetic theory to electrons in a solid. It assumes that the material con, 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 massIn solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields a.
A solid with filled bands is an insulator, but at finite temperature, electrons can be thermally excited from the valence bandIn solids, the valence band is the highest range of electron energies where electrons are normally present at zero temperature. In semiconductors and insulators, there is a bandgap above the valence band, followed by a conduction band above that. In metal 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.