Disentangling the intricate atomic short-range order and electronic properties in amorphous transition metal oxides

Solid state materials with crystalline order have been well-known and characterized for almost a century while the description of disordered materials still bears significant challenges. Among these are the atomic short-range order and electronic properties of amorphous transition metal oxides [aTMOs], that have emerged as novel multifunctional materials due to their optical switching properties and high-capacity to intercalate alkali metal ions at low voltages. For decades, research on aTMOs has dealt with technological optimization. However, it remains challenging to unveil their intricate atomic short-range order. Currently, no systematic and broadly applicable methods exist to assess atomic-size structure, and since electronic localization is structure-dependent, still there are not well-established optical and electronic mechanisms for modelling the properties of aTMOs. We present state-of-the-art systematic procedures involving theory and experiment in a self-consistent computational framework to unveil the atomic short-range order and its role for the electronic properties. The scheme is applied to amorphous tungsten trioxide aWO3, which is the most studied electrochromic aTMO in spite of its unidentified atomic-size structure. Our approach provides a one-to-one matching of experimental data and corresponding model structure from which electronic properties can be directly calculated in agreement with the electronic transitions observed in the XANES spectra.


Results and Discussion
Atomic short-range order of aWO 3 . As a starting point, standard nonlinear least-squares fitting [STF] of the experimental EXAFS spectrum, χ k k ( ) 3 , was implemented to obtain preliminary average values for interatomic distances, atomic coordination and σ 2 parameters [see Methods]. Results from standard fitting χ k k ( ) 3 STF displayed in Fig. 1a, show that structural disorder around the photoabsorbing W atoms reduces the oscillation amplitude in the χ k k ( ) 3 spectrum at high k, thus, main structural features can be identified in the interval Δk ≈ 2-10 Å −1 . Since the STF approach does not provide any 3D-structure of aWO 3 , the experimental χ k k ( ) 3 spectrum was compared against the ab-initio MD-EXAFS, χ k k ( ) 3 MD function, calculated directly from MD struc- tural trajectories of aWO 3 . To this end, energetically pre-converged ab-initio MD structural trajectories of aWO 3 comprising W = 64 and O = 192 atoms into cubic cells [V ≈ 4400 Å 3 , ρ ≈ 5.27 g/cm 3 ] were analyzed, and their associated χ k k ( ) 3 MD functions were extracted by ab-initio self-consistent real-space full multiple-scattering [FMS] 14 into the muffin-tin approximation [see Methods]. The structural averaging over twelve MD trajectories of aWO 3 leads to a main χ k k ( ) 3 MD function properly reproducing the phase, shape, and damping of the oscillations of the experimental χ k k ( ) 3 spectrum. However, spectral features at k ≈ 4.8 Å −1 and k ≈ 9.2 Å −1 are not well-resolved in the ab-initio FMS calculated χ k k ( ) 3 MD function [ Fig. 1a]. Next, in order to correct for those spectral discrepancies and to accurately reproduce the atomic short-range order of aWO 3 , those ab-initio MD structural trajectories were fitted to the experimental χ k k ( ) 3 spectrum through RMC-EXAFS simulations [see Methods] 15 . To incorporate the important multiple-scattering processes yielding explicit treatment of three-body correlations, and effects of static disorder due to the fluctuation of interatomic distances, atomic coordination and bond-angles, the following scattering paths between the photoabsorbing and scattering = where ijn are serial numbers of atoms, ξ the path-type (i)-(vii), θ is a parametrization angle for (k, θ), R eff is the effective path-length, the path-legs, (k, θ) is the EXAFS amplitude and (k, θ) the phase-shift, λ is the mean free path for the photoabsorber atoms. The amplitude (k, θ) and phase-shift (k, θ), for single-and multiple-scattering paths were self-consistently