Many popular experimental methods for determining the structure of materials rely on the periodic repetition of atomic arrangements present in crystals. A common example is X-ray diffraction. For amorphous materials, the lack of periodicity renders these methods impractical. Core-level spectroscopies, on the other hand, can give information about the distribution of atomic motifs in a material without the requirement of periodicity. For carbon-based materials, X-ray photoelectron spectroscopy (XPS) is arguably the most popular of these techniques.
In XPS a core electron is excited via absorption of incident light and ejected out of the sample. Core electrons occupy deep levels close to the atomic nuclei, for instance 1s states in oxygen and carbon atoms. They do not participate in chemical bonding since they are strongly localized around the nuclei and lie far deeper in energy than valence electrons. Because core electrons lie so deep in the potential energy well, energetic X-ray light is required to eject them out of the sample. In XPS, the light source is monochromatic, which means that all the X-ray photons have around the same energy, . When a core electron in the material absorbs one of these photons with enough energy to leave the sample, we can measure its kinetic energy and work out its binding energy (BE) as .
After collecting many of these individual measurements, a spectrum of BEs will appear, because each core electron has a BE that depends on its particular atomic environment. For instance, a core electron from a C atom bonded only to other C atoms has a lower BE than a C atom that is bonded also to one or more O atoms. And even the details of the bonding matter: a core electron from a “diamond-like” C atom (a C atom surrounded by four C neighbors) has a higher BE than that coming from a “graphite-like” C atom (which has only three neighbors). Therefore, the different features, or “peaks” in the spectrum can be traced back to the atomic environments from which core electrons are being excited, giving information about the atomic structure of the material. This is illustrated in the featured image of this post (the one at the top of this page).
What makes XPS so attractive for computational materials physicists and chemists like us is that it provides a direct link between simulation and experiment that can be exploited to 1) validate computer-generated model structures and 2) try to work out the detailed atomic structure of an experimental sample. 1) is more or less obvious; 2) is motivated by the fact that experimental analysis of XPS spectra is usually not straightforward because features that come from different atomic environments can overlap on the spectrum (i.e., they coincidentally occur at the same energy). In both cases, computational XPS prediction requires two things. First, a computer-generated atomic structure. Second, an electronic-structure method to compute core-electron binding energies.
While candidate structural models can be made with a variety of tools (a favorite of ours is molecular dynamics in combination with machine-learning force fields), a standing issue with computational XPS prediction is the accuracy of the core-electron BE calculation. Even BE calculations based on density-functional theory, the workhorse of modern ab initio materials modeling, lack satisfactory accuracy. A few years ago my colleague Dorothea Golze, with whom I used to share an office during our postdoc years at Aalto University, started to develop highly accurate techniques for core-electron BE determination based on a Green’s function approach, commonly referred to as the GW method. These GW calculations can yield core-electron BEs at unprecedented accuracy, albeit at great computational cost. In particular, applying this method to atomic systems with more than a hundred atoms is impractical due to CPU costs. This is where machine learning (ML) can come in handy.
Four years ago, around the time when Dorothea’s code was becoming “production ready”, I was just getting started in the world of ML potentials from kernel regression, using the Gaussian approximation potential (GAP) method developed by my colleagues Gábor Csányi and Albert Bartók ten years before. I also had prior experience doing XPS calculations based on DFT for amorphous carbon (a-C) systems. Then the connection was clear. GAPs work by constructing the total (potential) energy of the system as the optimal collection of individual energies. This approximation is not based on physics (local atomic energies are not a physical observable) but is necessary to keep GAPs computationally tractable. However, the core-electron BE is a local physical observable, and thus idally suited to be learned using the same mathematical tools that make GAP work. In essence, the BE is expressed as a combination of mathematical functions which feed on the details of the atomic environment of the atom in question (i.e., how the nearby atoms are arranged).
Coincidentally, it was also around that time that our national supercomputing center, CSC, was deploying the first of their current generation of supercomputers, Puhti. They opened a call for Grand Challenge proposals for computational problems that require formidable CPU resources. It immediately occurred to me that Dorothea and I should apply for this to generate high-quality GW data to train an ML model of XPS for carbon-based materials. Fortunately, we got the grant, worth around 12.5M CPUh, and set out to make automated XPS a reality.
But even formidable CPU resources are not enough to satisfy the ever-hungry GW method, so we had to be clever about how to construct an ML model which required as few GW data points as possible. We did this in two complementary ways. One the one hand, we used data-clustering techniques that we had previously employed to classify atomic motifs in a-C to select the most important (or characteristic) atomic environments out of a large database of computer-generated structures containing C, H and O (“CHO” materials). On the other hand, we came up with an ML model architecture which combined DFT and GW data. This is handy because DFT data is comparatively cheap to generate (it’s not cheap in absolute terms!) and we can learn the difference between a GW and a DFT calculation with a lot less data than we need to learn the GW predictions directly. So we can construct a baseline ML model using abundant DFT data and refine this model with scarce and precious GW data. And it works!
Four years and many millions of CPUh later, our freely available XPS Prediction Server is capable of producing an XPS spectrum within seconds for any CHO material, whenever the user can provide a computer-generated atomic structure in any of the common formats. Even better, these predicted spectra are remarkably close to those obtained experimentally. This opens the door for more systematic and reliable validation of computer-generated models of materials and a better integration of experimental and computational materials characterization.
We hope that these tools will become useful to others and to extend them to other material classes and spectroscopies in the near future.