Nondestructive Surface Depth Profiles from Angle ... - ACS Publications

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J. Phys. Chem. C 2009, 113, 21328–21337

Nondestructive Surface Depth Profiles from Angle-Resolved X-ray Photoelectron Spectroscopy Data Using the Maximum Entropy Method. I. A New Protocol M. Andrea Scorciapino,† Gabriele Navarra,‡ Bernhard Elsener,† and Antonella Rossi†,* Inorganic and Analytical Chemistry Department, UniVersity of Cagliari, Cittadella UniVersitaria di Monserrato, S.S.554, biVio Sestu, 09042, Monserrato (CA), Italy, and Chemical Sciences Department, UniVersity of Cagliari, Cittadella UniVersitaria di Monserrato, S.S.554, biVio Sestu, 09042, Monserrato (CA), Italy ReceiVed: July 4, 2009; ReVised Manuscript ReceiVed: October 8, 2009

The knowledge of the depth concentration profile of thin-layered surfaces a few nanometers thick is very important for research and applications in microelectronics, corrosion, wear, and tribology. In-depth profiling methods reported in the literature are either destructive (ion sputtering), based on severe approximations (concentration gradients are not taken into account, and electron inelastic mean free paths (IMFPs) are calculated for electrons traveling throughout pure elemental materials) or limited to relatively simple profiles (less than three components and constant IMFPs). A reconstructed depth profile should be consistent with the angleresolved X-ray photoelectron spectroscopy (ARXPS) data acquired, but transformation of XPS signal intensities vs emission angle into chemical species concentrations vs depth is an ill-posed mathematical problem which requires inversion of a Laplace transform. The main goal of this work was thus to develop a new, iterative protocol based on the maximum entropy method (MEM) that allows obtaining in-depth concentration profiles of layered surfaces from nondestructive ARXPS measurements. Numerical experiments were performed on a large series of computer generated, ideal, and error-containing ARXPS data from model depthprofiles with up to four layers and up to eight components. The new algorithm enabled us to reconstruct these depth profiles with a maximum uncertainty of (20% for layer thickness and of (30% for composition of the individual layers. Moreover, the new protocol involves an iterative procedure for calculating the IMFP values of the different components, taking into account the actual depth concentration profile of the sample surface under investigation. The new protocol proved to be more powerful than any of the existing algorithms since it has been successfully applied for reconstructing depth profiles with up to eight components. Introduction Quantification by X-ray photoelectron spectroscopy (XPS) relies on several factors such as knowledge of photoionization cross-section, electron inelastic mean free path (IMFP), influence of electron elastic scattering, and energy dependence of the spectrometer transmission function.1,2 The major contributing factor to the largest errors is, however, determination of the in-depth distribution of atoms.3,4 For a meaningful quantification, assumptions have to be introduced on the depth profile since the measured peak intensity depends critically thereon. Now, in practice the depth profile is never known, and usually the solid composition is for convenience but, quite arbitrarily, assumed to be homogeneous down to a depth of several nanometers. This assumption may result in enormous errors in quantification.3,4 In fact, it is precisely because samples are inhomogeneous at the nanometer depth scale that they are analyzed using XPS rather than other well-established, but less surface sensitive, techniques. To illustrate this fundamental problem assuming homogeneous composition with depth, Tougaard reported5 a clear example of model spectra calculated for different depth distributions of Cu in Au, showing that quantification based on peak intensities alone is affected by a large uncertainty.5 * To whom correspondence should be addressed. E-mail: [email protected]. Phone: +39-070-675-4464. Fax: +39-070-675-4456. † Inorganic and Analytical Chemistry Department. ‡ Chemical Sciences Department.

Thus, an accurate depth profiling method is needed to overcome these problems. Depth profiling with ion sputtering yields good in-depth resolution but is destructive and can produce several artifacts, including atom mixing at the sputter surface, preferential sputtering of some of the specimen components, and implantation of sputtered species.1,6 To avoid these artifacts, a nondestructive method is preferable. The threelayer model approach7,8 requires the acquisition of XPS spectra at just one angle, but the information on film composition is averaged and not in-depth resolved. The Tougaard method provides a quantitative estimate of atom depth distribution within the outermost surface region of the sample under investigation but requires spectra acquisition of highly pure reference compounds. This implies the assumption that the elements are present in the sample in the same chemical states as those in the reference compounds. Moreover, a maximum of three reference spectra can be used to simulate the sample spectrum,9 few model structures can be verified,9 and a maximum of six structural parameters can be determined.10 Otherwise, angleresolved X-ray photoelectron spectroscopy (ARXPS) is, in principle, a suitable method for nondestructive reconstruction of the composition depth profile of material surface with film thickness in the order of a few nanometers.11,12 However, the reconstruction of a concentration depth profile requires the inversion of a Laplace transform that does not have a unique solution, and it is strongly influenced by the noise of the ARXPS data (ill-posed problem). In practice, the reconstruction is based

10.1021/jp906326m CCC: $40.75  2009 American Chemical Society Published on Web 11/30/2009

