Using photoelectron spectroscopy to measure resonant inelastic X-ray scattering: A computational investigation

Resonant Inelastic X-Ray Scattering (RIXS) has emerged as a powerful technique to study elementary excitations Ament et al. (2011). RIXS probes excitations via core-valence transitions with element-specific energies, which allows one to tune the elemental locations being probed through the incident X-ray photon energy. Because of the large momentum of X-rays, RIXS is able to probe the dispersion of elementary excitations in solids, unlike lower energy optical photons. Further, in contrast to other X-ray-based spectroscopies, the energy resolution of RIXS is not limited by short core hole lifetimes. Because of these strengths, much effort has been devoted to developing RIXS capabilities. In the soft x-ray range, X-ray grating and synchrotron light source development has enabled RIXS measurements with energy resolution better than 100 meV Brookes et al. (2018). Having these high energy resolutions is particularly important for studies of solids where characteristic energies of many important excitations are around 100 meV or less Ament et al. (2011); Chaix et al. (2017). Leveraging these capabilities, RIXS studies have given new insights in wide-ranging topics including solid state physics Ament et al. (2011); Le Tacon et al. (2011); Schlappa et al. (2012); Chaix et al. (2017), nanoparticles Liu et al. (2017), interfaces Rajasekaran et al. (2012), batteries House et al. (2020); Firouzi et al. (2018), liquids Wernet et al. (2015) and gases Hennies et al. (2010).

Despite these strengths and much instrumentation development Brookes et al. (2018), grating spectrometers only detect one X-ray photon for every $\approx 10^{6}$ X-ray photons incident on a sample Ghiringhelli and Braicovich (2013); Dakovski et al. (2017). Thus, RIXS is a very photon hungry technique. In addition, the X-ray flux incident on a sample is constrained by radiation damage and X-ray source output. This limits achievable count rate, and thus accuracy, of RIXS measurements. Work has been done towards improving this. Transition edge sensors Uhlig et al. (2015) and off-axis zone plates Marschall et al. (2017) can make more accurate RIXS measurements than traditional instrumentation in certain cases, but so-far demonstrated resolutions are significantly more than 0.5 eV. More information could be gleaned from RIXS if one could make faster and more accurate measurements that maintained hundreds of meV or better resolution. For example, high resolution time-resolved RIXS measurements probe dynamics of elementary excitations and can discriminate well between different states that occur in sample evolution Wernet et al. (2015), but are very challenging due to their requirement of recording many accurate spectra.

Coinciding with the rise and development of RIXS, the capabilities of electron spectrometers have also greatly increased Damascelli et al. (2003). Photoelectron spectrometers can now have energy resolutions better than 20 meV for 500 eV kinetic energy electrons Seidel et al. (2017). The collection efficiency, and thus achievable signal-to-noise ratio with a given number of particles emitted from a sample, can be much higher for electron spectrometers than X-ray spectrometers with comparable resolutions. Further, the footprint of electron spectrometers with 10s of meV resolution when measuring several hundred eV electrons (few m²) is far smaller than current X-ray spectrometers with comparable resolutions for several hundred eV photons (few hundred m²) Brookes et al. (2018). These features are largely due to the ease with which electrons can be manipulated, owing to their charge, in comparison with neutral X-rays. Thus if one can transform the X-ray measurement problem of RIXS to an electron measurement problem, then there could be large gains in count rates, as well as instrumentation compactness and ease of implementation.

Recently, Dakovski et al. Dakovski et al. (2017) proposed doing exactly that to measure Resonant Inelastic X-ray Scattering (RIXS) with hundreds of meV or better resolution through Photoelectron Spectrometry for Analysis of X-rays (PAX) Krause (1965); Ebel (1975). Fig. 1 gives an overview of this technique, which makes use of sharp photoemission features that occur in the photoemission spectra of materials such as Ag, Au, Pt or Al when measured with monochromatic incident X-ray radiation. Fig. 1A shows an experimental schematic. X-rays to be measured (Fig. 1B) are incident on a converter system, where absorption of the X-rays generates photoelectrons. The converter material is assumed to give some photoemission spectrum when measured with monochromatic X-ray radiation, $xps(BE)$, as a function of binding energy, $BE$ (Fig. 1C shows an example Ag 3d photoemission spectrum, while Fig. 1D shows an example sharp Fermi edge). The emitted photoelectrons are then detected with a photoelectron spectrometer. We call the resultant electron spectrum the PAX spectrum (Fig. 1E or F). The expected shape of the PAX spectrum is given by a convolution with $h(E)=xps(-E)$ acting as an impulse response function. Convolving the spectrum of X-rays incident on the converter material (Fig. 1B) with $h(E)$ (Fig. 1C or D) approximately gives the expected value of the PAX spectrum (Fig. 1E or F),

where we only integrate over physically realistic positive photon energies. Here, $kE$ is electron kinetic energy, $s(\hbar\omega)$ is the X-ray spectrum incident on the converter material as a function of photon energy, $\hbar\omega$, and $m(kE)$ is a measured PAX spectrum.

