Electron Energy Loss Calculations
- Author:
Edward Tait, University of Cambridge
- Author:
Nicholas Hine, University of Warwick
- Date:
September 2015
- Date:
Updated and Converted by NDMH September 2022
Theory
Computation of Electron Energy Loss (EEL) spectra in the Kohn-Sham formal- ism relies on the application of Fermi’s Golden Rule to compute the imaginary part of the dielectric function,
Here \(\omega\) is the transition energy, \(\Omega\) the unit cell volume, the \(\psi_i\) are (all electron) conduction band states, the \(\psi_c\) are core states, with respective energies \(E_i\) and \(E_c\). \(\mathbf{r}\) is the position operator (defined as the displacement from the nucleus whose core electrons are being excited). The \(\delta\)-function conserves energy. \(\textsc{ONETEP}\) relies on the external tool OptaDoS[1] for the computation of \(\epsilon_2\) and only needs to supply matrix elements in a compatible form. The rest of this section discusses the calculation of these elements.
Projector Augmented Wave
In common with many plane wave codes onetep uses pseudopotentials to increase computational efficiency. A drawback of pseudopotentials is poor representation of the all-electron Kohn-Sham wavefunction close to the nucleus, this however is exactly the region in which the matrix elements used for EELS simulation are computed. To overcome this obstacle the projector augmented wave (PAW) formalism of Blochl[2] is adopted, which permits reconstruction of all-electron states \(\psi\) (and thus matrix elements) from pseudo-wavefunctions \(\widetilde{\psi}\):
This process is accomplished by decomposing the pseudo-wavefunction in the augmentation region into a sum of pseudo partial waves, \(\widetilde{\varphi}_i\), the weights in this sum are determined using the projectors \(\widetilde{p}_i\). These pseudo partial waves are subtracted off and all electron partial waves, \(\varphi_i\), added in their place. Simply put, within the augmentation regions (close to the nucleus) the pseudised part of pseudo-wavefunction is subtracted off and the all electron part is added back on.
Several PAW data sets are freely available (subject to the caveat that they should be thoroughly tested for a specific use case). onetep accepts PAW pseudopotentials in the abinit file format. Core wavefunctions, as produced by the pseudopotential generator are also required, these are less frequently available for download, but your PAW library should provide input files for a pseudopotential generator which can be used to produce core wavefunction data sets (again in the abinit format).
Generation of Position-Core Kets
To calculate the PAW matrix elements we must compute terms of the form: \(\langle\widetilde{\psi}|\mathbf{r}|\psi_\mathrm{c}\rangle\)
As an intermediate we compute expressions of the form: \(\langle\phi_\alpha|\mathbf{r}|\psi_\mathrm{c}\rangle\)
with \(\phi_\alpha\) an NGWF. The integral implied by this bra-ket must be computed on the grid. The bra term, an NGWF, is readily available in this form but the ket \(|\psi_\mathrm{c}\rangle\) must be constructed.
A Fourier space method is used, as it was found to offer superior numerical performance and correctly treats periodic systems without further modification. This method exploits the fact that differentiation in Fourier space is equivalent to multiplication by the position vector in real space:
Conduction Optimisation
Core loss calculations rely on an accurate description of conduction band states, in onetep it is necessary to perform a conduction optimisation calculation in order to obtain a second NGWF set and kernel which correctly represent the conduction manifold. Detail of the process may be found in the document “Conduction NGWF optimisation and optical absorption spectra in ONETEP”[3] and is published[4].
Two main parameters should be supplied: cond_energy_range which sets the
energy window above the HOMO in which conduction states will be optimised.
The second parameter cond energy gap
specifies the energy gap between the highest optimised state and the lowest un-
optimised state, this parameter is used to prevent attempts to optimise only
part of a set of degenerate states
Practical Example
In this section we will discuss the procedure for running an EELS calculation on a toy system: silene (the silicon equivalent of ethene). You will need to obtain PAW pseudopotentials for Silicon and Hydrogen and the associated core wavefunction data for Silicon. The author used the JTH pseudopotentials[5], though can’t offer any guarantee of their suitability for any particular use. A boilerplate input file for a onetep EELS calculation is provided in Appendix B.
This input file can also be downloaded from the tutorials section of the onetep site. There are a few differences between this input and the one for a single point calculation:
PAW is mandatory
We specify a second species for the atom whose core electrons we’re exciting
Because a conduction calculation is being performed we must provide a species cond block
We must provide a
species_core_wfblock and every species must be listed there.
The input file can be run swiftly on a single node and should produce a
large number of output files. Most of these files are .cube’s of wavefunctions
produced by default during the properties calculations. The files of interest
are the .elnes_bin files, which contain OptaDoS compatible matrix elements.
A little more setup is needed before we can run OptaDoS (using the silene example):
A dummy castep
silene-out.cellfile must be produced, and it must contain a symmetry blockBy default two
.elnes binfiles are produced, one based on Kohn-Sham wavefunctions represented using only the valence NGWFs (silene_val_grad.elnes_bin) and a second which makes use of the joint basis of valence and conduction NGWFs (silene_joint_grad.elnes_bin)As per the discussion above, you should choose the latter and copy it to
silene.elnes bin.A
silene.bandsfile must be produced, this is best done by copyingsilene.joint bandstosilene.bands.An OptaDoS input file,
silene.odiis needed.
To assist in these tasks a utility script, prep_optados_eels, is provided in the
utils folder of the onetep distribution. Run it with the calculation seed name
as its argument and the steps listed above will be completed automatically.
The .odi file produced should be regarded as a basic template, consult the OptaDoS documentation[6] if you wish to use more advanced features. Note that at the moment only fixed broadening is supported by onetep. When you are satisfied with your OptaDoS input file, execute OptaDoS with your calculation seed name as the argument. All being well, you should see a .dat file which you can plot with your favorite tool. Individual edges are listed sequentially in the file, so a little post processing with awk or python is needed to separate the edges for individual plotting
OptaDoS
The OptaDoS code provides a single tool to compute densities of states and optical spectra of various sorts. OptaDoS provides a number of smearing schemes for evaluating integrals over the Brillouin Zone including fixed and adaptive schemes. At present onetep only supports simple fixed broadening schemes. The prep optados eels utility script provides a template .odi input file:
TASK : CORE
# Method OptaDoS should use to work out
# the Fermi energy in the material .
# the insulator method relies on electron
# counting and is best for the systems
# onetep is most commonly used to study .
EFERMI : insulator
# Smearing scheme and width
BROADENING : fixed
DOS_SPACING : 0.01
# Use an average of x , y and z components of
# the position vector matrix elements
CORE_GEOM : polycrystalline
# Parameters below control lifetime and
# instrument broadening
CORE_LAI_BROADENING : true
LAI_GAUSSIAN_WIDTH : 0.6
LAI_LORENTZIAN_WIDTH : 0.2
LAI_LORENTZIAN_SCALE : 0.1