Energy dependence of the optical matrix element in hydrogenated amorphous and crystalline silicon

Abstract

The energy dependence of the average optical matrix element for both hydrogenated amorphous (a-Si:H) and crystalline silicon (c-Si) has been determined experimentally, with use of the density of states and the imaginary part of the dielectric function. The valence- and conduction-band density of states of a-Si:H and c-Si were measured with use of x-ray photoemission and inverse x-ray photoemission spectroscopy, respectively. Using the sub-band-gap density of states measured by isothermal capacitance transient spectroscopy, a nearly complete experimental density of states for a-Si:H has been determined. The imaginary part of the dielectric function was measured with use of ellipsometry and photothermal deflection spectroscopy. We find that the average dipole matrix element squared for a-Si:H is constant up to ∼3.4 eV with a magnitude of ∼10 A${̊}^2$ and decreases as $E^{\mathrm{-}5}$ at higher energies, where E is the photon energy. The energy dependence of the matrix element is the same as that calculated for complex forms of crystalline silicon (T-12). The matrix element does not depend on the localization of the initial state if the final state is delocalized, indicating that the random-phase approximation is valid for a-Si:H. The average dipole matrix element for c-Si, determined in a similar manner as a-Si:H, has three peaks at 3.4, 4.3, and 5.3 eV. At high photon energies the matrix element falls off as $E^{\mathrm{-}5}$. Finally, it was found that optical determinations of the band gap for a-Si:H tend to underestimate the mobility gap.