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, \(\bf{P}\):
where \({\bf -P}\) is the electric field from the internal charges, \(\rho_{int}\).
The polarization is related to the dipole moment of the material, \({\bf p}\),
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:
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 [Wis63]:
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, \(\chi\):
The derivation of the last relation can be found in [Mar04].
[Mar04] | Richard M. Martin. Electronic structure: Basic theory and practical methods. Cambridge University Press, New York, 2004. |
[Wis63] | N. Wiser. Dielectric constant with local field effects included. Phys. Rev., 129:62–69, Jan 1963. doi:10.1103/PhysRev.129.62. |