Practical guide for curve fitting in x-ray photoelectron spectroscopy

Over the past 30 years, x-ray photoelectron spectroscopy (XPS) has become the most widely used surface analysis tool and has been an essential component of many research studies.¹ Curve fitting has been widely used for more than 50 years for extracting chemical information from the overlapping features in high-resolution XPS spectra.² Despite computational advances and higher accessibility of software resources, it has been challenging to develop a chemically and physically meaningful approach to curve fitting. The absence of a distinct theoretical description of XPS fitting has led to the publication of erroneous conclusions about surface chemistry.^2,3 In an ongoing study of XPS data in three high profile journals,³ it was observed that roughly 70% of the papers using XPS analyzed the data using some type of curve fitting. Furthermore, errors, misconceptions, and bad curve-fitting practices accounted for most of the serious problems in both the measured XPS data and the spectral analysis that were identified in more than 30% of the papers analyzed. This guide is intended to help address this important problem.

Curve fitting, also known as peak fitting,⁴ is the process used to extract information from the spectral data for a number of techniques. Although the details of curve fitting depend on the technique in question, the curve fitted spectra generally contain overlapping peaks. Each of these peaks is represented by a function that reflects the physical process involved in generating the original signal. XPS data interpretation and representation range from a rudimentary understanding/extraction of the elements present in a material to advanced peak fitting and background analysis that reveal chemical states and sample morphologies.

In XPS, it is convenient to identify two spectral regions, namely, the core region (electrons with binding energies, BEs, greater than 30 eV) and the valence band region (BE < 30 eV). In the core region, the spectral features arise from photoelectrons generated from core energy levels (atomic orbitals), which are characteristic of the individual atoms in the sample.⁵ In contrast, the features in the valence band region arise from photoelectrons generated from energy levels that typically involve the chemical interactions between the atoms in chemical bonding (the molecular orbitals).⁵ The spectral features of both the core and valence regions are sensitive to the chemical environments of the atoms in a sample. In the core region, chemical shifts in the BEs yield a series of typically overlapping peaks when the sample has atoms in different chemical environments, e.g., oxidation states.⁶ The resulting data can be analyzed by curve fitting. Curve/peak fitting is often the only way to extract quantitative information from these spectra.

The objective of curve-fitting high-resolution core XPS spectra with a set of component peaks is to separate the photoemission signal originating from distinct elemental or chemical states. Parameters of importance extracted from curve fitting are the photoemission peak BE, full width at half maximum (FWHM), lineshapes, and area. The position of a component peak provides evidence for assigning it to an elemental or chemical environment. The FWHM and peak lineshapes are indicators of the chemical and physical environment of the atom. Photoemission peak lineshapes can vary from simple, narrow, symmetric shapes to complex structures characteristic of the different oxidation states of metals.⁵ Quantitative information about the concentrations of the chemical states identified is inferred by measuring the relative area of each component peak, the calculation of which requires the separation of zero energy-loss signals from the inelastically scattered background. The determination of accurate approximations to the background signal is also an integral part of the XPS peak fitting process.

Curve fitting is sometimes misdescribed as being deconvolution.⁴ Deconvolution is a different process to curve fitting. Deconvolution is a process that attempts to remove the broadening that results from instrumental factors, separating the “true” spectrum from the observed spectrum. The process results in a new spectrum with the data altered by the deconvolution process. Deconvolution will not be discussed in this paper.

Our goal is to demonstrate useful, practical approaches for analyzing XPS core regions (high-resolution or narrow scans) and what common mistakes should be avoided when curve fitting. Although there are several types of common errors in peak fitting, effective and useful approaches to peak fitting vary somewhat with the analysis objective, the type of sample, and the complexity of the spectrum. Fitting may be used to verify the composition of a pure sample (Sec. V), to separate overlapping peaks for quantitative analysis (Sec. VI B 1), to obtain the relative amounts of each phase from a sample with multiple phases of the same element (Sec. VI B 2), and to examine in great detail the chemical spectrum for an element or phase for comparison with theory. This guide describes multiple complications that can appear with fitting the core region, including multiplet splitting, conduction band interactions, and the use of achromatic X-radiation.

The following sections of the paper highlight important topics for consideration during peak fitting. Section II describes a fundamental approach to peak fitting, while Sec. III describes typical peak fitting errors observed in the literature and offers recommendations to avoid them. Section IV addresses issues to think about before starting to curve fit spectra (instrument setup, adequate data collection, charge correction, lineshapes, and phenomena impacting lineshapes and widths). Section V shows the fitting of poly(ethylene terephthalate) (PET) as an example of fitting a reference spectrum from pure material. Section VI summarizes a variety of tools and information that are critical for avoiding common mistakes in curve fitting. Important useful topics addressed include dealing with overlapping peaks, fitting transition metal spectra, appropriately constraining spin–orbit split peak ratios and energy separation, and others.

A peak fitting model is defined in terms of component peaks and a background algorithm. The component peaks are specified using lineshapes, which are mathematical functions, and fitting parameters that permit a component peak to vary in a variety of ways, including position, FWHM, area, Lorentzian character, degree of asymmetry, and Gaussian character (The first four of these parameters will apply to all peaks, the latter two in some specific cases.) Sets of component peaks are summed (sum of all component peaks) and then added to a background to form an approximation to the original data envelope.

The procedure of fitting component peaks to reproduce experimental photoemission spectra utilizes mathematical algorithms to minimize a figure-of-merit as a measure of the closeness of the mathematic model to the experimental data. The fitting algorithm incrementally alters the adjustable parameters, which leads from a current state to a termination state that represents a local minimum with respect to the figure-of-merit. The ability of the functions to produce a reasonable fit from an initial guess in the iterative curve-fitting process can vary. It is crucial to choose an initial guess that is appropriate based on the physics and chemistry of the instrument and the sample. A poor set of initial guesses can cause the fitting algorithm to direct the fit into a local minimum that is not representative of the sample physics and chemistry. That is, the curve-fitting process can give a mathematically good or even excellent fit even though the component peaks in the final fit are not chemically or physically reasonable.

