Sub‐Nanometer Depth Profiling of Native Metal Oxide Layers Within Single Fixed‐Angle X‐Ray Photoelectron Spectra

Many metals form nanometer‐thin self‐passivating oxide layers upon exposure to the atmosphere, which affects a wide range of interfacial properties and shapes the way how metals interact with their environment. Such native oxide layers are commonly analyzed by X‐ray photoelectron spectroscopy (XPS), which provides a depth‐resolved chemical state and compositional analysis either by ion etching or modeling of the electron escape depths. The latter is commonly used to calculate the average thickness of a native oxide layer. However, the measurement of concentration profiles at the oxide‐metal interface remains challenging. Here, a simple and accessible approach for the depth profiling of ultrathin oxide layers within single fixed‐angle XPS spectra is proposed. Instead of using only one peak in the spectrum, as is usually the case, all peaks within the energy range of a standard lab device are utilized, thus resembling energy‐resolved XPS without the need for a synchrotron. New models that allow the calculation of depth‐resolved concentration profiles at the oxide‐metal interface are derived and tested, which are also valid for angular‐ and energy‐resolved XPS. The proposed method not only improves the accuracy of earlier approaches but also paves the way for a more holistic understanding of the XPS spectrum.


Introduction
Over 75% of the known elements are metals, most of which tend to form nanometer-thin metal oxide layers on their surface upon exposure to atmospheric oxygen. [1]These native oxide layers determine a wide range of interfacial properties, including surface energy, crystallinity, permeability, conductivity, stability, and light interactions, to name just a few.They shape the way metals interact with their environment. [2]10] Their self-pacifying nature is fundamental for the corrosion resistance of metals. [11,12]The most common method to analyze thin oxide layers is, among others, photoemission spectroscopy, with X-ray photoelectron spectroscopy (XPS) being the most widely used.Figure 1a illustrates its working principle.Due to the low escape depth of photoelectrons (<10 nm), XPS is highly surface sensitive and provides information about chemical states, composition, and to some degree depth distribution. [13]arious techniques have been used to obtain compositional depth distributions from XPS data. [14]The most common method is incremental surface ablation using ion etching by Ar + sputtering (other ion sources are available but less common).However, its suitability for depth profiling of ultrathin oxide layers is questionable, because the determination of the exact ablation rate is prone to error, and ion bombardment can induce substantial chemical changes and intermixing of elements normal to the surface [15] ; the latter being especially problematic for ultrathin oxide layers.[19][20] However, these techniques account for only a tiny fraction of all published studies, owing to the large experimental effort and lack of accessibility.
Typically, instead of a spatially resolved depth profile, an average oxide layer thickness is calculated using XPS data from standard lab devices using different types of data analysis.One method is to compare the O 1s peak intensity, which depends on the oxide layer thickness, with that of a pure reference sample. [21]ougaard et al. established a formalism to estimate the overlayer thickness and surface texture using peak shape and background analyses. [22,23]26][27][28] Building on the latter approach, we demonstrate how the accuracy of a single-peak calculation can be improved through a holistic analysis of the entire XPS spectrum.We propose fit models for all peak regions within the energy range of a standard lab-XPS, thus resembling energy-resolved XPS without the need for a synchrotron.We then demonstrate that such an analysis yields not only more accurate information about the average oxide layer thickness but also about the depth-resolved concentra-tion distribution at the interface between oxide and metal.For this purpose, we derived a new model that we applied to experimental data from three widely studied metals -titanium, palladium, and tantalum-known to form self-pacifying nanometerthin oxide layers.The new model is also suitable for ERXPS and ARXPS data.Our approach can achieve high spatial resolution from single XPS spectra using standard lab devices with minimal experimental effort.Because only the peak with the highest intensity is usually used for the analysis, most peak shapes, chemical shifts, and backgrounds are virtually unexplored.Thus, this method may open the door to a more holistic understanding of the XPS spectrum, which may prove valuable for the analysis of heterogeneous surfaces.

