Solids with thermal or static disorder. II. Optical properties

The effect of temperature and impurities on optical properties of solids is discussed with particular emphasis on features absent for free electrons but present for Bloch electrons. Using an equation-of-motion method, a generalized Boltzmann equation valid for both intraband and interband conduction is obtained. This is formally solved to give an expression for the conductivity $\sigma \left(\omega \right)$ nearly identical to the usual Kramers-Heisenberg dispersion formula except for a complex, frequency-dependent function $M_{\mathrm{kn}{n}^{\prime }}\left(\omega \right)$ which serves to shift and broaden the energy denominators. When the external frequency $\omega$ is on resonance with an interband transition ${\epsilon }_{\overrightarrow{k}n}-{\epsilon }_{\overrightarrow{k}{n}^{\prime }}$, the function $M$ becomes to good approximation $\Sigma (kn, {\epsilon }_{\overrightarrow{k}n})-{\Sigma }^{*}(k{n}^{\prime }, {\epsilon }_{\overrightarrow{k}{n}^{\prime }})$, i.e., the one-electron self-energy shift. When $\omega$ is on resonance with an indirect transition ${\epsilon }_{\overrightarrow{k}n}-{\epsilon }_{{\overrightarrow{k}}^{\prime }{n}^{\prime }}$, the standard theory of phonon- or impurity-induced indirect absorption is obtained. New results are given for the shift of the dc dielectric constant. The connection with Holstein's and other quantum Boltzmann-like equations (for intraband conductivity) is described in an appendix.