Quantitative depth-dependent analysis using the inelastic scattering backgrounds from X-ray photoelectron spectroscopy and hard X-ray photoelectron spectroscopy

The inelastic scattering contribution to an X-ray photoelectron spectrum (XPS) reflects the distribution of elements within the sample depth. Varying the energy of the incident photons changes the inelastic scattering contribution. We present a standardless, automated method that exploits this change to enable compositional analysis within the XPS information depth (also known as amount of substance, AOS 3 λ ). In this method, the photoelectron intensities observed using two or more X-ray photon energies are normalised using correction factors. These correction factors are derived to determine the AOS 3 λ from the inelastic scattering signals collected from different samples. The AOS 3 λ are then compared with the ratio of the elastic XPS peak intensities calculated for different sample depths to find the layer thickness. The method has been applied to analyse spectra collected from thin amorphous carbon films to yield quantitative, standardless and automated XPS analysis.

element of interest. 11 Automated algorithms have been developed to enable inelastic scattering background analyses to be performed by non-specialists. 12 The automated method for inelastic scattering background analysis described by Tougaard determines the amount of substance, in atoms/nm 2 , within three IMFP lengths (AOS 3λ ). It does so by assuming all depth distributions can be approximated by an exponential with a characteristic decay length, L. This assumption has been demonstrated to be satisfactory for most types of sample. 12,13 The algorithm determines the most appropriate exponential function to approximate the depth profile based on the measurement of the photoelectron peak area, A p , and the intensity of the inelastic scattering background, B, at a point 30 eV higher than the peak binding energy. It has been remarked that it is somewhat strange that this simple method has not yet been incorporated in commercial software systems for routine analysis. 14 One barrier for its implementation may be the requirement for homogeneous reference samples with the same atomic density as the element(s) of interest to be measured for calibration. This is not convenient, nor is it always possible. It is therefore useful to develop a technique to calibrate the AOS 3λ with measurements obtained with the sample in situ and without the need for reference standards.
In this work, we describe a method for calibrating the depth of inelastic scattering background measurements by collecting spectra with two different, well-separated photon energies. The relative attenuation of the photoelectron peaks collected with each X-ray energy is used to determine the average layer thickness. This has been made possible in the lab by the recent supply of XPS and HAXPES systems with higher energy X-ray sources from instrument manufacturers. The method would also be applicable to synchrotron-based experiments where multiple X-ray energies are available.

| THEORY
The photoelectron peak intensity for a given element in a thin film sample measured with X-ray photon source 'X' can be described by, where t is the film thickness, z is the depth and λ X is the IMFP for electrons ionised by photon source 'X'. I 0 is the initial or 'unattenuated' photoelectron signal intensity.
Factors, including transmission function corrections and relative sensitivity factors, are used when performing quantitative XPS analysis to normalise the signal intensity from photoelectron peaks originating from different elements or chemical states so that their relative abundance can be compared. In essence, these factors are used to make I 0 in Equation (1) equal to 1. The stoichiometry of the material within the information depth can be calculated once this normalisation has taken place because the photoelectron intensity is contributed to by every atom within that volume.
If a similar intensity correction could be made for a photoelectron peak measured using two different X-ray energies, then it follows that the ratio of the number of atoms within the two different information depths will be equal to the ratio between the intensity-corrected peak areas. We can write, This concept can be used to quantify the amount of a particular material within the information volume of an XPS experiment.
The AOS 3λ algorithm published by Tougaard 15,16 provides a method to calculate the number of atoms within the information depth using a single X-ray energy. This algorithm requires inputs related to the intensity of the inelastic scattering background and the measured photoelectron peak intensity. As mentioned above, these values are usually calibrated using measured values from homogenous standards. The rationale is similar to Equation (2). The intensity of the inelastic scattering background and photoelectron peak intensity from samples with thickness different to 3λ are compared with the values measured from 3λ worth of material. The requirement for the use of standards can be removed by using two different X-ray energies and calibrating their photoelectron peak intensities using a correction factor, CF, which will be derived below. The ratio of the AOS 3λ values determined using different X-ray sources can then be compared with the theoretical values for the photoelectron peak intensities calculated using Equation (1), where I 0 has been set to 1, to determine the amount of material present.
