Sum rules in surface differential reflectivity and reflectance anisotropy spectroscopies

The sum rule for the imaginary part of the dielectric function epsilon" can help in discriminating surface and bulk-related optical transitions, by me ans of the number of electrons taking part in the transitions. This is part icularly simple in the region below the main critical points of the bulk ba nd structure (below 3.5 eV for Si, below 2.2 eV for Ge), since there the su rface differential reflectivity is roughly proportional to epsilon" of the surface. We discuss the cases of Ge(1 1 1)-2 x 1 and Si(1 1 1)-2 x 1 investigated by surface differential reflectivity in the energy range 0.3-3.5 eV, with lig ht linearly polarized parallel and perpendicular to the chains of the Pande y model. A key point is that the sum rule holds for each polarization separately (te rmed alpha and beta), implying that: integral omega epsilon (")(alpha)d omega = integral omega epsilon (")(beta) d omega. In these results, the effect of the sum rule can be easily recognized (cros sing of the curves and seemingly equal areas on both sides of the crossing point). Similar considerations should also hold for reflectance anisotropy spectros copy (RAS). It will be shown, however, that a compensation of positive and negative signals, in compliance with the sum rule, is actually expected onl y if RA spectra are multiplied by a suitable function. Moreover, the compen sation is not exact, when the bulk is strongly absorbing. The results of a numerical simulation valid for various materials and surface oscillators ar e reported. They help to understand why in some cases (for instance in GaAs (0 0 1)-2 x 4 and -4 x 2), the sum rule is apparently not obeyed. (C) 2001 Elsevier Science B.V. All rights reserved.