A figure-of-merit, e.g., $χ 2$ or the standard deviation of the residuals, is a single number that is often used to guide adjustments to the numerous fitting parameters in the peak fitting process. The fitting process may involve numerous paths though the parameter space with the possibility of hitting and getting stuck in a number of local minima. It is, therefore, possible to obtain a range of outcomes depending on the starting point of the fit. There are a number of additional checks that can be used to ensure that the physicochemical quality of a peak fit is adequate and that the current fit is not just the best mathematical solution. These include the monitoring of the quality of the data reproduction with respect to the noise; the applicability of the curve-fitting model to data of similar origin; the generation of accurate peak areas for spin–orbit components; and the accuracy of peak ratios for peaks from different elements or oxidation states of an element in a known chemical moiety.

Optimization algorithms mathematically fit peaks to data. They lack significant input from physics and chemistry, so optimization based on a single value figure-of-merit may not yield comprehensive, scientifically meaningful results. Physical and chemical information can be added to a peak model through the selection of the lineshapes, the number of component peaks within a peak model, and the parameter constraints, all of which present a means by which known relationships are imposed on otherwise independent component peaks. That is, the likelihood of achieving a chemically reasonable fit is increased if the initial guesses are based on solid scientific principles.

The art of fitting data with peaks is in selecting the appropriate number of component peaks, making use of appropriate lineshapes, and limiting the set of fitting parameters through parameter relationships to produce peak models capable of measuring physically meaningful quantities from XPS spectral forms. The challenging part arises in selecting the correct physical solution from the set of potential mathematical solutions. In practice, this is best done by obtaining verification of the fitting model by repeating these measurements on equivalent samples to confirm that the observed changes are representative of the samples. Data collected should include measuring data from standard samples to verify the performance of an instrument and understand the limitations of XPS when analyzing similar materials of known composition.

Following the principles outlined in this guide should help the analyst avoid many of the errors listed below. Nevertheless, these errors are regularly seen in the scientific literature.³ Of course, there are many examples in papers of proper XPS analysis and peak fitting that include appropriate considerations of the physical and chemical nature of the relevant spectra. Some of the errors listed below are superficial and do not impact the conclusions of a paper. However, some fundamentally alter the interpretation of the data and the conclusions drawn from it. Table I lists the errors in XPS peak fitting that we have observed in the scientific literature and suggests ways to avoid them. A more detailed discussion of the errors can be found in a report.⁷ 

The chemical information that can be determined from curve-fitting XPS spectra can only be physically meaningful if the data are appropriately collected, and proper consideration is made of all the physical and chemical contributions to the photoelectron spectra.

Energy calibration of the instrument is a prerequisite for accurate peak energy identification. Any instability within an instrument has the potential to alter lineshapes and accurate kinetic energy registration. Regular instrumental calibration and check-up are critical to ensure that the instrument is performing within the acceptable limits of reproducibility and accuracy⁹ needed for physically meaningful interpretation of curve fit results of spectra.

Figure 1 shows two Nb 3d photoemission spectra collected from a sample that was mainly metallic with a small amount of NbO and NbO₂ present on the surface. The bottom spectrum displayed an incorrect (compressed) spin–orbit splitting for the metallic component and a second component that did not correspond to one of the expected oxide components. These problems were identified by comparison with Nb spectra published in the literature (previously collected data and resources described in Sec. VI A) and were due to a malfunctioning pass energy (PE) control board. The correct spectrum in Fig. 1 (top) has all three expected components (Nb, NbO, and NbO₂).

Even though modern XPS instruments are becoming more stable and reliable, it is crucial to perform regular energy scale and linearity calibrations as prescribed by the standards.^11,12 Recently, a procedure that allows the performance and calibration of an XPS instrument to be checked rapidly and frequently was published.¹³ 

The quality of information extracted from curve-fitting XPS spectra can only be as high as the quality of the experimental data. The spectra acquired are not always of sufficient intensity and resolution to perform useful peak fitting. Instrumental contributions to the spectrally resolvable chemistries are discussed below. The analyst should also consider the signal-to-noise of the spectra produced. For elements with low sensitivities and/or that are present in low concentrations, overinterpretation of noise as chemistry is a common error.³ Another possible error is to acquire data that have a poor signal-to-noise (S/N) ratio and to smooth it before curve fitting. (Note that the S/N ratio varies across an XPS survey spectrum.³) Smoothing may introduce changes in the spectral shape and width;^14,15 changing the underlying physics, and may lead to erroneous interpretation of curve fitted spectra.

Accurate BE identification relies not only on the instrumental energy scale calibration but also on the accuracy of charge correction if charge compensation is employed during data acquisition. Charge compensation is often required when measuring XPS spectra from insulating materials. A charge builds up near the surface of these materials as the photoelectrons are emitted. This usually shifts the photoelectron energy and distorts peak shape. For details on charge compensation, please refer to the XPS Guide: Charge compensation and BE referencing for insulating samples.¹⁶ 

One of the most influential aspects of a successful peak model is the lineshapes or the shapes/functions appropriate for modeling the photoemission process after measurement by an XPS instrument. These lineshapes must accurately represent, or at least closely approximate, the signal measured by an instrument. Therefore, it is essential to understand the role of the instrumentation in modifying peaks.

The XPS process is relatively simple.¹⁷ X-ray light in the intrinsic process causes an electron to move from an initial, occupied energy level to a final, virtual energy level that has energy greater than the Fermi level of the solid under study. The electron is then lost from the virtual final state energy level and escapes from the solid at a specific energy. This energy is subsequently measured in the spectrometer. The uncertainty principle gives the initial energy level a Lorentzian energy distribution (discussed in Sec. IV E). The basic shape of the XPS peak recorded in a spectrometer is modified by instrumental and other factors (such as phonon broadening) to give a Gaussian contribution.¹⁸ This leads to a measured lineshape that is a convolution of a Gaussian and a Lorentzian function. This function, known as the Voigt function,¹⁹ most accurately reflects the nature of this intrinsic XPS process.

XPS peaks can be expressed by Voigt functions that vary in their amounts of Gaussian and Lorentzian character. This can be done by identifying a Gaussian (ghm) and a Lorentzian (lhm) FWHM.

In choosing a Gaussian–Lorentzian mix, it is important to note the following points:

• Typically, it is found that compounds have a more Gaussian peak shape than metals, which usually have considerable Lorentzian character. One reason for this is the presence of optical phonons in compounds, whereas metal has principally low-energy acoustic phonons leading to an increase in vibrational broadening in compounds and thus more Gaussian character. This topic is discussed in Sec. IV E.

