Experimental determination of ruthenium L-shell fluorescence yields and Coster–Kronig transition probabilities

Abstract

The L-shell fluorescence yields and the Coster–Kronig factors of ruthenium (and the corresponding uncertainty) were determined for the first time experimentally by applying radiometrically calibrated instrumentation of the Physikalisch-Technische Bundesanstalt. The resulting fluorescence yields (ω_L3 = 0.0459(20), ω_L2 = 0.0415(26), ω_L1 = 0.0109(9)) and the Coster–Kronig factors (f₂₃ = 0.177(32), f₁₃ = 0.528(90), f₁₂ = 0.173(73)) agree reasonable well with parts of the data from the literature.

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1 Introduction

Ruthenium is a versatile and widely used chemical element playing a crucial role in important areas of science and technology. Several applications in the area of semiconductor fabrication or catalysis can be identified, where ruthenium is essential. For extreme ultraviolet lithography masks^1,2 or as interconnect metal^3–5 either ruthenium or ruthenium-containing materials are very relevant. In catalysis, ruthenium-based catalysts provide remarkable properties in several different applications^.6 In addition to this, ruthenium is also of relevance for emerging applications for energy storage^7,8 and medicine.^9,10

But, if X-ray fluorescence (XRF) based techniques are to be used for determining the ruthenium content in such materials, one quickly finds that the knowledge of the relevant atomic fundamental parameter (FP) data for ruthenium is very limited: for ruthenium, especially its L-shell FP data and namely the L-subshell fluorescence yields and Coster–Kronig factors (CK), no experimentally determined data seems to exist so far. Available data in the literature is either purely theoretically determined or perhaps even less favorable, only interpolated employing adjacent chemical elements. As these FPs quantitatively describe the process of X-ray fluorescence generation, they are very crucial for most quantification approaches in XRF. Thus, they have a direct influence on the accuracy of the XRF quantification results.

As this is a highly inadequate situation, we applied the PTB's reference-free X-ray spectrometry toolset in order to experimentally determine the fluorescence yields and the Coster–Kronig factors of the L-subshells of ruthenium for the first time. Based on transmission and fluorescence experiments on thin film samples, such FP data can be derived as already demonstrated for a wide range of chemical elements.^11–15

2 Experimental procedure

For an experimental determination of L-shell fluorescence yields and Coster–Kronig transition probabilities, both fluorescence- and transmission experiments with a selective excitation of the three L-subshells on either a free standing thin foil or a thin coating on a carrier foil are required^.11–13,15 In the present work, these experiments were conducted on the four-crystal monochromator (FCM) beamline¹⁶ of BESSY II using a vacuum chamber that is in-house developed.¹⁷ This chamber was endowed with a silicon drift detector (SDD) of which the detection efficiency is radiometrically calibrated and the response functions are determined experimentally.¹⁸ The employed sample was a highly homogeneous 150 nm ruthenium deposition on a 500 nm Si₃N₄ membrane. To be able to isolate the ruthenium contribution from the total sample transmission, also a blank membrane of nominally identical thickness was used. Any potential moderate variation in the Si₃N₄ membrane thickness is only a second-order contribution to the uncertainties. Both samples were positioned in the chamber's center by using an x–y-scanning stage. The angle between the incoming beam and the sample as well as the angle between sample surface and detector was set to 45°.

The transmission measurements were conducted in an energy range around the Ru-L absorption edges between 2.1 keV and 4 keV. Furthermore, X-ray fluorescence measurements were performed in the incident-energy domain between 2.8 keV and 3.4 keV. The established methodology^11,12,19,20 to derive the relevant L-shell FPs from this experimental dataset is described in the following.

According to the Sherman equation,²¹ the measured count rate of fluorescence photons of a one-elemental foil, which is irradiated under 45°, is the product of the fluorescence production cross section σ_Li of the considered shell, the incoming Φ₀(E₀) as well as the fluorescence photon flux Φ_i^d(E0), the detection efficiency of the SDD, the mass deposition of that element, the attenuation correction factor M_i,E0 and the solid angle Ω of detection of the SDD.

