Electrical Conduction

The conductivity is one of the properties of materials that varies most widely, from 10⁷ (W-m) typical of metals to 10^-20 (W-m) for good electrical insulators. Semiconductors have conductivities in the range 10^-6 to 10⁴ (W-m).

Electronic and Ionic Conduction

In metals, the current is carried by electrons, and hence the name electronic conduction. In ionic crystals, the charge carriers are ions, thus the name ionic conduction.

Energy Band Structures in Solids

When atoms come together to form a solid, their valence electrons interact due to Coulomb forces, and they also feel the electric field produced by their own nucleus and that of the other atoms. In addition, two specific quantum mechanical effects happen. First, by Heisenberg's uncertainty principle, constraining the electrons to a small volume raises their energy, this is called promotion. The second effect, due to the Pauli exclusion principle, limits the number of electrons that can have the same property (which include the energy). As a result of all these effects, the valence electrons of atoms form wide valence bands when they form a solid. The bands are separated by gaps, where electrons cannot exist. The precise location of the bands and band gaps depends on the type of atom (e.g., Si vs. Al), the distance between atoms in the solid, and the atomic arrangement (e.g., carbon vs. diamond).

In semiconductors and insulators, the valence band is filled, and no more electrons can be added, following Pauli's principle. Electrical conduction requires that electrons be able to gain energy in an electric field; this is not possible in these materials because that would imply that the electrons are promoted into the forbidden band gap.

In metals, the electrons occupy states up to the Fermi level. Conduction occurs by promoting electrons into the conduction band, that starts at the Fermi level, separated by the valence band by an infinitesimal amount.

Conduction in Terms of Band and Atomic Bonding Models

Conduction in metals is by electrons in the conduction band. Conduction in insulators is by electrons in the conduction band and by holes in the valence band. Holes are vacant states in the valence band that are created when an electron is removed.

In metals there are empty states just above the Fermi levels, where electrons can be promoted. The promotion energy is negligibly small so that at any temperature electrons can be found in the conduction band. The number of electrons participating in electrical conduction is extremely small.

In insulators, there is an energy gap between the valence and conduction bands, so energy is needed to promote an electron to the conduction band. This energy may come from heat, or from energetic radiation, like light of sufficiently small wavelength.

A working definition for the difference between semiconductors and insulators is that in semiconductors, electrons can reach the conduction band at ordinary temperatures, where in insulators they cannot. The probability that an electron reaches the conduction band is about exp(-E_g/2kT) where E_g is the band gap and kT has the usual meaning. If this probability is, say, < 10^-24 one would not find a single electron in the conduction band in a solid of 1 cubic centimeter. This requires E_g/2kT > 55. At room temperature, 2kT = 0.05 eV; thus E_g > 2.8 eV can be used as the condition for an insulator.

Besides having relatively small E_g, semiconductors have covalent bond, whereas insulators usually are partially ionic bonded.

Electron Mobility

Electrons are accelerated in an electric field E, in the opposite direction to the field because of their negative charge. The force acting on the electron is -eE, where e is the electric charge. This force produces a constant acceleration so that, in the absence of obstacles (in vacuum, like inside a TV tube) the electron speeds up continuously in an electric field. In a solid, the situation is different. The electrons scatter by collisions with atoms and vacancies that change drastically their direction of motion. Thus electrons move randomly but with a net drift in the direction opposite to the electric field. The drift velocity is constant, equal to the electric field times a constant called the mobility m, v_d= – m_e E which means that there is a friction force proportional to velocity. This friction translates into energy that goes into the lattice as heat. This is the way that electric heaters work.

The electrical conductivity is: s = n |e| m_e where n is the concentration of electrons (n is used to indicate that the carriers of electricity are negative particles).

Electrical Resistivity of Metals

The resistivity then depends on collisions. Quantum mechanics tells us that electrons behave like waves. One of the effects of this is that electrons do not scatter from a perfect lattice. They scatter by defects, which can be: atoms displaced by lattice vibrations; vacancies and interstitials; dislocations, grain boundaries; impurities. One can express the total resistivity r_tot by the Matthiessen rule, as a sum of resistivities due to thermal vibrations, impurities and dislocations. The resistivity increases with temperature, with deformation, and with alloying.

Electrical Characteristics of Commercial Alloys

The best material for electrical conduction (lower resistivity) is silver. Since it is very expensive, copper is preferred, at an only modest increase in r. To achieve low r it is necessary to remove gases occluded in the metal during fabrication. Copper is soft so, for applications where mechanical strength is important, the alloy CuBe is used, which has a nearly as good r. When weight is important one uses Al, which is half as good as Cu. Al is also more resistant to corrosion.

When high resistivity materials are needed, like in electrical heaters, especially those that operate at high temperature, nichrome (NiCr) or graphite are used.

Intrinsic Semiconduction

Semiconductors can be intrinsic or extrinsic. Intrinsic means that electrical conductivity does not depend on impurities, thus intrinsic means pure. In extrinsic semiconductors the conductivity depends on the concentration of impurities.

Conduction is by electrons and holes. In an electric field, electrons and holes move in opposite direction because they have opposite charges. The conductivity of an intrinsic semiconductor is: s = n |e| m_e + p |e| m_hwhere p is the hole concentration and m_h the hole mobility. One finds that electrons move much faster than holes: m_e > m_hIn an intrinsic semiconductor, a hole is produced by the promotion of each electron to the conduction band. Thus: n = p