• In the curve-fitting process, the analyst starts the process by suggesting possible values for the various peak parameters such as BE, FWHM, Gaussian/Lorentzian character, and peak heights. It is usually necessary to fix some of these parameters in order for the iterative process to converge. The analyst will determine typical values for Gaussian–Lorentzian mix for their instrument, but typical values for many instruments are given below:

  • o Compounds: glmix below 10% for Voigt functions and M = 0.5 for pseudo-Voigt product functions and M = 0.2 for pseudo-Voigt sum functions.

  • o Metals and elements: glmix above 60% for Voigt functions and M above 0.8 for pseudo-Voigt functions and M above 0.6 for pseudo-Voigt sum functions.

The FWHMs of the Voigt function peaks used in core XPS peak analysis depend on several factors discussed below.

The photoelectron peak is broadened to give a Lorentzian shape that depends on the lifetime of the core hole resulting from the photoelectron process. The core hole corresponds to an excited state, and various processes fill this hole, especially x-ray or Auger emission. There are several tabulations (Refs. 22 and 23 and references therein) of natural linewidths, a number of which are based upon experimental data from x-ray emission spectroscopy and Auger spectroscopy and calculations. Krause and Oliver published²³ a tabulation of core energy levels from atomic number 10 to 110, and many other papers report natural linewidth values, including those using XPS data.²² 

The natural linewidth of core energy levels increases as the binding energy, BE, of the core energy level increases. Thus, the linewidth of the Kα_1,2 x-ray emission lines increases as the atomic number (Z) of the x-ray target increases (by a factor of Z⁴) for the K lines.²⁴ For example, aluminum Kα x rays have a much narrower linewidth than higher energy chromium Kα x rays. Higher energy x-ray sources are used in hard x-ray photoelectron spectroscopy, abbreviated as HX-PES or HAXPES. Photoelectrons from deeper core levels can be excited by higher energy x rays compared to soft Al Kα x rays. For example, for Al, a soft x ray can excite up to the 2s orbital only, but a hard x ray can excite up to the 1s orbital.

Figure 3 shows the Al 2p and Al 1s core regions obtained using a Cr x-ray source, which has an energy of 5414.7 eV. The kinetic energy of Al 2p photoelectron is ∼5336.8 eV, which is much higher than the kinetic energy of Al 1s electron (∼3850.3 eV). The spectra in Fig. 3 are overlayed to the same maximum and intensity for comparison. In Fig. 3(a), the Al 2p curve fitted with two asymmetric spin–orbit components separated by 0.44 eV and at 3:2 intensity ratio. The causes and treatment of asymmetry are discussed in Sec. VI F. In Fig. 3(b), Al 1s and Al 2p_3/2 components of Al 2p transition are overlayed to the same maximum and intensity for comparison. The FWHM of Al 1s spectrum is 1.35 eV while that of Al 2p_3/2 component is 1.08 eV, demonstrating the increase of lifetime broadening for the deeper core level.

The spectrometer that measures the kinetic energy of the photoelectrons adds additional width to the natural linewidth, as well as giving Gaussian character to the peak lineshape. This means that the observed linewidth is generally substantially different from the natural linewidth necessitating the use of the Voigt function described above. Tables of natural linewidth (e.g., Ref. 22) can be helpful in setting the Lorentzian component of the Voigt function in a curve fit of a spectrum but rarely describe the measured peaks themselves. Typically, the analyst should obtain the spectrum with the best resolution available from the specific spectrometer, consistent with a balance between the appropriate signal to noise ratio and analysis time available, used to get the best data for curve fitting.

Most XPS instruments use hemispherical analyzers. Usually, the photoelectrons have their energy adjusted in a lens system so that they enter the hemispherical analyzer with particular kinetic energy called the PE. The analyzer settings, including PE, contribute to the resolution function of the instrument and hence to the energy resolution and photoemission peak shape. Therefore, it is important to determine the energy resolution for the operating mode used to acquire spectra that will be curve fitted.

The chemical environment of the atom from which the core photoelectron was ejected has a substantial effect on the FWHM.⁴ In particular, the intrinsic photoelectron process couples with phonons in the solid, resulting in broadening of the core peak and giving the peak Gaussian character. Metal phonons are often dominated by acoustic phonons of low energy, whereas metal compounds such as oxides and carbonates have extensive optical phonons of much higher energy resulting from multiple solid-state vibrational modes.²⁵ Thus, metals normally have substantially narrower FWHM values than metal compounds and more Lorentzian character. Figure 17 in Sec. VI F for the Al 2p region illustrates this—it can be seen that the oxide peak has a substantially greater FWHM than the metal peak when measured at high resolution.

The photoelectron process can be simplified into a three-step model:²⁶ 

• Step 1: The excitation of the electron from the initial energy level to the final virtual orbital state. Processes associated with this step are called intrinsic processes.

• Step 2: The transfer of the photoelectron to the surface. Processes that occur after the first step are called extrinsic processes.

• Step 3: The escape of the photoelectron from the surface into the vacuum.

In addition to the key intrinsic process, there are intrinsic processes that involve the ejection of a photoelectron, as follows.

• Plasmons²⁵ can give rise to additional peaks in a spectrum. Both bulk and surface plasmons are possible. For example, aluminum metal has a rich plasmon structure.²⁷ Plasmons can also be extrinsic.

• Shake-up⁵ processes in which a valence electron and a core electron are excited. If the valence electron is promoted to another valence energy level in the sample, this is called a shake-up process. If the valence electron is ejected as an additional photoelectron, the process is called a shake-off process.

• Multiplet splitting⁵ arises from samples that have unpaired electrons because there is more than one final state depending upon the number of spin up and spin down electrons in these final states. Transition metal compounds are often paramagnetic and, in this case, display multiplet splitting in their core XPS data (discussed in Sec. VI C).

• High Binding Energy Assymetry²⁸ Conduction Band interactions give a tail of approximately exponential shape to the core XPS of conducting materials (Sec. VI F).

• Plasmons, which can also be intrinsic.

• Inelastic processes often occur between ejected photoelectrons and other atoms and molecules. These processes cause loss of energy of the photoelectron and represent the principal contribution to the background in XPS.²⁹ There are effective software models for the study of this loss structure, and these studies can provide valuable information. These processes can have a large effect on peak ratios as a function of emission angle.