For data collected using Al Kα and Ag Lα X-ray sources, the value for t is found by solving the following equation: Figure 1A shows the XPS signal intensity as a function of depth for a C 1s electron excited by different X-ray sources. The attenuation of electrons occurs at a much higher rate when excited by lower energy X-rays, such as Al Kα, when compared with higher energy X-rays like Ga Kα (9.25 KeV). The photoelectron peak intensity measured using each X-ray source can be determined using Equation (1) by integration of these curves. Figure 1B shows the ratio of the C 1s peak intensity measured using an Al Kα X-ray source and a higher energy X-ray source as a function of thickness. Each thickness has a unique value for this ratio. For experiments using Ag Lα X-rays, the curve is The calculation of AOS 3λ , as it is to be used for this method, will now be described. The first step is to normalise the measured XPS spectrum J(E) so that the intensity (min, max) is (0, 1).
The inelastic scattering background to be subtracted from J(E) is then calculated. The background-subtracted spectrum f(E) is described by where E is the energy at a point a few eV below the range of interest and B is a term used to scale the intensity of the inelastic scattering background. B is adjusted so that f(E) is zero at a point 30 eV higher than the peak binding energy (E Max ). K is the inelastic scattering cross section.
K(T), where T is the electron energy loss, can be described by the following expression: The B, C and D parameters are, generally, constants that can be looked up for different classes of material. 17 However, it is worth highlighting that, whereas B, C and D are material dependent, B is used as a fitting parameter in the AOS 3λ algorithm.
Next, the characteristic decay length of the exponential approximating the atomic depth distribution, L, can be calculated where λ is the IMFP, θ is the photoelectron emission angle and B H is the value for B measured from a standard homogeneous sample. B H can be approximated to 3200 eV 2 for calculation without the use of standards (see Discussion below for further details).
The AOS 3λ can then be calculated by where A p is the measured photoelectron peak intensity (distinct from I X , which is theoretically calculated).
The calculation of the CF values will now be explained. For a pure material, homogenous in z Therefore, if I 0 is set to unity, then A p = I X = λ. As shown below in Figure 2A, the value of I Al /I Ag is equal to λ Al /λ Ag for infinitely thick material. The CF required in Equation (3) so that we may normalise A p for comparison between spectra acquired using Al Kα and Ag Lα sources when analysing infinitely thick material is 1. For this type of sample, A p can be normalised to 1 without affecting the results. However, when analysing thin films of finite thickness, Therefore, if we wish to normalise A p , a CF must be applied to account for the change in elastic XPS peak intensity that would be observed if I 0 = 1 for all X-ray sources. The CF for films of thickness The CF required for a carbon layer of thickness t is shown in Figure 3B. The CF required for a buried layer is defined by Whereas the CF required for a substrate underneath an overlayer of thickness t is simply F I G U R E 1 (A) Calculated C 1s X-ray photoelectron spectrum (XPS) signal attenuation as a function of depth, z, for measurements with different X-ray sources. (B) Relative peak intensity (C 1s) measured with Al Kα (I Al ) and higher energy X-rays (I X ) as a function of t. Note that the X in the legend denotes the X-ray source and I X is its intensity. Figure 3 summarises the multi-photon AOS 3λ (MPA) algorithm, as it will henceforth be referred to. The first step is to normalise the spectra collected from the Al Kα XPS experiment and the chosen HAXPES source, so that their baseline intensity value is 0 and their peak intensity value is 1. Secondly, an appropriate K for the material of interest is calculated. This is then used to fit a background to the two measured spectra, and values for B are determined using Equation (4). Equation (7) is then used to calculate a relative value for AOS 3λ from both spectra. These values are then inserted into Equation (3), which is solved numerically to find t. It is important to note that, as the value for t is iterated, both the value of I Al /I X and CF change. Once a solution is found, the average atoms/nm 2 can then be found by multiplying t by the atomic density of the material c H if required. Table S1 shows the absolute values of AOS 3λ for the samples measured in this work.

