Surface analysis insight note: Straightforward concentration depth profiling by angle‐resolved X‐ray photoelectron spectroscopy using a Tikhonov regularization algorithm

Angle‐resolved X‐ray photoelectron spectroscopy (ARXPS) is a technique used for depth‐dependent analysis in the near‐surface region of samples. Calculation of a concentration depth profile using ARXPS requires an inverse Laplace transform, which adds considerable complexity to the analysis. In this insight note, the Tikhonov regularization algorithm for depth profile reconstruction from ARXPS data is examined. The steps required to produce a concentration depth profile are provided. The discussion includes strategies that deal with elastic scattering, electron attenuation, choice of regularization terms and optimization of the regularization parameters. The method is implemented in a Microsoft Excel spreadsheet that allows users to calculate a depth profile for data collected at up to five angles for five different peak components.

between the normal to the sample surface and the detector (referred to as the 'take-off angle'). The increased path length to vacuum at larger angles increases the attenuation of photoelectrons at greater depths, providing depth resolution.
The peak intensity from a given element and orbital can be described by, where c(z) is the concentration as a function of z, λ is the attenuation length and I 0 is the unattenuated photoelectron intensity. I 0 can be treated as unity if the transmission function and asymmetry factor corrections have been applied. This assumes that the X-ray flux and analysis area also remain constant. This equation allows the angleresolved peak intensity to be calculated if c(z) is known. However, the goal of ARXPS is to perform the inverse of this calculation, which is an ill-posed mathematical problem. 2 The use of a regularization algorithm to solve the problem ensures that a simple, physically plausible solution can be found.
The routine application of ARXPS methods for depth-dependent chemical analysis requires several assumptions including that the sample is atomically smooth, the composition of layers is homogeneous in the x-y plane, the attenuation of electrons from buried layers can be accurately described, elastic scattering can be neglected (or accounted for by simple means) and a reliable algorithm can be used to generate the concentration depth profile. Much has been written about issues related to attenuation length, 3 elastic scattering 4,5 and regularization algorithms. 6,7 On elastic scattering and attenuation length, the general consensus is that reasonable accuracy can be achieved if collection angles are limited to <60 and effective attenuation length (EAL) is used rather than the inelastic mean free path (IMFP) to describe the signal attenuation. 8 Although ARXPS has been available for approximately 50 years, the question of which regularization algorithm is the most stable and produces the most accurate concentration depth profile persists. 9 The most common types are those based on maximum entropy methods (MEMs) 6 and Tikhonov regularization. 7 In this Insight, we provide a straightforward method for generating concentration depth profiles using ARXPS data. It is simple enough to be automated and is designed to be approachable for non- Χ 2 represents the difference between the measured 'apparent' concentration and the calculated concentration at each angle using a peak intensity calculated by Equation (1) and an initial 'guessed' c(z).
Χ 2 is summed over j elements and k angles. σ 2 is the variance in each of the measured data points, representing the error in the experimental measurements, where c calc for element A is Note that I ∞ denotes the peak intensity from a standard of infinite thickness (i.e., c(z) = 100 at% for all depths). The goal of the algorithm is to alter c(z), such that χ 2 approaches zero, indicating that c(z) represents a concentration depth profile consistent with the experimental ARXPS measurements. In the Excel files (provided as Data S1 and S2), the total z range is separated into 'slabs' of thickness i, which serves as the step size of c(z).
R is the regularization term. It is used to penalize overfitting of the ARXPS data to ensure that non-physical features, such as spikes or noise, are not observed in the optimized c(z). α is a Lagrange multiplier, used to control the degree to which the overfitting of Χ 2 is suppressed by the regularization term. If α is too small, then overfitting is likely to occur. However, if α is too large, then c(z) will not agree with the experimental data. The optimization of α is discussed below.
The squared concentration gradient is suggested as the best regularization term because, in general, MEMs regularization penalizes spikes in the c(z) to a lesser degree than Tikhonov methods. Of the Tikhonov regularization terms, Ð ∞ 0 c z ð Þ 2 dz penalizes the total concentration of each element and forces c(z) towards zero as z approaches the bulk (MEMs methods also display this property 8 ). This leads to anomalies in c(z) when substrates are included in the analysis. Therefore, the most stable algorithm of those discussed is the following: The squared concentration has been found to be a superior regularization term compared with the squared concentration gradient when analysing discrete layers. 10 However, if a single optimization term is to be used for all elements in all layers (i.e., including the substrate), the squared concentration gradient will be the best choice.
To calculate c(z) from an ARXPS data set using RMMF ARXPS DP Modelling, users follow the steps on the 'Instructions' tab of the Excel files. The steps are as follows: 1. Insert the labels, binding energy and EAL(Å) (see Section 3 below for explanation of EAL(Å)) for each peak in 'Peak Labels' tab.
2. Insert measurement angles (in degrees) and measured atomic percent (at%) concentration in the 'Experimental inputs' tab. If the error in the measured concentration is known, it can be entered in the measured error (%) field.
3. In the 'DP data + init guess model' tab, construct a guess concen-