Surface Depth Profiles from ARXPS Data on the assumption of model structures. This might be misleading because experimental ARXPS data contain noise, and thus, a large number of very different model structures may exist that match the experimental data within the measurement precision.13 Consequently, simply minimizing the weighted sum-of-square differences between the calculated and measured (ARXPS) data is not always adequate for determining the correct sample depth profile, especially if the sample contains a large number of components. A depth profile that satisfies experimental data has to be found; however, it must contain the minimum amount of structural parameters necessary to do so (since the details of the noise must not be fitted), and a suitable algorithm has to be implemented to solve this problem. However, irrespective of the method applied, the achievement of accurate quantitative information from XPS analyses necessarily requires a suitable calibration procedure for determining signal intensity versus electron kinetic energy response function (IERF) of the spectrometer being used. The calibration procedure is more complex when the angle-resolved acquisition mode is used, since the signal intensity vs energy function also depends on electron emission angle (i.e., the angle between the normal to the sample surface and emission direction). Lastly, another very important issue in XPS quantitative analysis is the determination of the inelastic mean free path of photoelectrons. The inelastic mean free path (IMFP) is defined as the mean distance traveled by the electron between two consecutive inelastic scattering events.1,11 However, despite the importance of IMFP, experimental values for a given material are generally available only over a narrow energy range, and the measured values are affected by large uncertainties due to the experimental difficulties.14 From the theoretical viewpoint, the most widely used IMFP predictive formulas are the Seah and Dench,15 the TPP-2M,16 and the G-1,17 which for the same material and electron kinetic energy yield results often showing a large (.10%) relative difference. Thus, if the outermost surface region of the sample being analyzed cannot be considered to have homogeneous composition with depth profile, an IMFP predictive formula needs to be chosen that takes into account the differences in the material through which the photoemitted electrons travel along their escape path, while minimizing the error introduced into the final quantitative result. The main goal of this work was, therefore, to develop a new, iterative protocol based on the maximum entropy method (MEM) suitable to obtain accurate depth concentration profiles of layered surfaces from nondestructive ARXPS measurements. A suitable IMFP predictive formula had to be chosen and implemented in the computing protocol presented here to account for the potential constituents of the outermost surface region of the sample under investigation. In this first paper, a series of numerical experiments are presented to illustrate the new protocol performance, highlighting precision and accuracy of quantitative results. In a second paper, the results obtained applying this new protocol to the real case of a NiP alloy will be discussed.18 Maximum Entropy Method Goals of Surface Analysis of Thin Layers. The maximum entropy method (MEM), which became popular after its successful use for restoring astronomical images,19,20 has proven to be a powerful tool for reconstructing in-depth composition profiles from ARXPS measurements.6,13,21-24 The aim of the MEM, as mentioned above, is to find a depth profile that satisfies the experimental ARXPS data but contains the minimum “amount of structure” necessary to do so. The higher the number

J. Phys. Chem. C, Vol. 113, No. 51, 2009 21329 of structural parameters needed to define the system, the lower the corresponding entropy. Thus, the entropy can be used to evaluate the amount of information carried by the variables used to describe the depth profile of the sample. The reconstruction of a depth profile from ARXPS data is, for the reasons outlined above, an ill-posed mathematical problem.11 Thus, in an attempt to solve this problem, the entropy can be used as a regularizing function (which has to be maximized) to constrain the solution to obtain the simplest possible depth profile that matches the experimental data. The details of the theory of the ARXPS method and of the structural functions used in the following are reported in the Supporting Information and in the literature.24-30 Concepts of MEM. If we define entropy S as31 N

S)

p

∑∑ j)A i)0

( )]

[

nj,i - mj,i - nj,i log

nj,i mj,i

(1)

where nj,i is the atomic fraction of the element j in the ith layer and mj,i its initial estimate and the deviation of the calculated intensities versus emission angle from the experimental ARXPS data is defined by the chi-squared statistic misfit, as N

C)χ ) 2

Nth

∑∑

calcd obsd 2 (Xj,k - Xj,k ) 2 σj,k

j)A k)1

(2)

where Xj,kcalcd and Xj,kobsd are, respectively, the calculated and observed apparent concentration of the jth element at the kth emission angle and σj,k2 is the variance of the kth measurements for the jth elements. Finally, a depth profile that satisfies the experimental ARXPS data can be calculated by minimizing C. However, the profile must contain the minimum “amount of structure” necessary to do so; correspondingly, a maximum of S also has to be found. These two necessary conditions can be simultaneously satisfied by maximizing the so-called probability function

Q ) R·S -

C 2

(3)

where R is a Lagrange multiplier (also called the regularizing parameter). A large value for R will result in an oversmoothed solution that will thus not agree with the data, whereas a small value for R will lead to an overfitting of the data since the depth profile reconstruction will tend to fit and reproduce the noise in the data. Evolution of the MEM Algorithm. First Version of the MEM Algorithm. A first version of the algorithm published in a previous paper24 used a huge number of variables in the calculations, given by the product of the number of MEM layers (see Figure 1) into which the depth profile is divided and the number of components (chemical species). Each component had an individual concentration value for each of the MEM layers. The sum of all component concentrations within each MEM layer was normalized to 1. MEM layer thickness was taken as 1 Å. This version of the MEM algorithm was successfully applied to solve a five-component depth profile of iron oxyhydroxide surface films.24 However, it did not take into account the influence of composition changes on the IMFP values. Second Version of the MEM Algorithm. In the second version of the MEM algorithm, developed in this work, each

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Figure 1. Graphic illustration of the model used within the MEM.