The self-attenuation correction factor takes into account the attenuation of the incident radiation and of the fluorescence radiation on its way through the sample. The corresponding sample-specific attenuation correction factor M_i,E0 is determined by transmission experiments taking advantage of the fact, that the knowledge of the ruthenium deposition thickness d and its density ρ is not needed since they appear only in a product with the mass absorption coefficient μ_S or with the subshell photoionization cross section τ_S. The product μ_Sρd is derived from the transmittance data using the Lambert–Beer law.

For incident energies E₀ between the L₃ edge and the L₂ edge, the fluorescence production factor for the L₃-subshell is

where ω_L3 is the ruthenium L₃ fluorescence yield which should be determined. The sample-specific attenuation correction factor M_i,E0 is defined asHere, θ_in is the angle between the incident beam and the sample surface, θ_out is the angle between the sample surface and the SDD detector.

Due to the so-called Coster–Kronig effect, the effective photoionization cross section τ_eff,Li(E₀) for L₃ and L₂ is a linear combination with the higher bound shells since for photon energies above the excitation energy of the next subshell, created holes in L₂ can decay into L₃ by ejecting outer electrons. As a result, more than the directly created holes in L₃ exist. The CK-factor f₂₃ provides the probability for this to happen and similar transitions can occur between the L₁ and the L₂ and L₃ shells. So for an incident photon energy above the L₁ threshold, the fluorescence production factors are defined as:

τ_eff,L3(E₀) = τ_L3(E₀) + f₂₃τ_L2(E₀) + [f₁₃ + f₁₂f₂₃]τ_L1(E₀) (3)

τ_eff,L2(E₀) = τ_L2(E₀) + f₁₂τ_L1(E₀) (4)

τ_eff,L1(E₀) = τ_L1(E₀) (5)

Here, the τ_Li(E₀) are the photoionization cross sections of the respective L_i subshell,¹⁹ and f_ij are the Coster–Kronig factors. For incident energies below the subsequent subshell, the corresponding subshell photoionization cross section is zero (τ_Li(E₀) = 0 for E₀ < E_Li). Therefore, the fluorescence yields are determined for energies E₀ above the excitation energy of the considered and below the subsequent subshell.

All relevant observables are accessible from the experimental data as μ_Sρd are determined for the relevant energies by measuring the transmission of the ruthenium coating. Φ_i^d(E0) is determined by spectral deconvolution of the recorded SDD spectra considering the relevant fluorescence lines and relevant background contributions such as bremsstrahlung. Φ₀(E₀) and Ω are known because of PTB's calibrated instrumentation.²²

Taking into account the theoretical ratio of scattering to ionization cross sections, which one can take from databases,²³ the sample-specific total photoionization cross section τ_Sρd can be derived. To isolate the subshell contributions of the different τ_Li, Ebel polynomials²³ for each L_i contribution as well as a total cross section for lower bound shells are scaled into the data (see Fig. 1). For this scaling process, only the datapoints slightly above each absorption edge are used to minimize the effect of the fine structure.

Fig. 1 τ _S(E₀)ρd determined for the ruthenium thin film: separation of the contributions of lower bound shells (yellow), L₃ (green), L₂ (red) and L₁ (orange).

With these determined τ_Liρd, the equations for the fluorescence production cross sections can be solved for ω_Li. By replacing τ_Li by the effective photoionization cross section according to eqn (3)–(5), eqn (1) can be applied also for energies above the next subshell. Therefore, to determine f₂₃, energies between E_L3 and E_L2 were considered, see Fig. 2: with the already determined ω_L3, the modified version of eqn (1) can be solved for f₂₃. f₁₂ is determined in the same way but applied for the fluorescence of the L₂ shell and for E_L2 < E₀ < E_L1 with the already determined ω_L2. With these determined f₂₃ and f₁₂, from the fluorescence of the L₃ shell for energies above E_L1, f₁₃ can be determined.

