Optical response functions¶

The optical response functions couple an external electric field, \(E_\mathrm{ext}\), with the internal electric field arising from the response of the crystal. It is convenient to introduce the displacement field, \(\bf{D}\), which determines the electric field from external charges, \(\rho_\mathrm{ext}\). The displacement field is related to the internal electric field, \(\bf{E}\), through the polarization, \(\bf{P}\):

\[\bf {D} = \epsilon_0 \bf {E} + {\bf P},\]

where \({\bf -P}\) is the electric field from the internal charges, \(\rho_{int}\).

The polarization is related to the dipole moment of the material, \({\bf p}\),

\[{\bf P } \equiv {\bf p }/V,\]

where \(V\) is the volume of the material.

Linear response coefficients¶

The linear response coefficients \(\chi\) (susceptibility), \(\epsilon_r\) (dielectric constant), and \(\alpha\) (polarizability) relate the electrodynamic quantities outlined above to each other:

\[ \begin{align}\begin{aligned}{\bf P } = \epsilon_0 \chi {\bf E },\\{\bf D } = \epsilon_r \epsilon_0 {\bf E },\\{\bf p } = \alpha {\bf E }.\end{aligned}\end{align} \]

Optical conductivity¶

For a perturbation \({\bf E}({\bf r}) = {\bf E}_0 \exp({\rm i}{\bf q} \cdot {\bf r} )\), the linear response current in the long wave-length limit (\(q \ll 1/a\) , where \(a\) is the lattice constant) is given by [1]:

\[{\bf j}({\bf r}, \omega) = \sigma(q=0, \omega) {\bf E}({\bf r}, \omega),\]

where \(\sigma\) is the optical conductivity.

Units¶

The unit of the linear response coefficients are:

Coefficient	Unit
\(\alpha\)	C²/Nm⁵
\(\epsilon_r\)	1
\(\chi\)	1
\(\sigma\)	C²/Nsm²

Relation between the linear response coefficients¶

All the response coefficients follow from the susceptibility, \(\chi\):

\[ \begin{align}\begin{aligned}\epsilon_r(\omega) = (1 + \chi(\omega) ),\\\alpha(\omega) = V \epsilon_0 \chi(\omega),\\\sigma(\omega) = - i \omega \epsilon_0 \chi(\omega).\end{aligned}\end{align} \]

The derivation of the last relation can be found in [2].