Most XPS spectra have a significant background, and accounting for the background contribution is an important aspect of the curve-fitting process. The choice of the type of background and selection of end points has the most significant impact on the area under the curve, and hence the ratios of the individual peaks used to curve fit the spectrum. There have been numerous studies devoted to approaches for background inclusion, and a forthcoming paper⁸ is devoted to this topic. There are three major approaches:

• The removal of the background to yield a spectrum with an approximately horizontal background. This approach alters the experimental data with the subsequent curve fitting being made on the altered data. This can lead to errors depending upon the chosen background model. It also deprives the viewer of the ability to compare the original experimental data with the curve fit. This approach is not recommended.

• The addition of a fixed background to background subtracted spectrum before peak fitting, where this background is not subtracted from the spectrum, and it is shown in the final fit. This approach is mathematically identical to the previous one, but it allows the viewer to see the original data, the background, and the fit together.

• In the active approach, the background contains parameters that are included as potentially variable factors in the fit. This method also preserves the data in its original form but allows for different backgrounds in different regions of the spectrum. In some insulators and gases,³⁰ e.g., in near ambient pressure XPS,^31–34 the XPS background can be linear, but most XPS backgrounds in metals and semiconductors are nonlinear. The active approach allows for addressing these differences in complex materials.

We fit here a reference spectrum taken at high resolution from an XPS database of organic polymers—PET.³⁶ PET is an important reference polymer that is used for verifying instrument performance, including charge neutralization validation.^4,13,37,38 Monochromated Al K_α x rays (1486.3 eV) were used to collect these data on a Scienta ESCA300 spectrometer. The energy spread of the monochromator is predicted to be 0.26 eV. A PE of 150 eV was used, and high energy resolution was achieved by using an appropriate analyzer entrance aperture width. The FWHM of the C 1s envelope was determined to be 0.67 eV. Low-energy electrons were used for charge compensation.

The PET structure is shown in Fig. 4. Based on its chemical formula (C₁₀H₈O₄)_n, there is 71 at. % of C and 29 at. % of O in PET. There are 10 carbons in the repeating unit of the polymer, 4 of which are present in the aromatic ring (labeled 1) with a BE of 284.7 eV. Two other carbons (labeled 2) in the aromatic ring are secondary shifted carbons such as C*—C(=O)O, which will contribute to the BE of ∼285.3 eV. Two carbons that are singly bonded to oxygen (labeled 3) have BE of 286.2 eV. The other two carbons are in carboxyl (ester) groups (labeled 4), and their BE is 288.7 eV. Out of the four total oxygens, half are present in the O=C part of the carboxyl/ester group (labeled 1) at 531.6 eV, and the other half are in the O—C part of the carboxyl/ester group (labeled 2) at 533.2 eV.

The C 1s and O 1s spectra are accompanied by a survey scan confirming that no other elements are present at the PET surface. When a Shirley background is used for quantification of the high-resolution C 1s and O 1s spectra, 69 at.  % C and 31 at. % O are detected. Slightly different concentrations due to the possible presence of adventitious carbon³⁸ are observed.

Figure 4 shows C 1s and O 1s high-resolution spectra (narrow scans) with component peaks added to initiate a curve fit. The original data from the “High Resolution XPS of Organic Polymers: The Scienta ESCA300 Database”³⁶ have been fit using the commercial software MultiPak 9.9.³⁹ Four component peaks are added to the carbon spectrum, reflecting the number of different chemical environments that contribute to the resolvable BE shifts discussed above. Due to the aromatic structure of the benzene ring, a shake-up signal is observed above 290 eV, which should represent ca. 7% of the total carbon signal based on the reference data.³⁶ This shake-up contribution is fit with one or more peaks.

Once the necessary component peaks have been added into the envelope, constraints on the adjustable parameters need to be established. If the goal is to understand the chemical state information in the spectra, adding parameter constraints will improve the stability of the peak model with respect to the noise. However, with each new constraint, the output from the optimization converges ever more closely to the solution defined by these constraints until the usefulness of fitting peaks to the data is lost. If the goal is to measure the intensity changes for a series of samples from the same study and a peak model is well formed, then a rigid peak model that is heavily constrained will tend to return more consistent intensities for data with similar chemistry and experimental conditions. Somewhere near the middle of these two extremes is the domain for most photoemission peak fitting.

FWHM constraints and the indirect constraints imposed by lineshape selection are the two main parameters used in peak fitting. Other types of constraints, such as spin–orbit intensity ratio for spin–orbit splitting, are discussed in Sec. VI C. For spectra with well-defined peak resolution, e.g., the C 1s narrow scan of PET, the FWHM can be chosen based on the widths of the individual peaks to reproduce the experimental curve as best as possible. Some variation (ca. 10% of the FWHM or ±0.1 eV) might be justified among different peaks due to small secondary effects. As described above, a function having both Gaussian and Lorentzian character often best describes the experimentally derived spectra. The pseudo-Voigt function that is used to approximate the Voigt function is typically either a product function or a sum function. Both functions involve the mixing of equal width Gaussian and Lorentzian functions with a mixing ratio (M) defined in the analytical function. The mixing ratio, M, takes the value 0.0 for a pure Gaussian and 1.0 for a pure Lorentzian, though some authors have the reverse definition. During the iterative fit, a fixed GL mixing ratio can be chosen, or a range of GL mixing ratios can be used during optimization to find the best fit. In this example, a sum function of Gaussian and Lorentzian (GLS) with M = 0.1 (where M = 0.0 is for 100% Gaussian) was used.

Similarly, based on the sample chemistry, two peaks to be fit are added to the O 1s spectrum to best approximate the number and type of chemical moieties present. That is, as predicted by the chemistry of PET, the O 1s spectrum is fit with two distinct peaks of approximately equal height and similar FWHM. Due to the aromatic structure of the benzene ring, some shake-up signature is observed above 539 eV. One more peak is added at ca. 535 eV to account for surface contamination.

During an iterative fit, constraints of FWHM and lineshapes are used to minimize the residuals—the difference between experimental and fitted curves. It is good practice to show the residuals of a fit. Figure 4 and Table II show the final parameters used to fit the spectra. The percent area obtained is slightly different from the data published in the Handbook due to the different lineshapes used in the original Handbook.³⁶ The resulting peak model consists of the peak position and percent area that each component peak occupies out of the total carbon signal. The component peak position information is then translated into the chemical state of the sample. Curve fitting of the C 1s spectra results in 40.3% of the area coming from C¹ (284.8 eV), 20.2% from C² (285.3 eV), 20.2% from C³ (286.3 eV), and 19.3% from C⁴ (288.7 eV) (with shake-up peak at 291.3 eV excluded from the total area). Curve fitting of the O 1s spectra results in 42% of the area coming from O¹ (531.7 eV) and 58% from O² (533.2 eV) (with shake-up peak at 538.5 eV and contamination peak at 535.0 eV excluded from the total area).

