Electric-Field-Induced Indirect Optical Absorption in Solids and the Fine Structure of the Stark Splitting

Abstract

The problem of optical transitions in solids subject to a strong electric field ls discussed using time-dependent wave functions. Expressions for the change in transition rate for direct and indirect transitions are obtained in terms of integrals of Bessel functions. Moreover, using approximations for the Bessel functions, it is shown that the change in transition rate for direct transitions, ${T}_{d}$, and for indirect transitions, ${T}_{i}$, are given by ${T}_{d}={E}^{\frac{1}{3}}{F}_{d}\frac{({\ensuremath{\epsilon}}_{p}\ensuremath{-}{\ensuremath{\epsilon}}_{g0})}{{E}^{\frac{2}{3}}}$ ${T}_{i}={E}^{\frac{4}{3}}{F}_{i}\frac{({\ensuremath{\epsilon}}_{p}\ensuremath{-}{\ensuremath{\epsilon}}_{g0})}{{E}^{\frac{2}{3}}}$ where $E$ is the applied electric field, ${\ensuremath{\epsilon}}_{p}$ the excitation energy, and ${\ensuremath{\epsilon}}_{g0}$ the minimum energy gap. These formulas are valid both below and above the absorption edge. As a special case a machine computation of the change in the transition rate for Si with field applied in the [100] direction is presented. The results show a series of decaying peaks. In addition, a fine structure is predicted of the same kind as that predicted by Callaway for the Stark splitting. The details of the fine structure are analyzed and are shown to depend on the particular structure of the bands both near and far above the minimum energy gap. Difficulties in the experimental observation of the fine structure arising from inhomogeneity of the electric field are discussed.