Modeling of Oxide Depth-Distributions
The kinetic energy E kin of a photoelectron determines its inelastic mean free path (IMFP) (E kin ), which is defined as the average distance that electrons can travel between successive inelastic collisions.Its value determines the shape of the emission depth distribution function (DDF) (z), which is closely related to the Beer-Lambert law. [29] (E kin )⋅sin (1)   As illustrated in Figure 1a, the signal intensity I 0 is attenuated as a function of z, the distance from the surface, with  being the z coordinate of the integrals centroid.In other words, it describes the probability that an electron originates at a specified depth below the surface.When the electron emission angle  is smaller than 90°, the distance that electrons must travel to reach the surface increases, which is corrected for by the term sin.Each material and each of its peaks in the XPS spectrum has thus a different DDF.The function (E kin ) can be estimated using the predictive TPP-2M equation (named after Tanuma, Powell, and Penn). [30]Further discussion, equations, parameters, and results for the materials used in this study are shown in Tables S1 and S2 (Supporting Information).

Single Energy Model
When the DDF spans over both the oxide layer of thickness d and the metal underneath, it becomes discontinuous at their interface, as the IMFPs of the oxide  o and the metal  m are usually similar but not equal.In the oxide layer (0 < z < d) it can be written as In the metal layer (d ≤ z < ∞) it is written as For a given E kin , i.e., a peak position in the XPS spectrum (E kin = h − E bind − ϕ), the signal intensities originating from the oxide I o and metal I m can be determined by peak deconvolution.The metal cations Me X+ in the oxide layer have a higher binding energy E bind than the metal atoms Me 0 , whereby the chemical shift is greater with higher oxidation state.As shown in Figure 1c, I o and I m correspond to the integrals of  o (z) above and  m (z) below the oxide-metal interface at z = d.
As the density of metal atoms per unit volume is significantly lower in the oxide than in the metal, the signal intensity of the oxide is underrepresented, which is accounted for by the factors N m and N o , which are the number densities of metal atoms in the metal and oxide, respectively (sometimes replaced by an intensity ratio of pure reference samples).The intensity ratio I o /I m can be solved for d, which results in a widely used equation, that will be referred to as the single energy model, It is often attributed to Hill (1976) [24] or Strohmeier (1990). [31]owever, neither provided a derivation and both referenced earlier works.The first derivation of a version of this equation appears to be by Carlson & McGuire (1972). [32]o enhance the accuracy of the single energy model, Equation ( 5) is sometimes applied to ARXPS data as a function of 1/sin, for which the plot of ln( + 1) has d/ o as the slope. [24]It is often assumed that This single energy model is based on the assumption that the oxide layer is of uniform thickness and there is an atomically smooth interface between oxide and metal without intermixing.Further assumptions are that the X-ray attenuation, reflection, and refraction, as well as the effects of adventitious carbon and surface roughness, are negligible.These calculations focus strictly on the oxide-to-metal ratio; thus, they do not provide any information about possible hydroxide or carbon top layers.

Multiple Energies Model
Since Equation ( 5) is applied only to the peak with the highest intensity,  m and  o are treated as constants.It was developed and is most frequently used for thickness measurements of native oxide layers on silicon.As silicon shows only two significant peaks in the XPS spectrum (2s, 2p) with similar binding energies (thus, similar  values), not much accuracy is gained by using multiple peaks for analysis anyway.However, heavier metals usually have more than 4 emission peaks within the energy range of a standard lab-XPS (usually Al K radiation) with  values spread over more than 10 Å (see Figure 1b).
As illustrated in Figure 1d, the I o /I m ratio varies with .Since, in principle, the single energy model as defined by Equation ( 5) should yield the same thickness d for all core-level signals in the spectrum, it can be used as a fitting function I o /I m () to enhance its accuracy.
Thus, d is constrained by several peak deconvolutions instead of just one.This approach will be referred to as the multiple energies model.While the correction term N o /N m is indeed a constant, the ratio  o / m varies with E kin .To sidestep this rather slight inaccuracy, we used the average of  o and  m as the independent variable (weighted by intensity for spin-orbit split peak regions).The relative sensitivity factors can be ignored because the intensity ratios are determined individually within the same respective peak regions.