Given a PAX spectrum, the ground truth X-ray spectrum, $s(\hbar\omega)$, can be estimated directly through deconvolution or in a parameterized form such as a sum of peaks. (The ground truth X-ray spectrum is the X-ray spectrum that would be measured without noise and with perfect resolution.) The decomposition of RIXS spectra into a sum of peaks is a natural method already widely used for traditionally recorded RIXS spectra as the parameters of these peaks can be directly linked to physical characteristics of the matter under study Ament et al. (2011). For more complex RIXS spectra, or cases where less is known about the form of a RIXS spectrum before measurement, the more general case of deconvolution may be more appropriate or would be a first step in further decomposition of a RIXS spectrum into elementary features.

While Dakovski et al. Dakovski et al. (2017) demonstrated the possibility of recording PAX spectra for RIXS and estimating the corresponding RIXS spectra as a sum of peaks, a general algorithm for faithfully reconstructing X-ray spectra from PAX measurements, and a quantitative assessment of the potential of PAX for measurement of X-ray spectra are needed. Here, we fill these gaps by proposing an algorithm for analyzing PAX data using methods from statistical data analysis and machine learning Fister et al. (2007); Bertero et al. (2009); James et al. (2013), then characterizing the performance of PAX in the estimation of model RIXS spectra using the Ag 3d levels or a sharp Fermi edge as a model converter.

The rest of this report is organized as follows. In section II, we discuss the considerations for choosing a converter material for PAX, and explain why the Ag 3d lines and a sharp Fermi edge are compelling cases. In section III we describe and discuss the deconvolution algorithm we used to estimate RIXS spectra from simulated PAX spectra. In section IV we show the simulated performance of PAX in estimating model RIXS features. We find that, using the Ag 3d levels as a photoemission converter, PAX can accurately estimate the width of few hundred meV features when 10⁵ electrons are detected in the measured PAX spectrum. Using a sharp Fermi edge photoemission converter shows promise for estimating finer features. Finally, in section V we conclude and give an outlook for future investigations.

The converter material plays a key role in PAX measurements. Converter materials with high conversion efficiency and narrow photoemssion lines are desirable. Higher conversion efficiencies give higher numbers of detected electrons and thus, potentially, higher signal-to-noise ratios. Narrow photoemission lines enable retrieval of narrow X-ray spectral features with a reasonable number of detected electrons. For thick converter materials with near normal incidence of X-rays and emission of photoelectrons, the conversion efficiency of X-rays to photoelectrons can be approximated as Henke (1972)

where $\tau_{q}$ is the effective ionzation cross-section for the creation of photoelectrons from subshell $q$, $\rho$ is the number per unit volume of atoms within the sample which can emit photoelectrons from the subshell used for PAX, and $\lambda_{e}$ is the electron mean free path at the kinetic energy of the relevant photoelectrons.

In Fig. 2 we show the conversion efficiency estimated with Eq. 2 for some promising cases. For these calculations subshell photoemission cross sections were taken from Yeh and Lindau (1985), electron mean free paths for Ag and Au were taken from Tanuma et al. (2002), electron mean free paths for Al and Pt were taken from Shinotsuka et al. (2015), and atoms per unit volume were calculated from values in Rumble (2019). From Fig. 2, we see that, in the soft X-ray range, conversion efficiencies of nearly ten percent with photoemission linewidths of a few hundred meV are possible using the Au 4f or Ag 3d lines. Narrower photoemission features are available at the expense of a reduced conversion efficiency (such as Al 2p and Pt Fermi level photoemission shown in Fig. 2).

In principle, a similar convolution equation as Eq. 1 describes the measured signal in many X-ray spectroscopies. The signal of interest, such as a RIXS spectrum or X-ray absorption spectrum, is convolved with an instrument response function and an intrinsic broadening function to give the measured spectrum. Thus, more accurate X-ray spectra can often be retrieved through deconvolution of measured results Ebel and Gurker (1975); Fister et al. (2007); Laverock et al. (2011). It is not typical to analyze such spectra using deconvolution however. This is because these convolutions only broaden the measured spectra, and measured spectra are still interpretable as a simple blurring of the true spectrum. For PAX, however, the measured spectrum is typically convolved with a more complicated function than a single peak. The converter material photoelectron spectrum could consist of, for example, two narrow peaks and a non-uniform background, as is common for core levels. It may not be easy to infer the original X-ray spectrum from the measured PAX spectrum in these cases. Thus, while deconvolution is an optional step in traditional X-ray spectroscopies, it is important for PAX measurements.

Now that we have described an algorithm for analyzing PAX spectra, in this section we evaluate the performance of this algorithm on simulated data. We simulated PAX measurements using the Ag 3d photoemission lines to estimate features with structure on a 100 meV scale as well as a sharp Fermi edge such as seen in Au to estimate finer features.

