Analytical formalism for calculations of parameters needed for quantitative analysis by X-ray photoelectron spectroscopy
Quantitative analysis by X-ray photoelectron spectroscopy (XPS) requires knowledge of a theoretical model relating different features of recorded spectra with needed characteristics of a studied sample. An advanced theoretical approach describing an electron transport in condensed matter typically involved Monte Carlo (MC) simulations of electron trajectories since signal electrons undergo multiple interactions in a solid. The relevant algorithms are relatively slow and are burdened with statistical errors; thus they may be inconvenient in certain applications. However, much effort in the past was devoted to create models that describe electron transport by an analytical formalism with similar accuracy as Monte Carlo simulations. There are two major advantages of analytical approaches: (i) the computing time can be much shorter as compared with MC algorithms, and (ii) the relevant software can be easily included in external programs when large number of calculated parameters is needed. In the present work, the analytical formalism derived within the so-called transport approximation (TA) is described in detail, and implemented in the enclosed software TRANS_APPROX (Fortran 90). The formalism of quantitative XPS is based on an expression that provides a probability that a photoelectron emitted at a given depth reaches an analyzer without energy loss (emission depth distribution function – EMDDF). Consequently, the analytical expression for the EMDDF derived from the TA is discussed here. Stress is also put on parameters descending from the EMDDF: (i) the photoelectron signal intensity, (ii) the information depth, (iii) the mean escape depth, and (iv) the attenuation length for overlayer thickness measurements. The input parameters needed for calculations are briefly overviewed, followed by recommendations for use in the proposed program. Finally, it is indicated that the TA formalism requires calculations of numerous integrals with integrable singularities. It was proven here that these singularities do not need to be removed if the quadrature used is based on the so-called double exponential (DE) rule. This approach ensures high accuracy and fast convergence.