Depth Profile Models
The aforementioned models are based on the assumption that the interface between oxide and metal is atomically smooth.However, the analysis of multiple peaks also contains information about possible intermixing and roughness at the oxide-metal interface.To account for such a depth profile, a new model is derived that describes the intensities I o and I m by the integrals of two separate DDFs partially overlapping in the vicinity of Analogous to the multiple energies model, the intensity ratio I o /I m can be solved and used as a fitting function for the XPS data (further notes on the derivation can be found in the Supporting Information) Here, 2 F 1 (a,b; c; z) is the ordinary hypergeometric function, Γ is the gamma-function and k is the "sharpness" of the sigmoid function, i.e., the degree of interfacial mixing, which is an additional fitting parameter.Despite its daunting appearance, k is indeed the only additional parameter compared to Equation (6) (the C code for the implementation in the OriginPro software is provided in the Supporting Information).Figure 2a,b shows the behavior of Equation ( 8) for varying k and d values and the resulting sigmoid functions, which describe the respective concentration profiles.When k is large, the interface is sharp with little intermixing, whereas when k is small, it is broad with significant intermixing.It can be seen that with ever sharper interfaces, the fit function does not change much, so that the fits get less and less sensitive to slight data variations, i.e., arbitrary for atomically sharp interfaces.
Similar concentration gradients, as they occur in diffusion phenomena or that are used to describe interfaces with finite roughness, are often described by error functions.36] As an alternative solution, the intensities I o and I m can be written as convolutions of the DDF with complementary error functions erfc(z) which leads to the equation Both functions, Equation (8) and Equation (10), are equally suited to fit the data and result in almost indistinguishable concentration profiles.They can also both be used to model ERXPS data from a synchrotron or ARXPS data, when  is kept constant (i.e., for a single peak region) and  is used as the independent variable.The latter is illustrated in Figure 2c.

Modeling Experimental Data
[39][40][41][42][43] They have been characterized by different methods, including via sputtering and/or single energy models in XPS, and have been analyzed for various applications.The exact thickness and composition of these native oxide layers depend on various factors, such as surface texture, environmental conditions, and sample treatment.46] The presented results are just a showcase of the application of the new model to widely studied metals known to form native oxide layers with a thickness of a few nanometers.To the best of our knowledge, most of the examined peak regions are unexplored territories with virtually no peak models to be found in the published literature.It should be emphasized that the objective here is to explore the proposed methodology and not to substitute for dedicated studies on these materials.
The peak deconvolutions were guided by and optimized under the assumption that the I o /I m ratios obey Equations ( 8) and (10).While for some peak regions, a reasonably unique deconvolution is possible, some, especially low-intensity peaks attributed to s-orbitals, do not allow a clear distinction between oxide and metal.Peaks that can be fitted with a relatively high degree of cer-tainty, usually those attributed to p-, d-and f-orbitals, constitute the starting point for an iterative fit optimization.This results in a mutual benefit for the elucidation of unexplored peak regions on the one hand and improved depth resolution on the other.A detailed discussion of the peak fitting procedures, as well as calculations of the number densities and IMFP, is provided in the Supporting Information.

Tantalum
Charge-neutral Ta is consistent with the Aufbau principle and has a regular electron configuration of [Ar] 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 3 .Due to the ionization during the measurement, the peaks appear in the order [Ar] 3d 4s 4p 4d 5s 5p 4f 5d 6s, of which 4s to 4f are well resolved by XPS using Al K radiation (see Figure 3; 4p 1/2 and 5p are shown in Figure S2, Supporting Information). [47]The consensus is that the Ta 5+ oxide (Ta 2 O 5 ) is the most chemically stable oxide and the main component of native tantalum oxide layers. [48]However, clear evidence was found that the sample also contained nitrogen, which is likely to be constrained to the surface as it was not detectable by energydispersive X-ray spectroscopy.The intensity of the N 1s peak can be determined using the known area ratio between Ta 4p 1/2 and Table 1.Comparison of different thickness calculations: Single energy calculations were done using Equation ( 5) for the peak region given in brackets.The averaged single energy calculations give mean value and standard deviation for all peaks in the respective spectrum.Multiple energies calculations were done using Equation ( 6) as a fit function with d as the only fit parameter.The depth profiles were calculated using Equations ( 8) and ( 10), respectively, with d and k as fit parameters.Figure 3c shows the intensity ratios weighted by the number density ratio N Ta 5+ ∕N Ta 0 (see Supporting Information).While the thickness d ≈ 46 Å is close to the values obtained by single-and multi-energy modeling (see Table 1), the depth profile obtained by Equation ( 8) (equivalent to Equation ( 10)) indicates a gradual transition from oxide to metal over a range of about 30 Å.However, the data is similarly consistent with the simpler multiple energies model assuming a flat interface.