of the components of the system being studied is conveniently simulated with a four-parameter pseudo-Gaussian function to reduce the overall number of variables. Thus, whereas in the first version described above each component was described by a large number of variables, in this second version each component is simulated by only four parameters. This reduces calculation time. The sum of all component concentrations within each MEM layer was normalized to 1, and MEM layer thickness was taken as 1 Å. New Protocol for MEM Application. The first and second versions of the MEM algorithm were first checked independently for component structures (see in the following structures 7 and 8), to ascertain whether they were able to solve in-depth profiles for more than five components. The starting profile was always set with components having equal concentration at all depths, i.e., a perfectly homogeneous sample. Unfortunately, these first tests failed. Then, a new MEM protocol was developed for the combined use of the first and second versions of the MEM algorithm. This new MEM protocol was not entirely computer based but performed step by step as shown by the flow diagram in Figure 2. The starting profile was always based on simulator routine results. For the first calculation cycle (CC), quantitative data from the simulator were always disregarded when constructing the starting profile for this version of the MEM algorithm, i.e., only layer thickness was taken into account. At the end of each CC, i.e., when the first version of the MEM algorithm generated its output, the difference was calculated between these output results (i.e., layer thickness, component concentration, and depth profile curves trend) and the analogous output generated at the end of the previous CC. A 1% threshold was chosen as the criterion for deciding whether a new CC had to be started up. Regarding the second version of the MEM algorithm, neither the starting nor the final depth of each component was constrained at the estimated values but was allowed to range over a closed symmetric 6 Å interval. Once the probability function Q (eq 3) had attained a local maximum, parameter ranges were extended for all components whose parameters had reached an extreme of the constraining interval. Then, maximization of Q was allowed to continue. During processing of both versions of the algorithm, the regularizing parameter R was changed to keep the entropic RS term and the chi-squared -C/2 within the same order of magnitude. This regularizing parameter typically ranged between 100 and 1, always decreasing at each calculation cycle. The lowest value of 0.1 for R was “reached” only in the reconstruction of simple three- and four-component depth profiles. Moreover, in the simplest three-components cases, the maximum needed value for the regularizing parameter, i.e., in the first CC, was only equal to 35. The same range of 100-1 was found to be needed in MEM protocol application to real samples.18

Figure 2. Protocol for combining the first and new versions of the MEM algorithm.

Experimental Section Two different versions of the algorithm were implemented in a new MEM protocol as described above, to apply the MEM theory to the apparent concentration data derived from ARXPS data. Synthetic Structures for Numerical Experiments. First of all, note that MEM layers (Figure 1) do actually differ from real layers. Hereinafter, if a specimen has a layered structure, e.g., iron oxide formed on a pure metallic iron surface, these physically existing layers are simply referred to as layers, possibly accompanied by a distinguishing “name”, e.g., oxide layer, phosphate layer, intermediate layer, overlayer, sublayer, etc., whereas the parallel numerical layers of thickness t (Figure 1), down to a depth equal to XPS sampling depth, are referred to as MEM layers. The simulated structures were composed of three to eight components labeled A, B, C, D, E, F, G, and H. The IMFP values of the components were arbitrarily chosen as 40, 30, 20, 10, 9, 35, 25, and 27 Å, respectively. As the number of components in the numerical structures increased from 3 to 8, the “new” component was labeled in alphabetical order. In other words, all the three-component structures were composed of A, B, and C pseudospecies, all the four-component structures of A, B, C, and D, and so on. All profiles were composed of 150 MEM layers. Almost all possible combinations were checked.

Surface Depth Profiles from ARXPS Data

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TABLE 1: “Special” Synthetic Profiles Checked in This Work Are Listed with Their Labels, Layer Thickness, Components Involved, and Their Concentrations label

layer

7_a and 7_aerror overlayer 1st intermediate 2nd intermediate bulk 7_b and 7_berror overlayer 1st intermediate 2nd intermediate 1st bulk 2nd bulk 8 and 8error overlayer intermediate intermediate/bulk bulk

thickness concentration (Å) species (at. %) 7 12 6 / 7 12 6 16 / 7 12 12 /

B:G D:F:G A C:E B:G D:F:G A C:E C:E B:H D:F:G A:C:E C:E

80:20 23:15:62 100 82:18 80:20 23:15:62 100 65:35 82:18 80:20 23:15:62 40:36:24 82:18

Synthetic structures were unambiguously labeled as follows. As an example, let us consider the label “5_2 + 3”. The first digit indicates the total number of components throughout the entire depth profile of the synthetic structure. After the underscored blank, a series of digits are reported, all of which are separated by the sign “+”. Each digit indicates the number of components that together constitute one layer, moving from the surface to the bulk, respectively. Species were always included in alphabetical order. So, the label “5_2 + 3” indicates a five-component profile in which two species A and B form an overlayer, whereas the bulk is composed of three species, C, D, and E. In addition, three “special” synthetic structures were constructed, two of which were made up of seven components and one of eight. Composition and layered structure of these structures are shown in Table 1. Structure 7_a was built to assess the new protocol ability to solve a profile where one species (i.e., G in this case) was present in two adjacent layers. Another complicating feature was introduced into structure 7_b by generating an outer E-enriched bulk layer. Lastly, also the eight-component structure had an E-enriched outer bulk layer but was mixed with the nonbulk species A. Apparent Concentration Diagrams (ACD). Using eq 9 (see Supporting Information), a computer code was implemented to calculate the expected apparent concentration diagrams (ACD) data for a given depth profile. This code is hereinafter referred to as the simulator (see below). The apparent concentration diagrams (ACD) of the different synthetic structures were calculated without adding “experimental error”, and the numerical experiments using the MEM protocol (see below) were based on these error-free ACD data. To investigate the influence of error, the three “special” sevenand eight-component ACD data sets were modified adding to the ACD data a random error ranging from -10 to +10%. Thus, all the special seven- and eight-component numerical experiments were performed both on error-free and error-simulated ACD data sets. When starting from real experimental ARXPS intensities, the ACD diagrams are obtained correcting the experimental peak area by the photoemission cross-section, the asymmetry function, and the intensity/energy response function of the spectrometer being used. Finally, a relative depth plot (RDP) was calculated for all the ACD data sets. The relative depth plot is a histogram, which shows the so-called relative depth for each species. The relative

Figure 3. Simulator routine for apparent composition diagrams generation. This simulation routine was used to simultaneously search for the best layered depth profile (see text) and the best IMFP data set (for real sample analysis not reported in this paper).

