Conductivity is the most fundamental electrical property of a material. Conductivity, denoted by σ, is defined as the ratio of the electric current density **J** and the electric field **E**. Since **J** and **E** are vectors, σ is technically a matrix. For most materials, **J** and **E** can be safely considered collinear, in which case σ is a scalar. Assuming σ is a scalar, we arrive at an equation that is commonly called Ohm's Law.

J = σE (Ohm's Law)

Of course, it is nonsensical to define a quantity as the ratio of two quantities and call it a law! What Ohm's Law really says is that σ is a constant that is independent of the magnitude of E. Ohm's Law usually holds for homogeneous solids--solids made of a single type of material--under typical experimental conditions. Ohm's Law always fails eventually since if the current density gets too large a solid will melt! Ohm's Law is completely invalid when describing the current density vs. electric field relation at the junction of two dissimilar materials (see p-n junction).

One might wonder why Ohm's Law is valid and if it is possible to understand what determines the conductivity of a material. The standard *derivation* of Ohm's Law and the formula for conductivity is as follows.

Imagine that an electric field accelerates electrons (or holes). Newton's second law tells us that a = F/m = qE/m. Assume that on average an electron undergoes a "collision" every τ seconds, after which the electron's new velocity is zero and the electric field accelerates it again. Since v = qEt/m, the average velocity of an electron, v_{ave}, is qEτ/m.^{*} Now current density, by definition, is given by J = qnv_{ave}, where n is the concentration of electrons (in units of #/m^{3}). Thus J = q^{2}nτE/m. If we assume that n, τ, and m are independent of E, we have Ohm's Law. The conductivity is given by

σ = nq^{2}τ/m (siemen/meter).

^{*} The ratio of v_{ave} to E--qτ/m-- is called mobility.

This derivation of the formula for conductivity is obviously crude, at best. The physics of electron (or hole) transport was almost completely ignored. However, the equation is typically quite valid, as long as the parameters are correctly characterized. m is *not* the "real" electron rest mass. Rather, m is the effective mass of a material; m includes information on how the atoms in a solid affect electrical transport. τ is fairly difficult to characterize quantitatively. Qualitatively, collisions are due to vibrations of the atoms in a solid (phonons) or imperfect crystalline structure of a solid (philosophically phonons are imperfection of crystalline structure).

The most important factor that determines conductivity is n, the "carrier concentration." One might wonder if n should include every electron in a solid, or if it should include only one or more of the valence electrons. The determination of carrier concentration is fairly difficult and requires a background in solid state physics. The basic electrical difference between metals and semiconductors/insulators is that metals have much higher values of n. At room temperature, the conductivities of different material types are as follows:

Insulators: σ ~ 0

Semiconductors (undoped): σ ~ 10^{-2} siemen/meter

Metals: σ ~ 10^{2}-10^{8} siemen/meter

Superconductors: σ --> infinity

The inverse of conductivity is called resistivity.

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