Electric-Field Effects on Optical Absorption near Thresholds in Solids

The weak-field effective-mass-approximation calculations of the absorption coefficient in the presence of an electric field for direct and indirect transitions at a normal ($M_0$) threshold are extended to an arbitrary orientation of the electric field in an anisotropic solid. To do this, a systematic method of evaluating density-of-states integrals arising in the electroabsorption or Franz-Keldysh effect is presented. Certain integrals obtained in prior calculations at the $M_0$ threshold which have not been previously evaluated are given in closed form. This method is also used to derive the change in absorption coefficient, $\Delta \alpha \left(\mathcal{E}\right)$, occuring at the saddle-point edges $M_1$ and $M_2$, as well as the edge $M_3$, for an arbitrary field direction in an anisotropic solid. It is shown that there is a direct correlation between $\Delta \alpha \left(\mathcal{E}\right)$ for the $M_0$ and $M_3$ edges, and also for the $M_1$ and $M_2$ thresholds. Reduced masses of opposite signs, as in the $M_1$ and $M_2$ edges, give rise to two branches in $\Delta \alpha \left(\mathcal{E}\right)$.