depth is calculated as the ratio of the relative concentration at near-grazing to that at near-normal emission angle. The lower the relative depth, the farther the species from the surface. Simulator Routine. A simulation routine was used to search for the best-layered depth profile whose ACD reproduced the numerical ACD data (or experimental data for real sample analysis). The routine was not computer based but performed step by step as shown by the flow diagram in Figure 3. MEM layer thickness was taken as 1 Å. MEM layer number was set to 150. For the numerical experiments, IMFP values were assumed to be correctly known. Regarding the simulations of ACD data from experimental data (reported in a companion paper18), the number of MEM layers should be taken as ten times the maximum IMFP of the bulk components, considering the electrons to travel only through the bulk (i.e., overlayer and intermediate layers were not taken into account). However, as described in the next subsections, whenever the simulator profile is modified during the simulator routine, a new set of IMFP values (one for each component) is calculated accordingly. At the end of the simulator routine, the “minimum worth to be considered depth” or “minimum depth” of the simulator profile is investigated by gradually decreasing the number of MEM layers. As this total depth decreases, the ACD data and the difference with respect to their initial values (i.e., ACD data calculated before decreasing MEM layers) are calculated. The

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minimum number of MEM layers should be chosen so that ACD data difference from their initial values is no higher than 1%. Finally, the ultimate IMFP values set is determined in agreement with the best simulator profile and the minimum depth. This set of IMFP values is used in the protocol application to the experimental data for real samples, and the best-layered structure with the minimum depth is used for the starting profile. Determination of Electron Inelastic Mean Free Path. For real sample analysis, the best IMFP values set has to be determined so that it can be used within application of the MEM protocol. The best set can be determined within the simulator routine while looking for the best-layered structure (see previous subsection). Regarding our simulations of experimental data (not reported in this paper), IMFP calculations were performed with the G-1 predictive equation.17 The G-1 equation was applied using NIST “Standard Reference Database 71” software.32 IMFP values were plotted versus electron kinetic energy ranging from 200 to 2000 eV, considering an electron traveling through each of the materials that constituted the sample. The electron IMFP values were then calculated for all chemical species in the specimen (i.e., for photoelectrons which generated the corresponding components of the XPS signals) as the photoelectrons traveled through each of the materials considered separately. Then, starting from these values, the actual IMFP values were calculated with the simulator routine. For example, let us imagine a four-layer structure as shown in Figure 4. A formula has to be used for each of the four layers to calculate the IMFP of the chemical species located in that specific layer layerAIMFP(KE)

layerBIMFP(KE)

)

) IMFPmat.1(KE)

(4)

a · IMFPmat.1(KE) + b · IMFPmat.2(KE) a+b (5)

a · IMFPmat.1(KE) + layerCIMFP(KE)

)

b · IMFPmat.2(KE) + c · IMFPmat.3(KE) a+b+c

(6) a · IMFPmat.1(KE) + b · IMFPmat.2(KE) + layerDIMFP(KE)

)

c · IMFPmat.3(KE) + d · IMFPmat.4(KE) a+b+c+d

(7) where mat.1, mat.2, mat.3, and mat.4 are the four different materials considered, and a, b, c, and d are the thicknesses of layers A, B, C, and D, respectively. In other words, to determine the IMFP value of a photoelectron with a given kinetic energy, the mean value of its IMFP for each of the traversed materials has to be calculated, and each contribution to the mean has to be weighted by the layer thickness. Thus, in the simulator routine, whenever the thickness of the four layers and/or the composition of layers are modified, a new set of IMFP values is correspondingly estimated.

Figure 4. Example of a layered structure used in simulator routine for calculating IMFP.

Results The aim of the MEM is to convert the ARXPS data into the quantitative depth profile of the sample by fitting the recalculated data with the experimental points of the apparent concentrations diagram. On the contrary, in a numerical experiment, the ACD and the corresponding RDP are calculated on the basis of a given synthetic structure (i.e., a predefined depth profile), and as mentioned above, IMFP values are set and held constant. Thus, in the numerical experiments, the new MEM protocol was applied to these computed ACD data to verify whether it was able to reconstruct the depth profile used to calculate the ACD itself. In the following, the results of some numerical experiments will be presented, i.e., a quite simple four-component case as well as some of those performed on the special seven- and eightcomponent synthetic structures, in Table 1. Results of other interesting numerical experiments are available as Supporting Information (Figures S1-S8; Tables S1-S8). Synthetic Structure 4_2 + 2. Figure 5a shows the depth profile of the synthetic structure 4_2 + 2, together with its MEM simulation (dotted lines). This structure simulates a layered sample where the overlayer may be, for instance, adventitious contamination consisting of carbon (A) and oxygen (B), while the bulk could be a binary alloy (C and D). Figure 5b shows the ACD calculated on the basis of the synthetic profile using IMFP values of 40, 30, 20, and 10 Å for components A, B, C, and D, respectively, together with the MEM recalculated curves (dotted lines). Figure 5b also shows a schematic illustration of the depth profile model (right-top corner inset) as well as the RDP (right-bottom corner inset). Table 2 lists the results together with their relative errors. Very good agreement between the theoretical depth profile and the MEM calculations is found. Synthetic Structure 7_b. Parameters that characterize this synthetic structure are reported in Table 1. The MEM protocol was first applied to the theoretical ACD data using IMFP values of 40, 30, 20, 10, 9, 35, and 25 Å for components A, B, C, D, E, F, and G, respectively. Then a random error of (10% was added to the theoretical ACD data to simulate a set of experimental data. Thus, a new numerical experiment was performed on this new data set, referred to as 7_berror. The reconstructed depth profile obtained in the erroradded case is shown in Figure 6a (dotted lines) together with the depth profile model, while Figure 6b shows the ACD data and their MEM curve fitting (dotted lines). Figure 6b also shows a schematic illustration of the depth profile model (right-top corner inset) as well as the RDP of this synthetic structure (rightbottom corner inset). Table 3 shows the results obtained for both the 7_b and 7_berror numerical experiments, together with their relative errors.