Based on the area under each component peak, the quantification of PET can be derived, as shown in Table II. This direct, quantifiable comparison of the carbon and oxygen in the polymer can be used to confirm both the structure of the polymer and that the curve fit adequately reflects the chemical structure.

There are several resources available to assist in curve fitting, including complex spectra that contain several chemical components. The NIST XPS database⁴⁰ and XPS Spectra (Chemical Shift/Binding Energy)⁴¹ are searchable databases containing line positions, chemical shifts, values for doublet splitting, and energy separations of photoelectron and Auger-electron lines. X-ray Photoelectron Spectroscopy (XPS) Reference Pages⁴² and the XPS Reference Table of Elements⁴³ contain curve-fitting details, BEs, FWHMs, spin–orbit splitting values, references, and other practical notes for most of the elements in the periodic table. The XPS simplified website also contains useful information for narrow scan analysis.⁴⁴ 

There are various valuable databases that contain high-quality spectra of known materials that can be downloaded. These include:

• Representative spectra for each element in a book published by Physical Electronics and used by analysts for many years.¹⁰ 

• The journal Surface Science Spectra (SSS). This journal has an invaluable collection of spectral data with full experimental details. From volume 15 onwards, data can be downloaded as a text file. SSS contains many examples of peak fitting.

• eSpectra®⁴⁵ is a database and comparative tool that contains the complete collection of the spectra published in Surface Science Spectra as well as from other sources.

• The XPSSurfA database hosted by La Trobe University in Australia contains spectra from high-quality samples that were collected under precise conditions.⁴⁶ 

In this example, the N 1s spectrum of N-doped HfO₂ overlaps with the Hf 4p_3/2 region. The Hf 4p_3/2 peaks appear between 375 and 390 eV, while some higher BE components contribute to the energy range where a signal from N 1s is expected. To solve this problem of overlapping spectra, it is important to acquire the spectra from the reference material, a thick Hf oxide film without nitrogen doping in this case. Figure 5 shows a high-resolution Hf 4p_3/2 region in which several peaks are included to fit the experimental data.

The Hf 4p_3/2 region [Fig. 5(a)] has features that occur in the same region where N 1s signal is present. In this example, the purpose of curve-fitting Hf 4p_3/2 spectrum is not to extract the chemistry of Hf but rather to create a mathematically rigid model in which peaks overlapping with the N 1s region are constrained in position, width, and area to the higher intensity peak in the lower BE region which is free of overlap with the N 1s region. Two peaks due to Hf 4p_3/2 in the region where N 1s is located, i.e., between 390 and 405 eV, are constrained in intensity, position, and width with respect to the most intense peak at 383 eV. For an N-doped material with two different loadings, the rigid set of peaks is introduced into the high-resolution N 1s/Hf 4p_3/2 region [Figs. 5(b) and 5(c)]. The intensity of peaks overlapping with N is controlled by the height of the most intense peak, and this leads to a poor fit. This fit, though, is consistent with the Hf, so additional peaks must be added to complete a curve fit. These peaks are the contribution from N 1s photoelectrons. This approach is successful even in situations when N loading into the sample is very low, assuming that the data have been acquired with an appropriate signal to noise.

High-resolution spectra of transition metals are among the most challenging spectra to peak fit.^47–49 Indeed, it is common to see peak fitting of transition metals that oversimplify assignments of peaks to individual chemistries and subsequently have erroneous interpretations of the results. There are several contributing factors as to why peak assignments are difficult with transition metals. The first is the combinations of intrinsic and extrinsic processes, such as multiplet, shake-up, and plasmon loss structures, contributing to the overall photoelectron shape of the pure metals and their different coordination states. The second is the overlap of the various chemical states usually present in transition metal-based systems. The third is that exposing transient metal compounds to air always results in oxidation and formation of both stochiometric and substochiometric oxidation states. Another complication is that these oxides can have varying degrees of conductivity, which may affect the position and shape of photoelectron peaks due to charging. Early and even current databases, such as NIST and Handbook of X-ray Photoelectron Spectroscopy,¹⁰ usually indicate that there are multiple chemical states that can correspond to the same BE and, at the same time, they provide single BE for a unique chemical environment of the transition metal. However, as multiple reports of reference compounds have shown,^47–49 complex spectral structure, which is the result of an envelope of multiplet peaks and/or other intrinsic and extrinsic effects, is often present with the different chemical states of the element. Simply put, the multiple chemical states of a single transition element often give radically different spectra.

Figures 6(a)–6(c) show spectral signatures for pure metallic Ni, NiO, and Ni(OH)₂, as reported by Biesinger et al.⁴⁷ The complexity of the spectral signatures due to contributions of the multiple processes that can occur when the photoelectron escapes the 2p orbital is obvious. Therefore, the most accurate approach to solve this problem is to use a curve-fitting parameter either developed by the analysis from pure reference spectra, if they are available, or, to use curve-fitting parameters from reference compounds published in the literature. The first approach is preferred as it ensures that the same spectral and analyzer conditions are used to collect the data from the samples and reference materials. We emphasize that the series of peaks used to fit the narrow scans in Figs. 6(a)–6(c) are simply employed to fit the peak envelope to create an overall shape that will be used in subsequent peak fitting, more than they are intended to have specific chemical meaning.

Figure 6(d) overlays the photopeak shapes (or envelopes) that were created by Biesinger et al. based on fitting the narrow scans in Figs. 6(a)–6(c) for Ni, NiO, and Ni(OH)₂.⁴⁷ The ambiguity of finding a unique spectral signature for pure reference material is obvious. However, through a linear combination of the photoelectron signatures representing the pure chemical states, curve fitting of the mixed chemical states of the transition metals is possible. This is essentially a classical least squares (matrix algebra) problem in which one has the spectrum of the mixture of compounds as well as those of the pure components.⁵⁰  Figure 6(e) shows a curve fit of the spectrum from a mixed Ni-Mo oxide, which has had the background removed, using this type of complex photopeak model. The relative ratio of the various chemical states of Ni can be extracted from this information, assuming that one knows information on the physical form of the material (powder, spheres, layered surface structure, etc.) supporting the chemical composition derived by XPS fitting.