Fig. 2 Experimental determination of the Ru-L₃ (left image) and Ru-L₂ (right image) fluorescence yields: they are determined by averaging over all considered energies where only the respective shell is excited (below L₂ for ω_L1 and below L₁ for ω_L2). Using these ω_L3 and ω_L2, the Coster–Kronig factors are determined in such a way that the average in the higher energy domains matches the ω_Li value.

3 Results

The determined fluorescence yields are ω_L3 = 0.0459(20), ω_L2 = 0.0415(26) and ω_L1 = 0.0109(9). The resulting Coster–Kronig factors are f₂₃ = 0.177(32), f₁₃ = 0.528(91) and f₁₂ = 0.173(73). These values are compared with values from the literature in Table 1 and Fig. 3 and 4. The respective uncertainties were calculated via error propagation. The main contributions to the total uncertainty budget of the fluorescence yields were arising from the spectral deconvolution (∼2%) and from the photoionization cross sections (∼2%). The uncertainty budget is calculated by applying the reference-free XRF approach for the FP determination, discussed in more detail in ref. 24.

Table 1 Comparison of the experimentally determined Ru-L-subshell fluorescence yields and Coster–Kronig factors with the X-raylib database²⁵ [version 4.0.0] and other values (values from compilations (comp.) and theoretic values) from the literature

	Ru ωL3	Ru ωL2	Ru ωL1
This work (XRF)	0.0459(20)	0.0415(26)	0.0109(9)
X-raylib^25 (comp.)	0.043	0.040	0.012
Krause^26 (comp.)	0.043(9)	0.040(10)	0.012(4)
Perkins et al.^29 (comp.)	0.045231	0.043368	0.0084138
McGuire^28 (theory)	0.0450	0.0418	0.00774
Puri et al.^27 (theory)	0.045	0.043	0.0083
Xu et al.^30 (comp.)			0.015

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	Ru f23	Ru f13	Ru f12
This work (XRF)	0.177(32)	0.528(90)	0.173(73)
X-raylib^25 (comp.)	0.144	0.61	0.10
Krause^26 (comp.)	0.148(30)	0.61(7)	0.10(2)
McGuire^28 (theory)	0.136	0.779	0.057
Puri et al.^27 (theory)	0.140	0.766	0.057

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Fig. 3 Comparison of the experimentally determined Ru-L-subshell fluorescence yields with values from the literature.

Fig. 4 Comparison of the experimentally determined Coster–Kronig factor f₂₃ with values from the literature.

The X-raylib²⁵ and Krause²⁶ values of ω_L3 and ω_L1 are slightly outside of the error domain of the values determined in this work. The agreement with respect to the theoretically calculated data of Puri²⁷ and McGuire²⁸ is better in the case of ω_L3 but even worse for ω_L1. The data of Perkins²⁹ as well as the data by Xu³⁰ behaves very similarly. For ω_L2, all available data from the literature agrees well with the result obtained here. With respect to the Coster–Kronig factors, the tabulated data in X-raylib and the Krause compilation is in good agreement with our results. However, the results are on or slightly outside the boundary of our uncertainty budget for all three CK values. The data by McGuire and Puri is outside of our results considering their uncertainty budget.

4 Conclusion

The Coster–Kronig factors and the fluorescence yields of ruthenium are determined experimentally by applying PTB's radiometrically calibrated instrumentation. The values determined are in reasonably good agreement with the values from the literature, although some literature values are slightly outside the uncertainty ranges of this work. The magnitude of the determined uncertainties of this work is much lower than the estimated uncertainties of Krause²⁶ in the case of the fluorescence yield values. With respect to the Coster–Kronig factors, similar uncertainties were achieved here. In summary, this uncertainty reduction will positively influence the total uncertainties of fundamental parameter-based quantitative X-ray fluorescence experiments.As stated already in previous works of our group,^14,15,24,31 the X-raylib database is also in the case of the Ru-L shell fundamental parameters a reliable reference.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This project has received funding from the ECSEL Joint Undertaking (JU) IT2 under grant agreement No 875999. The JU receives support from the European Union's Horizon 2020 research and innovation programme and the Netherlands, Belgium, Germany, France, Austria, Hungary, the United Kingdom, Romania and Israel.

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