Palladium
The electron configuration of Pd, [Ne] 3s 2 3p 6 4s 2 3d 10 4p 6 (5s 0 ) 4d 10 , is irregular.In the spectrum, the peaks appear in the order [Ne] 3s 3p 3d 4s 4p 4d (5s), of which 3s to 4p are well resolved by XPS using Al K radiation (Figure 4; 4s is shown in Figure S3, Supporting Information). [47]The consensus is that the Pd 2+ oxide (PdO) is the most chemically stable oxide and is the main component of native palladium oxide layers.
The O 1s peak, consisting of C─O and lattice O 2− , overlaps with Pd 3p 3/2 .Its total intensity can be calculated using the 1:2 ratio due to spin-orbit splitting of Pd 3p.The expected C─O contribution has been calculated from the C 1s region.As for Ta 2 O 5 , the PdO layer also contains trace amounts of nitrogen (see Figure S3, Supporting Information) with an O 2− :N ratio of 6.6.As demonstrated in Figure S6 (Supporting Information), the MVV Auger peak shows no chemical shift between oxide and metal and was thus not included in the analysis.
Figure 4c shows the intensity ratios weighted by the numberdensity ratio N Pd 2+ ∕N Pd 0 .Here, the thickness of the oxide layer is only d ≈ 9 Å, corresponding to very few atomic layers, again in good agreement with single-and multi-energy modeling (see Table 1).The region of intermixing spans less than 10 Å, which is expectedly narrow.However, as mentioned in section 3.3, the k value becomes increasingly insensitive and unreliable to slight data or fit variations with ever-sharper interfaces.

Titanium
Ti has a regular electron configuration of 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 2 .In the spectrum, the peaks appear in the order 1s 2s 2p 3s 3p 3d 4s, of which only four, 2s to 3p, are well resolved (see Figure 5). [47]The most pronounced peak region, 2p, has been well-studied, and its fitting is uncontroversial.Native oxide on Ti is known to consist of multiple oxidation states, namely Ti 4+ , Ti 3+ , and Ti 2+ , of which Ti 4+ (TiO 2 ) is the most abundant.Individual depth profiles for different oxidation states cannot be calculated with the proposed models, because a necessary assumption is that there are exactly two homogeneous distinguishable phases that overlap only at their interface.Another challenge (that also applies to single energy calculations) is that no exact values for  o and N o can be determined for such heterogeneous oxides.For the data evaluation presented in Figure 5, we assumed that the Ti 4+ :Ti 3+ :Ti 2+ ratio, as determined rather unambiguously from the 2p region, is constant in all peak regions and that  o and N o are the same as in pure TiO 2 .Compared to Ta and Pd, Ti appears to exhibit larger chemical shift between oxide and metal, making the fitting of s-orbitals more reliable.The LMM Auger peak was included in the analysis as it exhibits a clear chemical shift between oxide and metal, as demonstrated in Figure S6 (Supporting Information).
The data evaluation resulted in an average oxide layer thickness of d ≈ 53 Å with a rather broad concentration profile spanning more than 100 Å -much broader than the Ta oxide layer.As seen in Table 1, the data from the Ti sample was the only one that showed significant deviations between the multiple energies and depth profile model.It can be concluded that the consideration of interfacial mixing becomes more important with an increasing degree of mixing and oxide layer thickness.For very thin oxide layers or sharp interfaces, the multiple energies model or even the classical single energy model might be sufficient to fit the data.However, the depth profile model is generally more versatile, particularly when the system is unknown, as it covers all possible homogeneous depth distributions.