Figure 5. (a) Depth profile of synthetic structure 4_2 + 2 and MEM simulation (dotted lines) and (b) apparent concentrations diagram of synthetic structure 4_2 + 2 (dots) and recalculated MEM data (dotted lines). Right-top corner inset: depth profile model. Right-bottom corner inset: relative depth plot.

TABLE 2: Depth Profile Parameters of Synthetic Structure 4_2 + 2 and Results of MEM Simulation thickness (Å) layer

model

simulation

|deviation| (Å)

overlayer

16.0

15.9

0.1

concentration (at. %) layer

species

model

simulation

relative error (%)

overlayer

A B C D

20 80 40 60

20 80 42 58

0 0 6 -4

bulk

The relative errors were generally below 5%; only the Eenriched bulk layer showed a concentration error of 9%, and its thickness clearly was much smaller than the layer thickness in the model. Synthetic Structure 8. Parameters of this structure are reported in Table 1. In this example there is a complex interface where the A component is mixed with an E-enriched phase of the bulk. The MEM protocol was first applied to the theoretical ACD data calculated using IMFP of 40, 30, 20, 10, 45, 35, 25, and 27 Å for components A, B, C, D, E, F, G, and H, respectively. Here too, a random error of (10% was introduced for each of the theoretical ACD data to simulate the random error which always affects any experimental data set, and a new numerical experiment was performed, referred to as 8error. The reconstructed depth profile obtained in the error-added case is shown in Figure 7a (dotted lines) together with the depth profile model. Figure

7b shows the ACD data and their MEM curve fitting (dotted lines), together with a schematic illustration of the depth profile model (right-top corner inset) and the RDP calculated for this synthetic structure (right-bottom corner inset). Table 4 shows the results obtained for both the 8 and 8error numerical experiments, together with the relative errors with respect to the depth profile model. In this eight-component system, the maximum error in a single component concentration reached 30%, and the thickness of the E-enriched layer was found to be too high compared to the model. Discussion To transform the intensity vs angle data from ARXPS into the concentration vs depth distribution of the elements in the different chemical environments, a new MEM protocol was developed in this work. Examples of MEM reconstruction from systems with three to eight components differently distributed within the layers are discussed. However, it is also clear that applying this new MEM protocol to real cases, any additional information about the sample, such as chemical state of the elements and a starting estimation of the depth profile (e.g., obtained with Tougaard’s method10), or gleaned from the literature, is essential for solving the problem of the nondestructive reconstruction of a depth profile from ARXPS data.11 Toward the New MEM Protocol. The maximum entropy method was originally developed with the aim of restoring astronomical images.19,20 After its successful application to that field, and thanks to the pioneering works of Livesey and Smith,21 several authors began to write suitable algorithms to apply MEM to concentration depth profile reconstruction from ARXPS

Figure 6. (a) Depth profile of synthetic structure 7_berror and MEM simulation (dotted lines) and (b) apparent concentrations diagram of synthetic structure 7_berror (dots) and recalculated MEM data (dotted lines). Right-top corner inset: depth profile model. Right-bottom corner inset: relative depth plot.

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TABLE 3: Depth Profile Parameters of Synthetic Structure 7_b and Results of MEM Simulation, Both With and Without Random Error in the ACD Data simulation with error-free ACD data

model

simulation with error-affected ACD data

layer

thickness (Å)

thickness (Å)

|deviation| (Å)

thickness (Å)

|deviation| (Å)

overlayer 1st intermediate 2nd intermediate E-enriched bulk

7.0 12.0 6.0 16.0

6.4 11.8 7.2 8.7

0.6 0.2 1.2 7.3

6.6 12.3 7.1 8.0

0.4 0.3 1.1 8.0

simulation with error-free ACD data

model layer overlayer 1st intermediate E-enriched bulk bulk


species

concentration (at. %)


relative error (%)


relative error (%)

B G D F G C E C E

80 20 23 15 62 65 35 82 18

78 22 24 15 61 65 35 constrained constrained

-3 10 4 0 -2 0 0 / /

79 21 22 14 64 62 38 constrained constrained

-1 5 -4 -7 3 -5 9 / /

intensity data. Some examples of these MEM applications are reported in refs 6, 13, 22-24, 33, and 34. Nevertheless, stateof-the-art MEM algorithms appear to be successfully applicable only to a morphologically limited range of depth profile. Up to now, MEM has been applied to a maximum of fivecomponent profiles. Cumpson29 showed that: (i) it is fundamental to have a reliable starting profile, (ii) depth resolution is limited by the signalto-noise ratio of ARXPS measurements rather than number of emission angles for which data are acquired, and (iii) depth resolution can be enhanced by narrowing as much as possible the depth reconstruction interval. As mentioned, when both versions of the algorithm written by the present authors have been separately applied to the synthetic profiles with seven and eight components reported hereinbefore, as well as to the experimental ARXPS data acquired on NiP alloys (seven components),18 the reconstruction failed. Running the algorithm from the same starting profile, rather different results were obtained at the end of each run. Now, since a better convergence was not achieved by narrowing the depth interval and both the algorithms “performed well” on profiles with up to five components, it was clear that the major problem derived from lack of a “good” starting profile.

