Optical conductivity

While the static electrical conductivity is vanishingly small in insulators (such as diamond or porcelain), the optical conductivity always remains finite in some frequency intervals (above the optical gap in the case of insulators). The total optical weight can be inferred from sum rules. The optical conductivity is closely related to the dielectric function, the generalization of the dielectric constant to arbitrary frequencies.

High frequency limit

In the simplest cases, this property can be considered as a complex valued scalar function of frequency only. This formulation applies in the limit of long wavelengths, coarse grained structure, and cubic material symmetry. In this approximation, the electric current density ${\displaystyle \mathbf{J}}$ (a three-dimensional vector), the scalar optical conductivity ${\displaystyle \sigma }$ and the electric field vector ${\displaystyle \mathbf{E}}$ are linked by the equation:

${\displaystyle \mathbf{J}(\omega )=\sigma (\omega )\mathbf{E}(\omega )}$

For comparison, the permittivity ${\displaystyle \epsilon }$ relates the electrical displacement to the electric field:

${\displaystyle \mathbf{D}(\omega )=\epsilon (\omega )\mathbf{E}(\omega )}$

In SI units, this implies the following connection between the two linear response functions:

${\displaystyle \epsilon (\omega )={\epsilon }_0+\frac{i\sigma (\omega )}{\omega }}$,

where ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity and ${\displaystyle i}$ denotes the imaginary unit.