Galore: Broadening and weighting for simulation of photoelectron spectroscopy

1 Dept of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK 2 Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, UK 3 Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, UK 4 Dept of Materials, Imperial College London, London SW7 2AZ, UK 5 Dept of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QR, UK DOI: 10.21105/joss.00773


Photoelectron spectroscopy
Photoelectron spectroscopy (PES) is a family of methods used to characterise the chemical nature and electronic structure of materials.PES is based on the photoelectric effect, which was discovered by Hertz. 1 It was explored extensively by Rutherford and colleagues 2 and within a few years researchers including de Broglie 3 and Robinson 4 were using the technique to measure electron binding energies through the relationship Photons with energies hν ranging from 5 eV up to 12 keV eject electrons (referred to as "photoelectrons") from the occupied orbitals of a sample.The kinetic energy E k of each photoelectron therefore depends on its binding energy E B .The names of various PES methods refer to the photon energy range used: • ultraviolet photoelectron spectroscopy (UPS): 5-100 eV • X-ray photoelectron spectroscopy (XPS): 0.3-2 keV • hard X-ray photoelectron spectroscopy (HAXPES, HE-PES, HXPS, HX-PES): above 2 keV These methods generate spectra that are directly related to the electronic density of states (DOS), a distribution which is routinely calculated in ab initio materials chemistry.When comparing the computed DOS with a PES measurement, it is often possible to identify general peak agreement simply by reversing the energy scale (i.e.replacing negative orbital energies with positive binding energies), applying a little broadening, and shifting the energy values to account for different references.This approach has been applied succesfully where peak positions are of interest. 5,6Broadening is generally applied by convolution with a Gaussian and/or Lorentzian function: intrinsic lifetime broadening causes a Lorentzian energy distribution of the photoelectrons, while instrumental factors, including the width of the X-ray source and analyser resolution, give rise to a Gaussian line shape.Franck-Condon phonon broadening is caused by relaxation of atomic positions in response to creation of a photohole, as well as thermal population of vibrationally excited states before photoionisation, and gives around 0.8 eV Gaussian broadening in metal oxides.Photoemission spectra for the same material will vary depending on the radiation source used.The probabilities of the underlying photoionisation events are based on the radiation and orbital energies, as well as the shape of the orbital.In order to account for this it is necessary to apply weighting to states according to their photoionisation cross-sections.This is accomplished by projecting the full DOS onto contributions from atomic s, p, d, f orbitals (PDOS), as is done routinely in analysis of ab initio calculations.It is then assumed that the contributions of these orbital-projected states to the total photoelectron spectrum will be proportional to the photoionisation cross-sections of corresponding orbitals in free atoms.1][12] ) The free atom cross-sections have been computed by several methods and are available as reference data (e.g.13).[16][17][18]

Known limitations and improvements
1][12] This model is expected to break down at longer wavelengths, and in the case of low-energy UPS it is likely that significant scattering effects would be neglected.
Asymmetry corrections may be applied to the photoionisation cross-sections to account for the polarisation of sources and angular acceptance range of electron detectors.This is especially relevant for HAXPES measurements where the experimental geometry may be deliberately changed in order to manipulate the spectrum and expose different contributions. 19 best-practice approach is to integrate the relevant equations over a range of angles depending on the equipment geometry. 20Currently this data is not included in Galore, but users are able to include corrected cross-sections from a JSON-formatted data file if available.
At high photon energies, it has been observed that intensity changes in oxides do not correlate with photon energy as predicted by the available tabulated data; in particular the intensity of O-2p states in CdO, PbO 2 and In 2 O 3 seem to vary more linearly than predicted. 20

Further reading
For further information about PES there are some helpful reviews in the academic literature, including Refs 21, 22 and 23.

Vibrational spectroscopy (IR and Raman)
In infrared (IR) spectroscopy, low-energy photons are absorbed corresponding to the energies of lattice vibrations and an absorption spectrum is obtained.In a highly-crystalline system, symmetry selection rules limit the absorption activity to a small number of possible excitations with zero crystal momentum ("Gamma-point phonons").In Raman spectroscopy another optical method is used to observe lattice vibrations and different selection rules apply; again, the resulting spectrum corresponds to a limited selection of Gamma-point movements.
It is possible to predict the frequencies and intensities of these vibrational modes by performing ab initio lattice dynamics calculations.Usually these will be performed within density-functional theory, either using variations of density-functional perturbation theory (based on the work of Gonze 24 ) or the frozen-phonon ("direct") method (see Refs 25,  26).When the underlying set of vibrational frequencies and mode intensities has been calculated it is typical to broaden the data by convolution with a Gaussian-Lorentzian function. 27,28This is necessary to correctly intepret the effect of overlapping peaks; for example, Figure 2 shows a case in which a group of peaks with low intensities combine to form a large peak in the broadened spectrum.

Galore
Galore provides a command-line tool and Python API to import data and resample it to a dense, regular X-Y series.This mesh can then be convolved with Gaussian and Lorentzian functions to yield a smooth output, in the form of a plot or data file.Numpy functions are used for data manipulation and convolution on a finite grid and Matplotlib is used for plotting. 29,30As well as simple tabular data files, the electronic DOS or PDOS may be imported directly from the output of the VASP or GPAW codes.
The Gaussian and Lorentzian functions employed have the forms: where f is the x-axis value, f 0 is the mid-point, γ is the full-width-half-maximum of the peak.Cross-sectional weights are included for some standard energy values (He(II) UPS and Al k-alpha) from tabulated ab initio calculations. 13Users may provide their own weighting values in the same human-readable JSON file format.Higher-energy (HAXPES) spectra may be simulated using cross-sections from fitted data over an energy range 1-1500 keV.Tabulated data 31 was fitted to an order-8 polynomial on a log-log scale, and coefficients for each element and orbital shape are stored in a database file.The fitting error is generally below 1%, with outliers in the region of 2-3%.The order-8 fit was selected based on cross-validation in order to avoid over-fitting (Figure 3).

Figure 1 :
Figure 1: Procedure (left to right) for simulated photoelectron spectrum from ab initio DOS

Figure 2 :
Figure 2: Schematic example of misleading peak intensities due to overlap

Figure 3 :
Figure 3: Cross-validation error of HAXPES data fitting over full energy range across all elements and orbitals