Supporting Information for:
“Electronic structure of lithium battery interphase compounds: comparison between
inelastic x-ray scattering measurements and theory”
Tim T. Fister,1 Moritz Schmidt,1 Paul Fenter,1 Chris S. Johnson,1 Michael D. Slater,1
Maria K. Y. Chan,2 Eric L. Shirley3
1.Chemical Sciences & Engineering Division, Argonne National Laboratory, Argonne, IL 60439
2. Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439
3. Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD
20899-8441
The LERIX instrument measures the intensity of NIXS as a function of energy loss (ħ)
by scanning the incident energy (E1) with a two-crystal monochromator while collecting
scattered x-rays using spherically bent silicon analyzers fixed to the (444) scattering condition
(E2=7911 eV).1 Nineteen analyzer/detector pairs are positioned at scattering angles ranging from
9° to 181°, resulting in simultaneous measurement at values of q ranging from 0.6 Å1 to 8.0 Å1.
Elastic scattering was used to calibrate the energy-loss scale and characterize the energy
resolution, which ranged from 0.9 eV to 1.0 eV FWHM (full-width at half maximum). Samples
consisted of commercial powders sealed in polyimide tubes (2mm outer diameter). Because the
penetration length at 8 keV in lithium oxides is typically about 1 mm, the 25 m thick polyimide
tubing was a negligible contribution to the NIXS. Each K edge was measured at least three times
and showed no systematic change that could be attributed to radiation damage.
While the overall NIXS describes scattering from all possible core and valence electrons,
we concentrate on excitations at (and slightly above) each individual core-shell threshold. For
the corresponding excitation spectrum, the dynamic structure factor describes the transition from
the ground state ( i ) to an energy- and momentum-conserving final state ( f ), subject to the
light-matter interaction that has an effective operator of the form
Oˆ   j el exp( iq  r j ) ,
(1)
with q being the momentum transfer, and rj being the coordinate of the j-th electron. From this,
Fermi’s golden rule immediately provides a formula for the dynamic structure factor:
S (q,  )   | f Oˆ i |2  ( E f  Ei   ) .
(2)
f
Considering Eq. (1), the selection rules associated with this operator are more apparent when we
expand a given term in the summation, because we have.
exp( iq  r )  1  iq  r  (q  r ) 2 / 2  ...
(3)
For small qa (with a being the 1s initial state’s radius), only the linear term in (3) contributes to
the NIXS appreciably. In this regime, the dynamic structure factor is proportional to the dipolelimited processes measured in XAS and small-q EELS.2 For larger q, the next-order, (q  r ) 2 / 2
term and even higher-order terms contribute more substantially, aiding observation of dipoleforbidden transitions, such as s→s and s→d, when considering angular momentum about a given
lattice site. For a relatively spatially extended core state, such as for the lithium K edge (e.g. 1s
initial state), this q-dependence is quite well-known in NIXS3 and has been used to
experimentally decouple the s and p-type final states in Li3N.4
In that case, the dynamic
structure factor was expanded in a spherical, partial-wave basis, which, for powders, results in an
expansion in angular momentum (l) over the density of states (l or ‘l-DOS’), weighted by
transition-matrix-element coefficients (Ml) derived from the easily calculated core wave
functions.5 Thus, one could equivalently write
S (q,  )   (2l  1) M l (q,  ) l ( ) .
2
(4)
l
Because antibonding states for lithium oxides are typically limited to s- and p-type states (0 and
1 respectively), the nineteen measurements in q overdetermine Eq. (4) and the unknown
components of the l-DOS must be found by a least-squares fit. Whereas EELS has been used
outside of the dipole limit in non-forward-scattering geometries,6 multiple scattering of an
electron is more prevalent, so that a symmetry-analyzed interpretation might not be as easily
applied.7
The BSE NIXS spectra were computed using the NIST BSE core-excitation program
(NBSE), and built on density-functional theory (DFT), plane-wave/pseudopotential calculations.
More details about the methodology and calculational approach to the DFT calculations are
discussed elsewhere.8, 9 An atomic structure program was used to facilitate projector-augmentedwave (PAW) reconstruction of Bloch states to recover all-electron wave functions used for
calculating transition matrix elements and electron-core hole interaction matrix elements.10 The
core-hole screening was implemented as described elsewhere.11 The BSE calculations were akin
to those in the work by Vinson and co-workers.12
In this work, orientational averaging necessary to simulate scattering from powder
samples was accomplished by averaging the spectra for q aligned along the three Cartesian
directions. The matrix-element-weighted l-DOS was calculated by setting the one-electron
matrix elements from a core level to PAW projectors to zero for all but the pertinent l value, for
which we only sampled l=0 and l=1.
After a given spectrum was calculated, a model self-energy was used to introduce
lifetime broadening of spectral features. The self-energy is based on a free-electron-gas for the
electronic Green’s function G and a q-dependent model dielectric function. The model dielectric
function used the Levine-Louie function for the  1 moment of the of the imaginary part of the
dielectric function, screened Coulomb interaction,  2 (q, ) , the numerically calculated density
matrix to obtain the  0 moment,13 and the f-sum rule to fix the 1 moment, with the assumed
form,  2 (q,  )    (  Eg )  (  Eg )  exp[  (  Eg )] , where Eg is the DFT band gap, and
,  and  depend on q.14 Additional broadening was added to match the apparent overall
broadening (including resolution effects) present in the measured spectra, including vibrational
broadening not considered in the calculations. Finally, the energy scale was also adjusted to
account for the core-level binding energy in each compound. This also entailed a relative
adjustment of different spectra in cases where there was more than one type of site, which in this
work occurred only for the O K edge in LiOH·H2O. To determine the relative core level binding
energies for those different types of sites, the method of Tinte and Shirley was followed.15
Charge density isosurfaces were obtained from DFT supercell calculations. For each
compound, a supercell was constructed and an explicit 1s core hole placed on a Li atom by
constraining the occupation numbers, such that core-level relaxation effects are included. Plane
wave DFT calculations were carried out using the Vienna Ab Initio Simulation Package
(VASP)16 and accompanying Projector Augmented Wave potentials.17 The partial charge
densities corresponding to orbitals in the different features in the l-DOS are summed and
isosurfaces plotted.
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