Discussion
The recorded spectra are qualitatively in good agreement with those of native oxide layers published in earlier studies and databases.However, as stated previously, specific thickness values are difficult to compare across studies.The primary goal was to compare the new multiple energies and depth profile calculations with the established single energy calculation.Single energy calculations yield roughly the same thickness value for all peaks in the respective spectrum, as indicated by the small standard deviation (see Single Energy Averaged in Table 1).
There is only ≈5-10% deviation between single energy and multiple energies calculations (applying Equation ( 6) as a fitting function).While for Ta and Pd, the depth profile models resulted in virtually the same average thickness as the multiple energy model, there was a substantial deviation of about 30% for the TiO x layer thickness.As illustrated in Figure S5 (Supporting Information), the goodness of fit is noticeably better using the new depth profile models than the multiple energies model ( R 2 equation (6) = 0.78 versus R 2 equation (10) = 0.99; see Figure S5c, Supporting Information).It can be assumed that the deviations between the models become larger as the degree of mixing at the interface increases because the existing models do not take this into account. [49]This might be why earlier studies have shown that the single energy model yields increasingly arbitrary results with increasing oxide layer thickness. [21]ompared to the single energy model, the analysis of multiple peaks necessarily increases accuracy.Thus, the more emission peaks lie within the recorded spectrum, and the more evenly they are distributed, the more reliable the result becomes.This is also the case for ER-and ARXPS, for which both depth profile mod-els are equally applicable.Since the signal is laterally averaged, in contrast to methods such as transmission electron microscopy or atom probe, it is not possible to distinguish between roughness and intermixing at the atomic level, as is the case for XRR measurements or alternative spectroscopic methods.XRR measurements, although insufficient to accurately resolve the discussed oxide depth profiles, are shown in Figure S1 (Supporting Information).The data is shown to be consistent with the XPS results.
Many elements exhibit well-resolved Auger peaks in the XPS spectrum, which are generally known to show oxidation statedependent chemical shifts. [50,51]Since Auger peaks often appear at high binding energies, i.e., low IMFP, they may provide valuable data to increase the accuracy of the models.However, fitting Auger peaks in the XPS spectrum is controversial because many overlapping transitions in the valance band with different probabilities (i.e., intensities) may require nontrivial peak deconvolution.Of the materials examined here, only Ti LMM showed a clear chemical shift and was used for the evaluation.
The models could be tested, applied and refined in subsequent studies for example to investigate corrosion, i.e., thicknessvariable oxide layers, 3D surface structures such as nanoparticles, multilayered samples, or oxide composites.In this study, as is typically the case, a homogeneous depth distribution of oxidation states was assumed.Native oxide layers have been modeled as multi-layer systems with different compositions in other studies (e.g., including an outer hydroxide layer [52,53] ), and in a similar manner, the depth profile models could be extended.However, such approaches are problematic because of uncertainties in number densities and IMFP.The former is quantitatively more impactful, whereas the latter is vertically unknowable in complex layered and intermixed systems.The application of the depth profile models to alloyed oxide layers would certainly suffer from the same uncertainties.However, the consideration of multiple peaks might prove advantageous for modeling XPS data more broadly, as it multiplies the number of available data points.Furthermore, this approach increases the accuracy of IMFP-based models, benefits the validation of peak deconvolutions, especially in overlapping peak regions, and generally contributes to a more holistic understanding of XPS spectra.

Conclusion
We introduced a novel approach for sub-nanometer depth profiling at the interface between metals and their native oxide layers.We began by deriving the most widely used equation for the calculation of the oxide layer thickness, which we refer to as the single energy model.We then demonstrated how its accuracy can be improved by fitting multiple peak regions in the XPS spectrum, thus emulating an energy-resolved approach without the need for a synchrotron.On this basis, a new model was derived to calculate depth-resolved concentration profiles at the oxide-metal interface from single fixed-angle XPS spectra (or alternatively, ER-/ARXPS data).Compared to previous approaches, it reduces ambiguity, increases accuracy, and requires no calibration.However, most peak regions have backgrounds, line shapes, and chemical shifts that have rarely been studied with no established fitting procedures in the existing literature.Conversely, this approach may elucidate these rarely analyzed and poorly understood peak regions, thus utilizing the entirety of the XPS spectrum.There are more than 10.000 papers published annually containing XPS data. [54]Large amounts of the data accessible to standard lab devices have never been put to use.Here we showed that such data contain valuable information for potential applications in a wide range of disciplines.At the time of measurement, the Ta and Ti samples were exposed to the ambient atmosphere for about 1 week.The Pd sample was sealed in a vacuum bag for about 12 months and the time of exposure to the atmosphere was less than 2 days in total between film deposition and loading into the XPS.

