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Galore: Broadening and weighting for simulation of photoelectron spectroscopy |
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24 April 2018 |
paper.bib |
Galore simplifies and automates the process of simulating photoelectron spectra from ab initio calculations. This replaces the tedious process of extracting and interpolating cross-sectional weights from reference data and generates tabulated data or publication-ready plots as needed. The broadening tools may also be used to obtain realistic simulated spectra from a theoretical set of discrete lines (e.g. infrared or Raman 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.[@Hertz1887]
It was explored extensively by Rutherford and colleagues
[@Rutherford1914]
and within a few years researchers including de Broglie [-@deBroglie1921]
and Robinson [-@Robinson1923]
were using the technique to measure electron binding energies through
the relationship
- 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.[@Veal2015;@Savory2016] Broadening 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.[@Cox1983; @Nelin2011; @Regoutz2015]
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. (This is the Gelius model.[@Gelius1972; @Gelius1972a; @Gelius1974]) The free atom cross-sections have been computed by several methods and are available as reference data (e.g. @Yeh1985). Implemented with ad-hoc scripts and spreadsheets, this method has already been used in many academic studies.[@Dou1998; @Morris2000; @Glans2005; @Farahani2014; @Sathasivam2017]
The Gelius model is based on photoelectrons with short wavelengths, such that the photoinionisation matrix elements are dominated by regions of space close to the atomic nuclei.[@Gelius1972; @Gelius1972a; @Gelius1974] 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.[@Regoutz2016] A best-practice approach is to integrate the relevant equations over a range of angles depending on the equipment geometry.[@Mudd2014] Currently 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.[@Mudd2014]
For further information about PES there are some helpful reviews in the academic literature, including Refs @Huefner2005, @Fadley2009 and @Fadley2010.
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[@Gonze1997]) or the frozen-phonon ("direct") method (see Refs @Parlinski1997, @Togo2008). 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.[@Hills1975; @Grevels1998] This 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 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.[@Numpy2011; @Hunter2007] As 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: $$ y = \exp \left( \frac{-(f - f_0)^2}{2 c^2} \right) \quad \text{where} \quad c = \frac{\gamma}{2 \sqrt{2 \log 2}} $$ and $$ y = \frac{0.5 \gamma}{\pi (f - f_0)^2 + (0.5 \gamma)^2} $$
where
Cross-sectional weights are included for some standard energy values (He(II) UPS and Al k-alpha) from tabulated ab initio calculations.[@Yeh1985] Users 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[@Scofield1973] 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).
Development work by AJJ took place in the course of research into new transparent conducting materials, led by DOS and funded by EPSRC (project code EP/N01572X/1). DOS acknowledges support from the European Research Council, ERC (grant no. 758345). Work by AMG was supported by a studentship co-sponsored by the Diamond Light Source at the EPSRC Centre for Doctoral Training in Molecular Modelling and Materials Science (EP/L01582/1). AR acknowledges the support from Imperial College London for her Imperial College Research Fellowship.
We are grateful to Francesco Borgatti for his work transcribing the HAXPES cross-section data. The polynomial fits were derived from these tables.
We acknowledge useful discussions with Alexey Sokol (who proposed that a code such as this would be useful), Katie Inzani and Tim Veal. Feature requests and user testing came from Benjamin Williamson, Christopher Savory and Winnie L. Leung.
This would have been much more painful if not for the excellent scientific Python ecosystem, and the Python Materials Genome project spared us the pain of writing Yet Another Vasp Parser.