| METHODS
The amorphous carbon (a-C) samples measured in this work were deposited onto Si wafers with thermally grown SiO 2 layers using a filtered cathodic vacuum arc. The deposition conditions have been described in detail elsewhere. 18 Thin film samples deposited onto Si/SiO 2 were chosen for this analysis because they form ultra-smooth F I G U R E 2 (A) I Al /I X for C 1s electrons. The value of I Al /I X tends to λ Al /λ X for infinitely thick materials. Correction factors for analysing C 1s spectra using the multi-photon AOS 3λ (MPA) algorithm when calculating (B) thin films, (C) substrates and (D,E) buried layers with different overlayer thickness.
F I G U R E 3 Flowchart describing the multi-photon AOS 3λ (MPA) algorithm that utilises spectra with two different, well-separated photon energies. In this diagram, the input data are collected using a conventional Al Kα X-ray photoelectron spectroscopy (XPS) source and a higher energy hard X-ray photoelectron spectroscopy (HAXPES) source labelled X. See text for details.
films but with variable density depending on the deposition conditions. Pure a-C films can be deposited with densities ranging from 1.9 to 3.5 g/cm 3 depending on the fraction of diamond-like sp 3 bonds present. The inelastic scattering background of carbon films depends heavily on the electronic density of the films. This makes it impracticable to use a reference standard in most cases, with diamond and graphitic samples being the exceptions. 19 Therefore, a standardless method is required for the inelastic scattering background to be used for quantification of these samples. formula, 21 could also be used. The algorithm was able to be automated to the point where only the XPS data, the X-ray energies and the C and D values for Equation (5) were required as inputs. The C and D parameters for the C 1s and O 1s spectra analysed in this work were calculated such that the absolute intensity of f(E) was minimised in the 10-to 30-eV range. This was necessary for a-C samples and the O 1s signal of SiO 2 because they have not previously been reported in literature. The optimal C and D values for the a-C films were found to vary greatly because the inelastic scattering background is highly dependent on c H , which varied from sample to sample (see Table S1). A Python script to optimise the C and D values for samples with unknown inelastic scattering cross sections is included in Data S4. Figure 4A,B shows the inelastic scattering backgrounds from normalised C 1s spectra collected from (nominally) 3.5-and 10-nm-thick films, respectively. Note that these thickness measurements were determined by surface profilometry of a step-edge in the film at a site away from the XPS analysis site. The spectra collected using Al Kα and Ag Lα sources are shown overlaid. The difference in the intensity at 30 eV is clearly observable when comparing the two spectra in Figure 4A, indicating that the inelastic scattering is reduced in the thinner sample when the Ag Lα source is used. The analysis of the inelastic background for this sample using Equation (4) is shown in Figure 4C,E, whilst the fitting parameters are displayed in Table 1.

| RESULTS
Here, the B parameter values were determined to be 57 and 15 eV 2 , for spectra collected with the Al Kα and Ag Lα sources, respectively.
However, in the data from the thicker film shown in Figure 4D,F, the difference in the background intensity is less pronounced, and the B parameters were found to be 148 eV 2 (Al Kα) and 117 eV 2 (Ag Lα).
The implication is that due to the higher electron energy and hence, F I G U R E 4 (A,B) X-ray energy-resolved C 1s spectra collected from films of thickness $3.5 nm (top row) and $10 nm (bottom row). Corresponding inelastic background fitting from Al Kα X-ray photoelectron spectroscopy (XPS) is shown in (C) and (D). Inelastic background fitting from Ag Lα XPS is shown in (E) & (F).
higher IMFP, there will be a reduced number of inelastically scattered electrons from spectra collected using the Ag Lα source when measuring thinner films. This can be used as a reference to compare against the inelastically scattered signal in spectra collected using a lower energy X-ray. If the A p values are the same, then the difference in the B parameter between the two spectra will be indicative of the thickness. Figure 5 shows thickness measurements from several samples using the MPA algorithm. These results were compared with those obtained using a single overlayer thickness calculator (Equations SE1-SE5; see Data S1). A correlation is observed, confirming experimentally that the MPA algorithm can be used to estimate thickness. It also shows that the thicknesses of films slightly above 3λ may be estimated as long as I Al /I X is >λ Al /λ X , although increased error is expected due to the lower signal from this depth.