In certain situations, an analyst is confronted with the need to curve fit spectra from complex multicomponent materials where finding appropriate references may not be possible and limited input from other spectral features exists. In this case, care should be taken in deciding physical constraints to be used during the optimization of curve fit parameters.

For most elements, participation in chemical bonding with other elements causes an increase in BE, especially when the elements to which they are bonded are more electronegative. That is, for elements having multiple possible oxidation states, more oxidized chemistries will usually show higher BEs. This pattern is well established for carbon, where an increased number of oxygen atoms bonded to a carbon atom leads to a steady increase in its BE.⁶ Here, the presence of more electronegative oxygen atoms deshields the carbon atom, effectively increasing the nuclear charge felt by its electrons, which raises their BEs (and lowers their kinetic energies in XPS). In many cases, the least oxidized (lowest BE) form of an element will produce the “cleanest” peaks in an XPS spectrum that can then be used to select the best FWHM value for the fit. However, this will not always be so—it will depend on the sample chemistry.

Figure 7 shows a high-resolution C 1s spectrum in which the highest intensity component results from the contribution of carbon in the lowest oxidation state in the material, which is also the most abundant. In Fig. 7(a), the FWHM of the component peak is too narrow, which results in part of the spectral area being under fitted. In Fig. 7(c), the FWHM value is too broad, which results in overfitting the right shoulder of the spectrum. The best match between the component peak and the experimental data is shown in Fig. 7(b). The FWHM value obtained from this match is probably an adequate representation of the physical, chemical, and instrumental contributions to the data. To a good first approximation, this FWHM could be used for other component peaks that might be considered in the fit. Nevertheless, the lone peak at ca. 289 eV in this spectrum is another reasonable choice for determining the FWHM values of the fit components—it seems to have far less overlap with other signals than the larger peak around 284.5 eV. It would be advisable to consider both FWHM values in this curve fitting (they should be quite similar). Further analysis would be needed to determine which would be most appropriate, or their average, range of values near the average, etc.

Area parameter constraints are particularly useful for spin–orbit peaks, which are doublet peaks originating from the same electron configuration initial state where the final state electronic configuration is split between two possible outcomes. Physics for the photoionization of electrons dictates these double peaks appear offset in energy, representing the difference in energy between the final states and at a defined ratio in terms of peak area (inset, Fig. 8). Scofield cross sections⁵¹ were calculated for photons with energies corresponding to Al and Mg x-ray gun anodes, and these cross sections include relative intensity values for both peaks in a doublet pair. Thus, the ratio for these peaks can be readily fixed by making use of Scofield cross sections to estimate the relative area imposed by area constraints within a peak model. Figure 8 shows the S 2p spectrum that was fitted with two sets of doublet peaks. The separation between the 3/2 and 1/2 components of the S 2p orbitals is fixed to 1.16 eV, and so is the area ratio to 1:2.

It is worth noting that in the case of p, d, and f levels, spin–orbit splitting occurs, and in a number of cases, the FWHM value of the higher BE component is greater than the lower BE component, an effect which has been described as a Super Coster–Kronig effect.⁵² When the FWHM of spin–orbit split components differs, it is essential that the curve-fitting program has the capability of setting the FWHM of the spin–orbit components to an adjustable ratio while keeping the peak areas in the expected area ratio.

Table III shows energy separation for a few common transitions. For some transitions, such as Ti 2p and Si 2p, there may be a difference in separation for different chemical states. For example, spin–orbit energy separation is smaller for Ti oxide than for metal, while spin–orbit splitting is only used in fitting metallic Si and not higher oxidation states. The separation may be smaller than energy resolution or as large as 20 eV.

For elements like the transition metals with a rich range of oxidation states, e.g., Mo, there may be challenges using a single doublet for each chemical state. Photoemission from Mo⁴⁺ is an example of where the relationship between the chemical state and a pair of synthetic components representing a doublet is not appropriate. Following Scanlon et al.,⁵³ Mo⁴⁺ is represented by two doublets, while Mo⁶⁺ is modeled using a single doublet (Fig. 9). It is also an example of where complex data envelopes for a specific oxidation state are essential for correlating the amount of substance with intensity derived from synthetic components. That is, the amount of material calculated for Mo⁴⁺ requires the summation of the signal corresponding to four synthetic components labeled [1] Mo 3d⁴⁺ and [2] Mo 3d⁴⁺.

In some cases of spectral overlap between two different transitions, the spin–orbit splitting ratio and separation constraints can help resolve overlapping peaks. Figure 10 shows a region in which both Sb 3d and O 1s contribute. That is, the signal from the oxygen photoelectrons overlaps with a larger Sb3d_5/2 component, while the smaller Sb 3d_3/2 component is free of overlap. In this case, peaks included in the 3d_3/2 region can be a constraint in position and intensity to the peaks contributing to the 3d_5/2 region. Figure 10(a) shows two doublets from the metallic and oxide forms of Sb constrained rigidly between each other. The intensity of the two peaks in the 5/2 part of the spectrum is limited by the intensity of the two peaks in the 3/2 part. A large deviation in the residuals points to where a peak corresponding to O 1s photoelectrons should be included. When the peak due to O 1s photoemission is included [Fig. 10(b)], the contributions of overlapping transitions can be determined quantitatively.

When selecting lineshapes for use in modeling spectra, one should be aware that the choice of lineshapes has an influence on the measured relative area for components in a peak model, and therefore how the signal is allocated to a chemical state. The choice of background also influences the computed areas of the components. A regularly encountered example that illustrates these factors is that of a silicon oxide film on elemental silicon. The Si 2s and Si 2p signals are a useful example because both photoemission lines exhibit features that can be identified as silicon oxide and elemental silicon. They are relatively close in BE and therefore show similar instrumental responses, but they are different in terms of component FWHM and lineshapes. Notably, the Si 2s signal measured on modern instruments with monochromatic x-ray sources shows characteristic lineshapes with wings typically associated with a Lorentzian shape. One way to allow for extended wings in spectral features is to fit basic background curves as part of the optimization step. The Si 2s data in Fig. 11 are fitted using Voigt lineshapes representing elemental and oxide signal plus two lineshapes with background characteristics. A component in the peak model is used to emulate a Shirley response to model inelastic scattering due to elemental silicon. At the same time, an offset summed with the Shirley background is defined by making use of a top-hat lineshape of width greater than the energy interval shown in Fig. 11 that is adjusted to allow the use of Voigt lineshapes when the fitting is performed.

