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.X射线光电子能谱（X-ray photoelectron spectroscopy, XPS）的定量分析，需要建立理论模型以关联谱图特征与样品的待测特性。凝聚态物质中电子输运的高级理论方法，通常采用蒙特卡洛（Monte Carlo, MC）模拟电子轨迹，因为信号电子在固体内部会经历多次相互作用。这类相关算法的计算速度相对较慢，且受限于统计误差，在部分应用场景中实用性欠佳。过往研究已投入大量精力，开发出与蒙特卡洛模拟精度相当的解析形式电子输运模型。解析方法具备两大核心优势：其一，相较于MC算法，计算耗时可大幅缩短；其二，当需要处理大量计算参数时，相关软件可便捷地集成至外部程序中。本研究详细阐述了基于输运近似（transport approximation, TA）的解析形式，并将其实现于附带的Fortran 90程序TRANS_APPROX中。定量XPS的理论基础为下述表达式：该表达式可给出某一深度处发射的光电子无能量损失地抵达分析仪的概率，即逸出深度分布函数（emission depth distribution function, EMDDF）。据此，本文对基于TA推导得到的EMDDF解析表达式展开讨论。此外，本文还重点分析了由EMDDF衍生出的四类参数：（i）光电子信号强度；（ii）信息深度；（iii）平均逸出深度；（iv）用于覆层厚度测量的衰减长度。本文先简要概述了计算所需的输入参数，随后给出了该程序的使用建议。最后指出，TA形式需要对大量含可积奇点的积分进行计算。本文已证实，若采用双指数（double exponential, DE）求积规则，则无需移除这些奇点。该方法可保证极高的计算精度与快速收敛性。