As with the conventional AOS 3λ algorithm, quantitative nondestructive depth profiling can be performed using the MPA algorithm. However, it must be used in conjunction with the correct CF value for the layer of interest. Using inelastic background analysis allows thickness measurements to be made on multilayer films in a 'top-down' fashion. That is, the thickness of the topmost layer can be determined without the need to consider the attenuation of XPS signal from underlying layers (e.g., using the Beer-Lambert law). This process can be repeated for subsequent layers in the stack down to a thickness of 3λ, allowing depth profiles to be calculated from spectra collected at normal incidence. In fact, for the MPA algorithm to perform quantitative analysis on buried layers in multilayer samples, knowledge of the thickness of the overlayers, as well as K(T) for these layers, is required. This is because the I 0 values in Equation (1) can no longer be treated as unity for data collected with X-rays of different energy. The relative attenuation by the overlayers will be higher for lower kinetic energy electrons. Therefore, in the case of buried layers, Equation (1) becomes where t 1 and t 2 are the thicknesses of the overlayer and buried layer, respectively.
Knowledge of K(T) for overlayers is required to account for inelastic scattering of photoelectrons emitted from the buried layer by the material in the overlayer. In this case, an effective inelastic scattering cross section can be used to analyse the inelastic scattering backgrounds of peaks originating from a buried layer. 22 where was set to 1.6 nm. Figure 6C shows the value of AOSAl = AOSAg on the IAl = IAg calibration curve (indicated by AOS 3λ in the annotation). The inset shows a schematic of how the calibration curve is calculated for a buried layer. I Al and I Ag for the O 1s peak are calculated using Equation (13). Note that this is equivalent to using Equation (1) and shifting the z origin in the positive direction by the value of t 1 . This method of depth profiling is quite simple when compared with angle resolved XPS because we can ignore effects such as elastic scattering and asymmetry factors. It also avoids complications due to Laplace transform inversions.
T A B L E 1 Fitting parameters used to analyse the inelastic scattering backgrounds within the spectra shown in Figure 4. F I G U R E 5 Thicknesses of continuous a-C thin films computed using MPA. The line Y = X is used to guide the eye.
Analysis of the AOS 3λ in the Si substrate will now be shown for completeness. Si 2p data from the same sample are shown in Figure 6.  Figure 7A,B, respectively.
Equation (14) was used to calculate K eff for the SiO 2 layer, which will have contribution to the inelastic scattering background from the ta-C and SiO 2 layers. To calculate K eff for the Si substrate, the inelastic scattering cross section needs to account for scattering in the ta-C, SiO 2 and Si layers. The inelastic scattering cross section from the overlayers was calculated by This reflects the fact that the electrons from the substrate will be transported through the entire thickness of both overlayers, in contrast to those from the SiO 2 layer. Equation (15) was then used to calculate the total K eff for the Si substrate, with K 1 = K eff,over and K 2 being the inelastic scattering cross section for elemental Si. 17 This modelled inelastic scattering cross section could then be used to analyse the experimental Si 2p spectrum and quantify the AOS 3λ in the Si substrate. It is important to use the total background when fitting the experimental data (i.e., but only use the A p value for the Si substrate when calculating AOS 3λ .
The inelastic scattering backgrounds fitted to Si 2p spectra collected by Al Kα XPS and Ag Lα XPS are shown in Figure 7C,D, respectively. The calculated backgrounds fail to describe the sharp plasmonic signal from the Si substrate, but nevertheless, the thickness calculated using the MPA algorithm was equal to 5.7 nm, which is ≈3λ Ag minus the total thickness of the ta-C and SiO 2 overlayers.