Thus, the idea was conceived of having to resort to an apparent concentration diagram simulator to look for the simplest wafer model that reproduced “experimental” data as fine as possible and then combining the two versions of the MEM algorithm together in a unique protocol, the new MEM protocol (Figure 2). The second version of the MEM algorithm, that is characterized by a smaller number of variables, is used to improve the raw guess derived from the simulator, whereas the first version, characterized by a higher number of variables, performs the refinements of the depth profile at the end of the protocol. In addition, here a novel iterative approach is adopted for estimating IMFP values. Whereas they are perfectly known in a numerical experiment, in the real case they have to be calculated. The resort to IMFPs derived from predictive formulas applied to homogeneous materials is clearly a very strong approximation when the surface region of a sample is known to be inhomogeneous on the nanometer depth scale. In this work, for the first time, an attempt to overcome this problem is addressed. Accuracy of the New MEM Protocol. Results of the numerical experiments bring evidence that our new MEM

Figure 7. (a) Depth profile of synthetic structure 8error and MEM simulation (dotted lines) and (b) apparent concentrations diagram of synthetic structure 8error (dots) and recalculated MEM data (dotted lines). Right-top corner inset: depth profile model. Right-bottom corner inset: relative depth plot.



TABLE 4: Depth Profile Parameters of Synthetic Structure 8 and Results of MEM Simulation, Both With and Without Random Error in the ACD Data model



layer

thickness (Å)

thickness (Å)

|deviation| (Å)

thickness (Å)

|deviation| (Å)

overlayer intermediate E-enriched bulk + A species

7.0 12.0 12.0

6.6 12.1 11.8

0.4 0.1 0.2

6.8 12.7 15.0

0.2 0.7 3.0

model layer overlayer intermediate E-enriched bulk + A species bulk



species



relative error (%)


relative error (%)

B H D F G A C E C E

80 20 23 15 62 40 36 24 82 18

78 22 27 14 59 40 40 20 constrained constrained

-3 10 17 -7 -5 0 11 -17 / /

81 19 25 13 62 34 11 19 constrained constrained

1 -5 9 -13 0 -15 31 -21 / /

protocol is able to resolve the depth profile of a sample, composed of between three and eight components. Deviation of the calculated layer thickness from the input values of the synthetic structure was found to increase with increasing depth (Figure 8a). Within the investigated range up to 45 Å, the relative error was found to be e10% for the majority of the examined synthetic structures (Tables 2-4). A relative error of 10-20% was only found in a few cases, all of which can be considered “secondary structural parameters”.10 The superposition of a 10% random error in the ACD data resulted in slightly higher errors in the layer thickness (Tables 3 and 4). The relative error in the concentration of the individual components increased with the number of components present (Figure 8b). This is not surprising, as structure complexity increases with the number of species contained therein. As can be seen from Tables 3 and 4, the relative error in the calculated concentration is higher for buried layers and increases also when noise is added to the data. Overall, it was found to be e10% for most of the cases examined. A relative error of 10-30% in concentration was found only for a few cases, all of which can be considered “secondary structural parameters”.10

Summarizing, accuracy of the new MEM protocol was found to be very satisfactory. Errors increased with increasing layer depth and with increasing number of species in the sample (Figure 8). Comparison with Other Methods. As mentioned in the Introduction, other nondestructive methods for calculating the layered structure or reconstructing in-depth profiles are reported in the literature.7,8,10 These methods, e.g., the Tougaard method10 and the three-layer-model,7,8 will be compared to the new MEM protocol in a subsequent paper.35 It can be anticipated that the new MEM protocol presented here is at least as accurate as the other methods but overcomes the inherent limitations and drawbacks in the Tougaard or the three-layer model. However, our protocol has not been tested on purely exponential depth profiles while Tougaard’s method allows reconstruction of this type of structures.9 Nevertheless, even if further experiments are needed to prove this point, the MEM protocol is believed to accurately reproduce also exponential profiles, the algorithm being able to properly account for concentration gradients.18,24 Future Developments. In this work, the new MEM protocol was numerically tested on structures with up to eight components, but this “limit” could be extended to nine or more

Figure 8. (a) Absolute deviation of layer interface depth vs depth and (b) relative error of species concentration vs number of species involved in the structure.

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components performing additional numerical experiments. Further experiments are also required to investigate the role of surface roughness on MEM protocol accuracy, especially on overlayer thickness determination. However, even if in this work this issue has not been addressed, an important consideration can be derived from works of other authors that focused on this significant and interesting aspect. Gunter and co-workers33 reported that the thickness of an overlayer on a rough substrate may be accurately determined without taking into account the roughness in the analysis of the XPS intensities; neglecting roughness effects results in an average error less than 10% in the determined thickness. Oswald34 concluded that rough surface layers often are considered homogeneous and smooth, especially when the structure sizes are in the range of (or below) the attenuation length of the measured photoelectrons. This makes it nearly impossible to interpret the character of the roughness on the basis of MEM reconstruction. Nevertheless, at least the amount of the overlayer material and its effective thickness can be satisfactorily calculated for a wide range of structures. For the quantification it can be concluded that an underestimation in the range of up to 5% systematically occurs.34 The IMFP calculation accuracy can also be improved taking into account surface excitations. In fact, it is known that electron excitation and photoemission processes are affected by the competitive surface excitation; i.e., impinging and escaping electrons suffer losses in the solid surface region, producing surface plasmons.36 Thus, knowledge of inelastic scattering cross-section is of primary interest in a surface-sensitive technique such as ARXPS. Surface effects result in IMFP variation with depth and might also be important for the properties of photoelectron angular distribution.37 Dielectric response theory is often used to study the inelastic interaction of electrons near solid surfaces.37 For instance, Yubero and Tougaard38 proposed a dielectric model to demonstrate the depth dependence of the IMFP. They showed that in the case of XPS there are interference effects between fields caused by the creation of the static core hole upon photoemission and movement of the photoemitted electron.38 Therefore, IMFP cannot be separated as bulk and surface contributions, and a method to determine the dielectric function from ReflectionElectron-Energy-Loss (REELS) spectra was suggested.38 On the other hand, Chen37 found that the IMFP inside a solid can be considered as a constant equal to the bulk IMFP because surface excitation probability is compensated by a reduction of the bulk excitation probability. Nevertheless, surface effects in the vacuum side result in additional energy-loss probability and may cause a significant influence on peak intensity.37 The additional inelastic-scattering probability due to surface effects of the vacuum side can be conveniently described by a surface excitation parameter (SEP).37 However, it was shown that surface effects lead to a reduction of photoelectron intensities especially at large angles (>75°) since surface excitations are most probable for glancing electrons37 and that the difference between the results obtained with and without neglecting surface effects for emission angles less than 60° are small.37 Finally, other than our simulator, the NIST database for the “Simulation of Electron Spectra for Surface Analysis” (SESSA)39 is available to simulate both standard mode and ARXPS spectra of a given sample in a set of given experimental conditions. The approach is different and intriguing. The simulation algorithm is based on the partial-intensity approach (PIA) for the electron-solid interaction.39-41 The key quantities in this approach are the so-called partial intensities that describe the number of electrons arriving at the detector after having

