Band nonparabolicities, broadening, and internal field distributions: The spectroscopy of Franz-Keldysh oscillations

Abstract

The complex dielectric function of a semiconductor or insulator in a uniform electric field is expressed asymptotically as the sum of a zero-field, a third-derivative, and an oscillatory (or exponential) term by appropriately deforming a contour of integration in the complex plane. The latter term describes separately the field-induced change in the dielectric function arising from Franz-Keldysh oscillations (or exponential edges). The present expansion extends the region of validity of the previously developed low-field approximation to a wide range of energies and applied fields. The stationary-phase formalism by which this term is evaluated allows the asymptotic behavior of the Franz-Keldysh oscillations to be calculated by a series of algebraic steps, a simple procedure that permits the influence of various perturbations on the Franz-Keldysh oscillations to be studied generally. We find by application of the theory that the period of the oscillations is relatively unaffected by band nonparabolicities, broadening, or field-distribution effects, showing that the value of the interband mass at the critical point determined from them is highly accurate. By contrast, the envelope of the oscillations is shown to depend strongly upon these interactions. Since Franz-Keldysh oscillations appear to be relatively unaffected by exciton effects and can be measured for several different critical points of the same crystal, measurement of their field dependence can provide a stringent means for evaluating current theories of internal-field distributions and represent and attractive alternative to exponential-absorption-edge studies for systematic investigations of these effects.