In samples consisting of a single overlayer of uniform coverage supported by a substrate, the MPA algorithm can use the XPS signal from the substrate to calculate the thickness of the overlayer. The thickness of the overlayer will be equal to 3λ Ag (or the IMFP of electrons measured using the highest photon energy) minus the thickness of the substrate within 3λ Ag . Examples of this measurement were performed on synthetic spectra and are shown in Figure S5. It can be observed from Figure S5e,f that consistent results are obtained using spectra from the overlayer and substrate, respectively.

| DISCUSSION
The the MPA algorithm detailed above provides an automated method for quantitatively analysing the inelastic scattering background signal from samples of finite thickness where bulk analogues are not available. Although the relative attenuation of photoelectrons excited by different X-ray energies has been used for thin film metrology in the past, 23 only the elastic signal from the substrate peak was considered, which precludes the method from being applied to samples with discontinuous overlayers.
In order for quantitative measurements to be made, Tougaard's AOS 3λ algorithm requires a reference sample to obtain values for both B H (used in Equation 6) and the calibrate the photoelectron peak  Figure S4 shows simulated spectra (see Data S5 for method) from polymer, metal oxide and silicon sample types. The quantification results comparing Tougaard's AOS 3λ algorithm with the MPA algorithm for each of these materials are shown in Figure S4c, The accuracy of the AOS 3λ values calculated will usually be limited by the signal to noise ratio of the spectrum collected using the higher energy X-ray source due to the lower photoemission cross section. 24 To improve the noise tolerance of the algorithm, the B parameters were calculated by averaging the signal across 10 data points (±0.5 eV). The monochromated Ag Lα X-ray source used in this study utilises the second order diffraction from the quartz monochromator mirror, so the incident X-ray intensity is also reduced. It is expected that the accuracy can be improved by using higher X-ray flux sources. Another source of error at depths >2λ is the decreasing value of d I Al = IX À Á =dt. This effect is exacerbated by the lower signal from >2λ and makes it increasingly difficult to discern the thickness value at larger depths, although this is an issue for all non-destructive XPS measurements. One strategy to increase d IAl = IX À Á =dt is to increase the energy difference between the two X-ray sources, as can be observed in Figure 1B. Tougaard used noise-free synthetic spectra to calculate the accuracy of his original AOS 3λ algorithm. The conclusion is that the algorithm is accurate to within 10-20%, depending on the thickness of the films analysed. 13 Although they are benchmarked against the thickness determination made using the method in Data S1, which will have its own uncertainty, our experimental results from Figure 5 also show agreement to within 20%.
Altering the X-ray energies can also be of use if an experiment requires that either higher binding energy peaks or thicker samples are analysed. The binding energy range is dictated by the lower in energy of the two X-ray sources used, whereas the maximum thickness range is determined by the highest energy X-ray source used.
The results obtained from the MPA algorithm will be very similar to those achievable with the commercial software package Quases-Analyse and with similar ease of use. It is expected that the MPA algorithm can also be modified to include analysis of surface island structures, as is possible with Quases. However, because there are few steps beyond fitting a common Tougaard background to the input spectra, the MPA algorithm may be preferable for users (including software developers and instrument manufacturers) to implement. It must be said that Quases will retain an advantage for quantitative analysis for lab-based HAXPES spectra because it will be easier to find peaks free of interference from Auger peaks. This could be overcome by using a synchrotron-based HAXPES system or, possibly, by building lab-based systems with multiple hard X-ray sources.

| CONCLUSION
In conclusion, an algorithm has been described that provides an automated method for quantitatively analysing samples of finite thickness where bulk analogues are unavailable. The algorithm retains the advantages of Tougaard's AOS determination methods. That is, by analysing the inelastic scattering background in addition to the XPS peak intensity, limitations due to sample inhomogeneity, surface roughness, elastic scattering and asymmetry parameters can be avoided when characterising samples with a varying concentration profile in the z direction. This method removes the requirement for standards when quantification is performed, which will broaden the applicability of AOS 3λ measurements. The MPA algorithm will be straightforward to implement for the growing number of practitioners using instruments with dual XPS/HAXPES capabilities. Given that several manufacturers now offer these instruments, and their software packages already contain spectral analysis routines, this algorithm may be a straightforward and beneficial addition to incorporate.