Scorciapino et al. participated in a given number of inelastic collisions in the solid between their point of generation and escape from the surface.39 The new MEM protocol developed in this work can be interesting for its application to the study of a very large variety of real samples but now needs to be fully implemented. The input/output linkage between the two algorithms involved, as well as between the simulator routine and the second version of the algorithm, needs to be computer coded. Furthermore, it is possible to also implement the G-1 predictive formula for calculating the appropriate IMFP set during protocol runs instead of working with a fixed values set. These improvements will make the overall simulator-protocol routine enormously less time-consuming and should also improve accuracy of the results. Once this code has been implemented, it could be incorporated into a software package for XPS spectra analysis, to simplify the passage from ARXPS spectra curve fitting to their intensity correction for sensitivity factors and their input into the MEM simulator-protocol code. Conclusions In this work, a new MEM protocol was developed for the nondestructive reconstruction, from ARXPS data, of in-depth composition profiles for the outermost surface region of layered samples. Numerical experiments on a series of synthetic structures demonstrated that this MEM protocol allows reconstructing the depth profile of a sample composed of between three and eight components with multiple layers. The accuracy of the protocol is good: errors are e10% for the primary structural parameter and 10-30% for the secondary ones. Accuracy decreases with increasing depth. The in-depth composition profiles reconstructed using the new MEM protocol describe a layered interface in detail. For widespread application in XPS analysis of thin-layered structures, several operations that are performed manually have to be implemented in a computer-based routine. Acknowledgment. This work was undertaken with the financial support of the University of Cagliari. The Italian Ministry of the University and Research (MIUR) is acknowledged for the Ph.D. fellowship to M.A. Scorciapino. Supporting Information Available: Results of other numerical experiments. The synthetic structures are: 4_1 + 1+1 + 1, 5_1 + 1+1 + 1+1, 5_3 + 2, 6_1 + 1+1 + 1+1 + 1, 6_1 + 2+1 + 2, 6_1 + 3+2, 6_3 + 1+2, 6_3 + 3. The codes are available upon request to the corresponding author ([email protected]). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Briggs, D.; Seah, M. P., Eds.; Practical Surface Analysis by Auger and X-Ray Photoelectron Spectroscopy, 2nd ed.; Wiley & Sons Ltd.: West Sussex, UK, 1990; Vol. 1. (2) Seah, M. P. The Quantitative Analysis of Surfaces by XPS: A Review. Surf. Interface Anal. 1980, 2, 222–239. (3) Tougaard, S. Inelastic background correction and quantitative surface analysis. J. Electron Spectrosc. 1990, 52, 243–271. (4) Tougaard, S. Formalism for quantitative surface analysis by electron spectroscopy. J. Vac. Sci. Technol. A 1990, 8, 2197–2203. (5) Tougaard, S. Surface nanostructure determination by x-ray photoemission spectroscopy peak shape analysis. J. Vac. Sci. Technol. A 1996, 14, 1415–1423. (6) Opila, R. L.; Eng, J. J. R. Thin films and interfaces in microelectronics: composition and chemistry as function of depth. Prog. Surf. Sci. 2002, 69, 125–163. (7) Rossi, A.; Elsener, B. XPS analysis of passive films on the amorphous alloy Fe70Cr10P13C7: effect of the applied potential. Surf. Interface Anal. 1992, 18, 499–504.

