Universal analytical formula for the emission depth distribution function for photoelectrons with kinetic energies up to 5000 eV

Numerous parameters needed for quantification of X-ray photoelectron spectroscopy can be derived from the emission depth distribution function (EMDDF) known for a given photoelectron line, an experimental configuration, and a solid. This function describes the probability that a photoelectron emitted at a certain depth enters an analyzer without energy loss. The EMDDF turns out to be distinctly affected by photoelectron elastic scattering effects. A useful measure of influence of elastic scattering is a closely related function named the correction factor (CF). This function is a useful tool for correcting the in-depth concentration profiles obtained from the procedure of non-destructive depth profiling. Due of complexity of the theoretical models describing transport of photoelectrons emitted by unpolarized X-rays in the surface region of solids, a typical computational approach involves Monte Carlo algorithms with different simulation strategies. However, convenient sources of EMDDFs (and CF functions) are analytical formulas that can be implemented in the software dedicated for practical surface analysis. The present report evaluates accuracy of different analytical formulae. A new expression is proposed which can be considered as a predictive formula, i.e., a simple analytical expression of reasonable accuracy for typical measurement conditions. This formula is applicable to photoelectrons emitted by unpolarized or circularly polarized X-rays. Furthermore, stress is put on the influence of non-dipolar parameters that define a high-energy photoemission cross section on the EMDDF for photoelectrons with kinetic energies exceeding 1500 eV. Eventually, a photoelectron kinetic energy of 5 keV was established as an upper limit of applicability of the presented formalism. Finally, applicability of the analytical formalism, derived for unpolarized X-rays, to photoelectrons emitted by linearly polarized radiation is analyzed. Due to the considerable influence of the position of the polarization vector on the photoelectron signal, the analytical formalism developed for unpolarized radiation cannot be recommended for use if the of X-ray beam is linearly polarized or has partial polarization.