Experimental Section
Characterization: XPS was measured in an Omicron Multiprobe Ultra High Vacuum system (Scienta Omicron) at 7 × 10 −11 mbar using monochromatic Al K irradiation (1486.7 eV).The electron emission angle (angle between the sample plane and analyzer axis) was 77°, and the source-to-analyzer angle was 54.7°.The core-level spectra were recorded with an acquisition time of 0.6 s per 0.05 eV in constant analyzer energy mode with a pass energy of 20 eV (survey scans at 0.1 s per 0.2 eV and pass energy of 50 eV).The samples were grounded via a frame on the sample holder, and no charge compensation was used.The size of the X-ray spot on the sample was about 0.75 mm full width at half maximum.Peak deconvolutions were carried out in CasaXPS (version 2.3.22PR1.0),and the depth profiling models were fitted using OriginPro 2023.

Figure 1 .
Figure 1.a) Schematic illustration of the working principle of XPS.b) Exemplary XPS wide-scan spectrum (of Ta) with multiple core-level signals associated with different values of .The upper diagram shows the intensity ratio I o /I m as a function of  (red) and the relative oxide concentration as a function of depth (blue).The inset shows  as a function of E kin calculated by the TPP-2M equation for an exemplary metal and one of its oxides (here Ta and Ta 2 O 5 ).c) Illustration of a discontinuous DDF for  o ≠  m .The insert shows an exemplary core-level peak deconvolution.d) Three different DDFs for different averaged  values.The insert illustrates changing I o /I m ratios for different core-level regions.e) The shown DDF is now separated by a sigmoid function at the interface, resulting in a partial overlap of the areas.

Figure 2 .
Figure 2. a) Equation (8) is shown for varying k values with constant d = 10 Å and (b) varying d values with constant k = 1 as well as the resulting sigmoid functions.c) Equation (10) is shown as a function of emission angle  for varying k values and constant d = 10 Å (bottom x and left y axis) with the resulting sigmoid functions (top x and right y axis).

Figure 3 .
Figure 3. XPS spectrum and depth profiling of native Ta 2 O 5 : a) narrow-scans (magnified by the factor given in brackets) of the peak regions highlighted in the (b) survey-spectrum.Residuals are shown in pale grey below the fitted data (their magnification is half that of the corresponding data).c) Intensity ratios Ta 5+ :Ta 0 weighted by number densities fitted by Equation (8) and resulting depth distribution of Ta 5+ relative to Ta 0 .

Figure 4 .
Figure 4. XPS spectrum and depth profiling of native PdO: a) narrow scans (magnified by the factor given in brackets) of the peak regions highlighted in the (b) survey-spectrum.Residuals are shown in pale grey below the fitted data (their magnification is half that of the corresponding data).c) Intensity ratios Pd 2+ :Pd 0 weighted by number densities fitted by Equation (8) and resulting depth distribution of Pd 2+ relative to Pd 0 .

Figure 5 .
Figure 5. XPS spectrum and depth profiling of native TiO x : a) narrow-scans (magnified by the factor given in brackets) of the peak regions highlighted in the (b) survey-spectrum.Residuals are shown in pale grey below the fitted data (their magnification is half that of the corresponding data).c) Intensity ratios Ti X+ :Ti 0 weighted by number densities fitted by Equation (8) and resulting depth distribution of Ti X+ relative to Ti 0 .
N ratio of 3.8 was calculated (the anionic O 2− of the oxide lattice obtained from the O 1s peak after correction for O attributed to adventitious C as determined from C 1s).The chemical state and depth distribution of N remain unclear.However, because the assumption of a single oxidation state of Ta was sufficient to fit the data, a possible contribution of Ta x N y was neglected.