Intensity of Optical Absorption by Excitons

The intensity of optical absorption close to the edge in semiconductors is examined using band theory together with the effective-mass approximation for the excitons. Direct transitions which occur when the band extrema on either side of the forbidden gap are at the same K, give a line spectrum and a continuous absorption of characteristically different form and intensity, according as transitions between band states at the extrema are allowed or forbidden. If the extrema are at different K values, indirect transitions involving phonons occur, giving absorption proportional to ${\left(\Delta E\right)}^{\frac{1}{2}}$ for each exciton band, and to ${\left(\Delta E\right)}^2$ for the continuum. The experimental results on ${\mathrm{Cu}}_2$O and Ge are in good qualitative agreement with direct forbidden and indirect transitions, respectively.