Fig. 6 shows the performance of PAX with the Ag 3d lines as a photoemission converter (A) in estimating the model RIXS spectrum shown in Fig. 1A. Part B shows ground truth and deconvolved X-ray spectra for the number of detected electrons increasing from top to bottom. As the number of detected electrons increases finer details of the X-ray spectrum are accurately estimated. Features with a few hundred meV width are already estimated well with $\approx 10^{5}$ detected electrons, and the width of well estimated features decreases further with increasing number of detected electrons. The RMSE of the deconvolved spectrum decreases as the number of detected electrons increases, as shown in Fig. 6C. Fig. 6D quantifies the ability of the method to accurately estimate fine features through the FWHM of the lowest energy loss peak of the deconvolved spectrum as a function of the number of detected electrons. This FWHM decreases as the number of detected electrons increases, and, after $10^{6.5}$ electrons have been detected, the width of the feature in the deconvolved spectrum is within 10 percent of its true width. We note that the integrated intensity of this first loss feature is only 2.4 percent of the total integrated intensity of the entire model RIXS spectrum. Thus one only needs to detect less than $10^{4}$ photoelectrons that were emitted from RIXS photons originating from this feature in order to accurately estimate the shape of this feature. We accurately estimated this feature despite it being much sharper than the the 233 meV FWHM widths of the Ag 3d photoemission peaks of the model impulse response function.

Fig. 7 shows an analysis of the performance of PAX in estimating spectra with structure on scales finer than 100 meV by using a sharp Fermi edge. For this analysis we used an energy separation of 2 meV between points. The photoemission spectrum was modeled with a constant density of states near the Fermi level and a temperature of 4K (boiling point of He). PAX spectra were simulated for $10^{7}$ electrons detected from photoemission within 0.4 eV of the Fermi level and variable separation of the X-ray doublet peaks. For each panel of Fig. 7, we compare three deconvolved simulated PAX spectra to the ground truth spectrum. By deconvolving more than one simulated PAX spectrum we get an indication of whether features in the deconvolved spectrum are reproducible with the experimental accuracy. This check has the drawback that not all of the data is used in forming a single deconvolved spectrum. In order to get an idea of whether features are reproducible without dividing the data into subsets, we can use the bootstrap method Efron and Tibshirani (1986); James et al. (2013). In this method, we approximate the distribution of PAX spectra that we would obtain from averaging $n$ recorded PAX spectra that compose a PAX data set (in this case $n=1000$). We do this by forming bootstrapped PAX spectra from the data by sampling $n$ times from the $n$ recorded PAX spectra and averaging the results. The sampling is done with replacement so that some samples may be chosen more than once and others not at all. We can then tell if a feature is unlikely to be a random occurrence by confirming whether it consistently occurs in a large fraction of the deconvolved bootstrapped PAX spectra. For clarity, only three such spectra are shown in Fig. 7, but to make scientific conclusions one should look at several times more spectra Efron and Tibshirani (1986).

As the peak separation decreases from top to bottom in Fig. 7, our ability to tell that the ground truth spectrum is two peaks rather than a single peak decreases. With the 70 meV peak separation (A) as well as the 45 meV peak separation (B), it is clear that the deconvolved spectra are not well represented by a single peak, and this is reproducibly the case between independent measurements or bootstrapped measurements. In contrast, for the 25 meV peak separation (C), it is no longer clear that the ground truth spectrum consists of more than one Gaussian peak, and a measurement with better statistics would be required to tell this.

We have demonstrated the potential of PAX for measurement of RIXS spectra. Using the Ag 3d levels as a photoemission converter for PAX, few hundred meV FWHM features can be accurately estimated with $10^{5}$ detected electrons. Even finer features can be accurately estimated with higher numbers of detected electrons. Details with aspects much smaller than 100 meV could be estimated using a sharp Fermi edge photoemssion converter, albeit at the expense of reduced conversion efficiency of RIXS photons to electrons.

In this report, we proposed and tested one algorithm which can be used to estimate a RIXS spectrum from measured PAX data. Our algorithm is simple and closely linked to the classic Lucy-Richardson algorithm. Recently, however, more sophisticated algorithms have shown much promise on achieving more accurate deconvolution results and reducing computational time Ikoma et al. (2018); Zhang et al. (2017). Applying such techniques to PAX could improve on the performance described here. In addition, using uncertainty quantification methods would enable more robust interpretation of X-ray spectra estimated from PAX data. This has been done for similar problems by assessing the sensitivity of deconvolved spectra to artificially added noise Fister et al. (2007) as well as more sophisticated methods Kaipio and Somersalo (2006).

Additional experimental development could also push the capability of PAX further. The model photoemission converters highlighted here were chosen based on a survey of literature photoemission data. A systematic investigation of other materials may provide photoemission features better suited for PAX measurements. Finally, the PAX measurement method could be applied to other situations where high signal-to-noise ratio estimates of X-ray spectra are desired without using traditional grating-based technology. An example could be transmissive soft X-ray spectrometers.