| NOTES ON ATTENUATION LENGTH
The surface sensitivity of XPS is explained in terms of electron attenuation. For homogeneous samples, photoelectron signals measured at normal take-off angle will have a zero (or negligible) contribution from elastic scattering. The electron attenuation can then be described by the IMFP, which is defined as the average distance along an electron trajectory between inelastic collisions for an electron of given energy. 11 However, the description of electron attenuation becomes more complex at higher take-off angles. Elastic scattering may alter the trajectory of photoelectrons. This means that a portion of the emitted electrons reach the detector having travelled through <z/cos (θ) worth of material. Hence, the ARXPS measurements will have lower surface sensitivity than calculated using equation 1 if the IMFP is used for λ. Instead, it is recommended that EAL be used. The EAL is defined as the decay length in the probability that an emitted electron originated from depth z. This value takes elastic scattering into account. There are published methods that relate IMFP to EAL and allow the EAL to be calculated from material properties. 12 The EAL for many materials can also be found in the NIST Electron Effective-Attenuation-Length Database. 13 In Data S1, RMMF ARXPS DP Modelling, the EAL is an editable field, allowing the user to choose their own calculation method. For some materials, neither the IMFP nor material properties will be well characterized. Therefore, we have also included RMMF ARXPS DP Modelling_Seah as Data S2. This program uses an approximation devised by Seah and Dench 14 that estimates the EAL based on photoelectron kinetic energy alone. This will allow users to compute a semi-quantitative c(z) from their ARXPS measurements, which will be sufficient for certain applications. The data were collected using a Kratos Axis Supra with a monochromated Al Kα X-ray source. A charge neutralizer was used. The pass energy was set to 40 eV, and the energy step size was 0.1 eV. The apparent concentration of each element was determined using

| EXPERIMENTAL EXAMPLES
CasaXPS. Figure 1A shows the apparent concentration measured as a function of angle, along with the calculated peak intensity at each angle using the c(z) displayed in Figure 1B. The α value was set to 1.0 Â 10 À6 , and the i thickness was set to 0.1 nm. The generalized reduced gradient (GRG) nonlinear solver was used to minimize Q. The only constraint applied during minimization was that c (at%) Variables must be non-negative values. The c(z) profiles were then scaled after minimization of Q so that the total at% equals 100 at each step i. The c(z) profile reveals 3 distinct layers corresponding to the carbonaceous contamination, the WS 2 and the Al 2 O 3 . The stoichiometry of the inorganic compounds has been predicted with reasonable accuracy. The width at half of the S2p and W4f intensities is $0.7 nm, which is in close agreement with a monolayer thickness of WS 2 . 15 The expected sample structure was also modelled using simulation of electron spectra for surface analysis (SESSA). 16 The results are displayed in Figure S1. The c(z) from the simulated sample closely resembles those from the experimental data in Figure 1. Data from a second experimental sample are also included in Figure S2. This shows the ARXPS data and c(z) from a thin Ga 2 O 3 film sputtered onto SiO 2 . The experimentally determined angle-resolved apparent concentrations for both samples are included as Tables S1 and S2.