Surface Depth Profiles from ARXPS Data (8) Elsener, B.; Rossi, A. XPS investigations of passive films on amorphous Fe-Cr alloys. Electrochim. Acta 1992, 37, 2269–2276. (9) Tougaard, S. QUASES-Tougaard Software Package, Ver. 5.1; http:// www.quases.com/. (10) Tougaard, S. Accuracy of the non-destructive surface nanostructure quantification technique based on analysis of the XPS or AES peak shape. Surf. Interface Anal. 1998, 26, 249–269. (11) Cumpson, P. J. Angle-resolved X-ray photoelectron spectroscopy. Surface analysis by Auger and x-ray photoelectron spectroscopy; Briggs, D., Grant, J. T., Eds.; Surface Spectra and IM Publications: UK, 2003; p 651. (12) Haupt, S.; Strehblow, H. H. Combined electrochemical and surface analytical investigations of the formation of passive layers. Corros. Sci. 1989, 29, 163–182. (13) Smith, G. C.; Livesey, A. K. Maximum entropy: A new approach to non-destructive deconvolution of depth profiles from angle-dependent XPS. Surf. Interface Anal. 1992, 19, 175–180. (14) Penn, D. R. Electron mean-free-path calculations using a model dielectric function. Phys. ReV. B 1987, 35, 482–486. (15) Seah, M. P.; Dench, W. A. Quantitative electron spectroscopy of surface: a standard data base for electron inelastic mean free paths in solids. Surf. Interface Anal. 1979, 1, 2–10. (16) Tanuma, S.; Powell, C. J.; Penn, D. R. Calculations of electron inelastic mean free paths. Surf. Interface Anal. 1993, 21, 165–176. (17) Gries, W. H. A universal predictive equation for the inelastic mean free pathlengths of x-ray photoelectrons and Auger electrons. Surf. Interface Anal. 1996, 24, 38–50. (18) Scorciapino, M. A.; Navarra, G.; Elsener, B.; Rossi, A. Nondestructive surface depth profiles from angle resolved XPS data using the Maximum Entropy Method (MEM). II. Characterization of surface film formed on Ni-18P alloy after polarization in 0.1 M Na2SO4. J. Phys. Chem. C, submitted. (19) Gull, S. F.; Daniel, G. J. Image reconstruction from incomplete and noisy data. Nature 1978, 272, 686–691. (20) Cornwell, T. J.; Evans, K. F. A simple maximum entropy deconvolution algorithm. Astron. Astrophys. 1985, 143, 77–83. (21) Livesey, A. K.; Smith, G. C. The determination of depth profiles from angle-dependent XPS using maximum entropy data analysis. J. Electron. Spectrosc. 1994, 67, 439–461. (22) Chang, J. P. Profiling nitrogen in ultrathin silicon oxynitrides with angle-resolved x-ray photoelectron spectroscopy. J. Appl. Phys. 2000, 87, 4449–4455. (23) Champaneria, R.; Mack, P.; White, R.; Wolstenholme, J. Nondestructive analysis of ultrathin dielectric films. Surf. Interface Anal. 2003, 35, 1028–1033. (24) Olla, M.; Navarra, G.; Elsener, B.; Rossi, A. Nondestructive indepth composition profile of oxy-hydroxide nanolayers on iron surfaces from ARXPS measurements. Surf. Interface Anal. 2006, 38, 964–974.

J. Phys. Chem. C, Vol. 113, No. 51, 2009 21337 (25) Werner, W. S. M.; Gries, W. H.; Sto¨ri, H. A Monte Carlo study of the angular dependence of the depth distribution function of signal electrons in electron spectroscopies. J. Vac. Sci. Technol. A 1991, 9, 21–26. (26) Jablonski, A.; Zemek, J. Experimental and theoretical tests of elastic scattering effects in XPS. Surf. Sci. 1997, 387, 288–299. (27) Jablonski, A.; Powell, C. J. The electron attenuation length revisited. Surf. Sci. Rep. 2002, 47, 33–91. (28) Jablonski, A. Quantitative AES: Via the inelastic mean free path or the attenuation length. Surf. Interface Anal. 1990, 15, 559–566. (29) Cumpson, P. J. Angle-resolved XPS and AES: depth-resolution limits and a general comparison of properties of depth-profile reconstruction methods. J. Electron. Spectrosc. 1995, 73, 25–52. (30) Paynter, R. W. Modification of the Beer-Lambert equation for application to concentration gradients. Surf. Interface Anal. 1981, 3, 186– 187. (31) Skilling, J.; Gull, S. F. Maximum entropy method in image processing. IEE Proc.-F 1984, 131, 646–659. (32) Powell, C. J.; Jablonski, A. NIST Electron Inelastic-Mean-FreePath Database, v.1.1; 2000. http://www.nist.gov/srd/nist71.htm. (33) Gunter, P. L. J.; Gijzeman, O. L. J.; Niemantsverdriet, J. W. Surface roughness effects in quantitative XPS: magic angle for determining overlayer thickness. Appl. Surf. Sci. 1997, 115, 342–346. (34) Oswald, S.; Oswald, F. Modeling of surface roughness for ARXPS. Phys. Status Solidi C 2007, 4, 1817–1821. (35) Scorciapino, M. A.; Navarra, G.; Elsener, B.; Rossi, A. Comparison of the Tougaard, ARXPS-MEM and three-layer method to determine the compositional depth profile of surface layers in the nanometer range. In preparation. (36) Gergely, G.; Menyhard, M.; Gurban, S.; Sulyok, A.; Toth, J.; Varga, D.; Tougaard, S. Surface excitation effects in electron spectroscopy. Solid State Ionics 2001, 141-142, 47–51. (37) Chen, Y. F. Surface effects on angular distributions in X-rayphotoelectron spectroscopy. Surf. Sci. 2002, 519, 115–124. (38) Yubero, F.; Tougaard, S. Model for quantitative analysis of reflection-electron-energy-loss spectra. Phys. ReV. B 1992, 46, 2486–2497. (39) Werner, W. S. M. Electron transport in solids for quantitative surface analysis. Surf. Interface Anal. 2001, 31, 141–176. http://www.nist.gov/srd/ nist100.htm. (40) Werner, W. S. M. Electron Transport in Solids. Surface analysis by Auger and x-ray photoelectron spectroscopy; Briggs, D., Grant, J. T., Eds.; Surface Spectra and IM Publications: UK, 2003; p 235. (41) Werner, W. S. M. Simulation of electron spectra for surface analysis using the partial-intensity approach. Surf. Interface Anal. 2005, 37, 846–860.

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