In the case of the silicon dioxide overlayer on silicon, the Si 2p and Si 2s core level measurements should return equivalent corrected areas for component peaks. An equivalent model for Si 2p to that of Si 2s is used for curve fitting. The principal difference between Si 2p and Si 2s is that Si 2p involves doublet peaks with significantly less Lorentzian character. Nevertheless, both peak models in Figs. 11 and 12 provide excellent data reproduction, and the use of these lineshapes is supported by observing the consistency achieved for chemical state component areas calculated from these peak models (inset table in Fig. 12). The asymmetric Lorentzian (LA) lineshape used in this example is described in detail in recent reports.^54,55

For unsaturated carbon, which is often in aromatic rings, a shake-up feature due to π–π* (HOMO–LUMO) transitions coming from the ring excited by the exiting photoelectrons must be considered.

A peak model for PEEK (Fig. 13) includes features assigned as shake-up signals typical of ring polymer spectra. The use of shake-up components in a peak model for PEEK is particularly important for modeling the O 1s signal since a shake-up feature is potentially beneath signal attributed to O—C O 1s. PEEK spectra are comprised of core photoelectron peak, shake-up, and inelastically scattered background signal. Accounting for these three sources of the signal is fundamental to accurate quantification by XPS.

Asymmetry is observed in photoemission data for a number of reasons. The origin of this asymmetry is multifaceted, including instrumental influences, sample/measurement artifacts, the band structure of the sample, and other physical reasons associated with photoemission itself. Modeling asymmetric components is arguably one of the more difficult problems related to peak fitting.^54,57,58

Historically, asymmetry has been included in lineshapes via a so-called exponential tail. The necessary form for including an exponential tail is produced by blending the influence of an exponential function with a lineshape with characteristics of a Lorentzian and/or a Gaussian resulting in a well-defined component area. Figure 14 illustrates one possible form for exponential tail modification to a Lorentzian lineshape, which was extensively used when fitting polymer spectra within the Beamson and Briggs XPS of Polymers Database.³³  Figure 15 is an example of a peak model consisting of a single component making use of an exponential tail lineshape to fit an O 1s peak measured from Nylon 6 prepared by cleaning with an argon cluster source ion gun.

There are two aspects to fitting data using asymmetric lineshapes, namely, data reproduction and quantification via a comparison of component area. The example shown in Fig. 16 illustrates one of these aspects.

The problematic lineshape used in Fig. 16 is a Doniach–Sunjic profile convoluted with a Gaussian, which certainly allows good data reproduction, but fails to permit accurate quantification as a consequence of an extended asymmetric tail. Care should be taken when using the Doniach–Sunjic lineshape, as component area changes as a function of the asymmetry parameter, and contributions from higher BE components may overlap with an extended tail.

An example of where both data reproduction and quantification are important is measuring intensities for an oxide film over an elemental substrate, such as aluminum oxide on aluminum or silicon oxide on silicon. Separation of oxide signal from an elemental signal is best performed using lineshapes for which adjustments to component parameters do not have significant consequences for the component area, which is not the case for Doniach–Sunjic profiles. For this reason, asymmetry in photoemission peaks is typically modeled using lineshapes other than the Doniach–Sunjic profile.⁵⁹ 

The case of aluminum oxide on aluminum metal (Fig. 17) is offered as an example of where relating metal and oxide intensities measured by fitting components to data is the basis for estimating a thickness for an oxide film on a metal substrate. In this example, a lineshape is modified to simulate geometric aberration induced asymmetry within measured photoemission lines. Figure 17 demonstrates the use of a subtle but important asymmetry within narrow metallic components. The peak model also includes a background generated by these metallic components, which is modeled with the Shirley algorithm applied to the metallic components only. The oxide background is modeled as linear. The model in Fig. 17 represents a systematic tool for comparing intensity from aluminum oxide and aluminum metal, from which film thickness can be estimated using the Hill Equation.⁶⁰ 

The mathematics of optimization suggests that making use of lineshapes that best approximate photoemission shapes within a peak model is important to chemical state determination. Using lineshapes closest to observed peak shapes reduces the need to enforce parameter constraints during optimization, bearing in mind that the application of parameter constraints represents an input of user bias to a peak model. A number of components and lineshapes are essential inputs to peak models, while parameter constraints should be considered nonessential inputs but required to compensate for imperfections in lineshapes, backgrounds, and issues with signal-to-noise. There is, therefore, just cause to investigate asymmetric lineshapes other than the traditional forms, particularly for situations where narrow components are involved, and instrumental artifacts have increased influence over observed peak shapes.

In this section, the impact of FWHM on a peak model is considered. The data analysis in Fig. 18, performed using a so-called uniqueness plot,⁶¹ shows that a range of plausible figure-of-merit values is obtained through optimization involving specific FWHM values for the Ga 3s component. The uniqueness plot demonstrates how one can select an informed, fixed FWHM parameter for Ga 3s for use in fitting curves to an individual spectrum as the only means of establishing a physically meaningful peak model.

Each peak model includes parameters for peak position, FWHM, and area. FWHM differs from position and area in the sense that FWHM depends upon instrumental energy resolution, data acquisition conditions, as well as on the chemical state of the sample. Ideally, the BE and relative area for the component peaks will be independent of the instrumental energy resolution. Furthermore, there is correlated information between the lineshape and FWHM in the sense that lineshape parameters chosen also alter FWHW for the curve being fitted using the FWHM fitting parameter. By making a uniqueness plot, one can assess how a particular fitting parameter changes the outcome of curve fitting. This technique is applied to the FWHM for the data and curves shown in Fig. 18.

Optimization of the peak model shown in Fig. 18 was performed using optimization parameters that were free to adjust within physically meaningful intervals. The standard constraint (discussed in Fig. 8) on the component areas of the S 2p doublet components was utilized in the original optimization model. One means of testing the influence of FWHM on a peak model is to specify a fixed FWHM value close to the value for the FWHM resulting from the original optimization, then refitting the new model to the same data. If repeated for a range of fixed FWHM values, a set of solutions with a new figure-of-merit result. The plot in Fig. 19 is that of a figure-of-merit plotted as a function of fixed FWHM values for the Ga 3s component in Fig. 18. The curve in Fig. 19 clearly shows a minimum in the figure-of-merit and significant deviation away from the minimum. The flatter this curve, the smaller the effect of the parameter on the fit. A flat curve in the uniqueness plot can often indicate that the peak that it belongs too is not appropriate for use in the peak model.