| OPTIMIZATION OF Α
The data obtained from ARXPS have a much lower information density with respect to depth than the number of adjustable parameters (i.e., c (at%) Variables in the Supporting Information) used to model c (z). It has previously been shown that the maximum number of adjustable parameters that can be used to accurately determine c(z) is just three per peak. 8 Using a greater number of variables risks overfitting the data as the calculated c(z) tries to account for the noise in the experimental data. However, it is obvious that if we compute the depth profile in Figure 1B with a 0.1-nm step size, then the number of c (at%) Variables will be much greater than three. This is where the regularization is used. The goal of the R parameter in Equation (2) is to reduce the complexity of the model to the point where c(z) constructed from all the c (at%) Variables could have been described by a three-parameter model. Hence, c(z) becomes smoother, and any structure in c(z), such as spikes or sharp steps, is removed. The bias towards a smooth model is controlled by the α parameter. Figure 2A shows c(z) for S 2p data from the WS 2 sample computed using different log(α) values. For log(α) > À10 (α = 1 Â 10 À4 ), the c(z) shows barely any structure because the model is too regulated. The S 2p layer only becomes resolvable at lower values of α.
Ideally, α should be optimized using some 'unbiased' criterion.
One method that has been previously used is the S-curve method. 17 It is a basic, automatable approach, where the χ 2 value is minimized by varying α. An example is shown in Figure 2B. The minimum χ 2 value is observed where log(α) is between À14 and À10 (α = 1 Â 10 À4 -1 Â 10 À6 ). Χ 2 increases slightly below this range of α, indicating that overfitting occurs as the model complexity increases. The authors suggest that this is the simplest method for users to assess the F I G U R E 3 Average concentration depth profiles for (A) C 1s and (B) S 2p computed from multiple initial guess models. The C 1s data were acquired from 20 different initial guess models by varying the carbon layer thickness from 0.1 to 2 nm thick (WS 2 thickness = 1 nm). The S 2p data were computed from 30 different initial guess models by varying the WS 2 thickness from 0.1 to 3 nm (carbon thickness = 0.2 nm). Α was set to 1 Â 10 À6 for all calculations.
appropriateness of the α value using RMMF ARXPS DP Modelling.
Users should note that the initial guess model should not be altered when calculating S-curves or α will need to be re-optimized (see section below for details).

| INFLUENCE OF INITIAL GUESS MODEL
Some prior knowledge is necessary for the calculation of physically plausible depth profiles using regularization methods. The set of initial guess values for c(z) can introduce operator bias into the concentration depth profile reconstructions. Prior knowledge can also be inserted into the calculation by increasing α, which restricts the number and complexity of possible solutions. Figure S3 shows c(z) calculated for the WS 2 sample without any prior assumption given for the shape of the depth profiles but with α set to 100. A result for c(z) is achieved that is comparable with the data in Figures 1 and 2. However, it is clear by comparing the results in Figure S3 with those using higher α values in Figure 2A that there is some dependency on the initial guess model used as to which α value will be appropriate. This may lead users to ask whether it is better to attempt to provide a more accurate estimation for c(z) or to simply raise α. The authors suggest that it may be more intuitive for users of RMMF ARXPS DP Modelling to provide the required prior knowledge in the form of an initial guess for c(z) and begin with an α value in the 1 Â 10 À4 -1 Â 10 À6 range, as demonstrated above. If users rely on α alone for prior knowledge, it may be difficult to determine an appropriate starting point without calculating a full S-curve, which is time consuming, because α cannot be determined directly from the experimental data. Conversely, it is assumed that if a sample is suitable for ARXPS analysis, the relative depth of the elements within the sample will be able to be deduced by inspection of the XPS data prior to depth profile reconstruction thus allowing a reasonable guess model to be constructed.
As observed in Figure 2A, α still provides the necessary regularization to prevent overfitting.
The effect of varying the thickness of the carbon contamination and WS 2 layers in the initial guess models was tested using the data from Figure 1A. Figure 3A shows the average depth profile from the C 1s signal calculated from 20 different initial guess models by varying the carbon layer thickness from 0.1 to 2 nm, whereas the guessed WS 2 thickness remained at 1 nm. Figure 3B shows the S 2p depth profile from a similar experiment where the guessed WS 2 layer thickness was varied from 0.1 to 3 nm (initial carbon thickness = 0.2 nm).
The results show that, while some variability is introduced by the initial guess model, the deviation in the calculated layer thickness is smaller than the theoretical depth resolution. 8 The depth profiles converge to have a reasonably precise shape regardless of the thicknesses used in the initial guess model. The error in the at% concentration is most pronounced at depths where the concentration gradients are highest due to shifts in the interfaces. There is an obvious correlation between the error in calculated thickness and the error in the at% concentration near the interfaces. Users should be cognizant of this and of the depth resolution limits in ARXPS.

| CONCLUSIONS
The RMMF ARXPS DP Modelling program provides users with an easyto-follow method of producing concentration depth profiles from ARXPS measurements. Tikhonov regularization using the squared concentration gradient parameter allows users to extract depth profiles from both the substrate and overlayers in a single, stable calculation.
Beginning with an α value in the 1 Â 10 À4 -1 Â 10 À6 range and using the method of Seah and Dench allows users to quickly calculate semiquantitative depth profiles. However, the authors suggest that the α parameter should be varied to find the minimum χ 2 value for their model and EAL values derived from material properties should be used to describe the attenuation length. This will ensure that the most accurate and reproducible concentration depth profiles are calculated.