Lineshapes used in peak models are not necessarily transferable between data collected from different instruments. The most apparent differences occur for data collected using achromatic x-ray sources compared to monochromatic x-ray sources. XPS spectra collected with achromatic x rays are more complicated to fit due to the presence of an x-ray satellite structure. While many modern XPS instruments use monochromatic x rays, there are still many instruments using achromatic x-ray sources. The energies of the x-ray satellites are well known, and to be effective, the curve fits need to include additional peaks for photoelectrons excited by these typically higher energy x rays.

In most cases, up to seven peaks can be included for each of the principal XPS peaks, these peaks corresponding to photoelectrons ejected by Kα_1,2, Kα′, Kα₃, Kα₄, Kα₅, Kα₆, K_β x rays. Since curve fitting is generally conducted over a narrow energy range (e.g., 10–25 eV), only the Kα′, Kα₃, Kα₄ x-ray satellites typically need to be included. Peaks excited by Kα′, Kα₃, Kα₄ x rays have intensities representing about 1%, 8%, and 6%, respectively, of the intensity of the main peak and are at 4.6, 8.4, and 10.2 eV lower BE than the principal peak. The other higher energy x-ray satellites need to be included when a wider energy range is fitted, for example, for a 50 eV range to cover the two spin–orbit split Ag 3d peaks.⁶² Information for x-ray satellite energies and relative intensities can be obtained from x-ray emission spectra^29,63 or from XPS studies.⁶² 

The valence band region provides complementary information to the core region since the appearance of the spectrum in this region is determined by the effect of chemical bonding and, in the case of many materials, especially metals, the extensive dispersion of the electron energy levels. This leads the valence band region to have a complex spectral shape where curve fitting is not as simply performed for the analysis of valence band as it is in core-level XPS. Overlays of the ground state density of states calculated using first-principles band structure theory,⁶⁴ or in suitable cases by cluster calculations, can be quite useful in interpreting valence band spectra. A study of the often complex shape of the valence band region can assist identification since it is the pattern that is characteristic of the material so that errors in calibration and thus BEs are less of an impediment to the correct interpretation of the data than they are in the core region. In other words, the valence band region can be a useful “fingerprint” for identifying materials.⁶⁵ Listed below are some examples where the valence band region can be especially effective in the identification of different chemical species, where this is particularly true when monochromatic X-radiation is used:

• Metals often have very distinct features in the valence band region, for example:

  • o Molybdenum metal has a distinct four feature pattern at BEs below 5 eV, which can be explained by the calculated spectrum.^21,66

• Silver metal has a distinct four feature pattern at BEs below 8 eV, which can be explained by the calculated spectrum.²¹ 

• Oxides often have distinct valence band features, including a multiple peak pattern in cases where the core region exhibits one peak with little or no BE shift between different oxidation states, for example:

  • o Molybdenum oxides have distinctly different valence band spectra.^21,67

  • o Aluminum oxides and oxyhydroxides cannot reliably be distinguished in the Al 2p core region but show substantial differences in the valence band region.^64,68

• Molecular ions such as sulfates, carbonates, and phosphates have distinguishable valence band features, for example:

  • o Carbonates and bicarbonates can be distinguished.⁶⁹ 

  • o Different types of phosphates show no difference in the P 2p core region, but substantial differences in the valence band region (Fig. 20).^70,71

• Polymers typically give a valence band that is rich in features. Thus, organic polymers with little or no difference in the C 1s region often show a valence band with very characteristic features.³⁶ 

To enable others to understand and assess the results obtained from fitting of spectra, it is necessary that both the results and information about the fitting approach, constraints, and assumptions and final fitting parameters be included in journal articles or reports. ISO Standard 19830 describes the minimum reporting requirements for peak fitting in XPS.⁷² Important parameters include spectrometer and x-ray characteristics, the use of charge compensation and charge correction, the type of spectral pre-preprocessing such as satellite subtraction, smoothing, and deconvolution, the type of background used, fitting parameters such as lineshapes, FWHM, and χ², as well as the software used for curve fitting.

For the full list of minimum reporting parameters, please refer to ISO,⁷² while here we include some of the recommended parameters. On the experimental side, the photon energy and the type of source (monochromatic or achromatic) should be listed. The type of analyzer and the geometry of the measurement should be described, including the expected resolution of the measurement. Finally, the form of the sample (solid, thin-film, powder) should be given. On the fitting side, the fitting parameters should be given in detail, including a description of the backgrounds used in the fit. Lineshapes and linewidths should be documented. If linewidths used vary significantly across a spectral fit, an explanation of why this is physically reasonable should be included. The software used in the analysis should be listed. Spectra should have a complete core level spectral range to include background and spin–orbit components. Tables of parameters, including error bars, will make the fitting information more accessible to the reader.

As the use of XPS data in many areas of science and technology has become increasingly important, and the complexity of advanced materials has increased, the extraction of important physicochemical information increasingly requires the fitting of XPS data. Unfortunately, the processes used and the information obtained are often incorrect or incomplete,³ and information associated with fitting processes and results is often not adequately reported. The good practices and information presented in this guide are provided as a guide to improving the quality of the XPS curve fitting reported in the literature.

One of the main objectives of the curve fitting of XPS data is to extract chemical information about the sample being examined. Consequently, curve fitting needs to be done in a manner that is consistent with the physics and chemistry of both the XPS process and the specimen being examined. It is also important to remember that the quality of chemical information extracted depends on the quality of data acquisition and data processing.

Many errors related to fitting and peak identification in the literature arise from authors not recognizing the impacts of not only intrinsic but also extrinsic phenomena, such as spin–orbit splitting or the effects of plasmons, multiplet splitting, or shake-up processes. The XPS peak positions and lineshapes are significantly affected by these processes. Such effects highlight the importance of relevant reference spectra in the understanding and fitting of peak structures.

In the application of curve fitting to XPS spectra, reference materials, consistency of approach to both reference and target materials, paying attention to the physics and chemistry of the photoelectron peaks, and appropriate reporting of the approach, software used, and fitting parameters are all critical. This guide has provided a summary of important issues and described tools and strategies that represent good practices important to useful, informative, and reproducible peak fitting of core-level XPS spectra.