calculated by ab-initio FMS considering atoms up to R ≈ 7 Å from the photoabsorbing atoms according to (2) (3) where (k, θ), Λ k ( ) red , 2Δ(k) and θ Φ k ( , ) denote the amplitude magnitude, reduction factor, k-dependent phase correction and phase 2 . From a first RMC-EXAFS run, the structural averaged χ k k ( ) 3 RMC spectrum [Δk ≈ 2-10 Å −1 ], is calculated as a sum of the (k) contributions from the different single-and multiple-scattering paths (i)-(vii), according to (4) Results of spectral fitting from the RMC-EXAFS simulated spectrum, χ k k ( ) 3 RMC , is in excellent agreement with the experimental χ k k ( ) 3 spectrum [ Fig. 1a], and as also outlined above, include multiple-scattering events. After convergence, a minimum residual of ≈1-5 × 10 −3 is attained, which confirms that the RMC-EXAFS refinements correctly reflect the atomic short-range order of aWO 3 . Thermal damping of the RMC-EXAFS signal associated to the structural disorder σ 2 , is given by the statistical averaging of χ k k ( ) 3 RMC signals obtained by summing over the ensemble of atomic configurations 3 . The Fourier-Transform of the experimental χ k k ( ) 3 spectrum into the real-space FT χ k k ( ) 3 and its corresponding wavelet transform [WT; 2D-contour plot] are shown in Fig. 1b Table 1 summarizes the main interatomic bond-distances, atomic coordination and σ 2 factors, obtained from nonlinear least-squares spectral fitting of the measured χ k k ( ) 3 , FT χ k k ( ) 3 spectra, and those calculated directly from the RMC-EXAFS optimized structures of aWO 3 . The data show an excellent agreement to each other, which confirms that averaging over non-equivalent WO x atomic-environments yields accurate reproduction of the χ k k ( ) 3 , FT χ k k ( ) 3 spectra and correlated structural parameters of aWO 3 . The σ 2 factors in Table 1 reflect the attenuation of χ k k ( ) 3 and FT χ k k ( ) 3 due to the mean-square static disorder in the distribution of interatomic bond-distances and ionic displacements 3 . Thus, the calculated values for σ 2 correspond to the degree of disorder in aWO 3  After refinements, all the RMC-EXAFS optimized structures of aWO 3 attained a similar atomic-bonding distribution and displayed small variation in the atomic coordination and bond-angle distributions. This is of course, due to the use of pre-converged MD trajectories, and the structural constraints applied in simulations, which force the input structures to achieve a similar atomic short-range order to properly fit the χ k k ( ) 3 and FT χ k k ( ) To compare our results with X-ray and neutron diffraction experiments, the atomic short-range order of our RMC-EXAFS optimized structures of aWO 3 was further analyzed in terms of the structure factor S(Q), defined according to Table 1. Main interatomic bond-distances, atomic coordination and 2σ 2 factors of aWO 3 calculated from nonlinear least-squares fitting of the experimental χ k k ( ) 3 and FT χ k k ( ) 3 spectra to single-and multiplescattering theory [ ≈ .
Scientific RepoRts | 7: 2044 | DOI:10.1038/s41598-017-01151-2 where b j is the X-ray/neutron scattering length, R j the position of the atom j, and N the number of atoms. The X-ray S Q ( ) X , neutron S Q ( ) N structure factors and their associated reduced structure function Q[S(Q)−1], exhibit interference maxima around scattering wavevector magnitudes in the interval ΔQ ≈ 1.5-8.2 Å −1 . Beyond that range the oscillation amplitude in S Q ( ) X and S Q ( ) N is damped out due to the structural disorder [ Fig. 3a,b]. S Q ( ) X and S Q ( ) N exhibit distinct interference patterns caused by different scattering processes. The scattering due to W atoms contributes more to S Q ( ) X than that from O atoms, because X-rays interact mainly with the electron cloud surrounding the atoms [Z-dependent]. Contrary, the scattering due to O atoms contributes more to S Q ( ) N than that of W atoms, because neutrons interact with the atomic nucleus. These distinct features, the relative amplitude, peak position and the line-shape in our calculated S Q ( ) X , S Q ( ) N and Q[S(Q)−1] patterns agree with previous data reported for stoichiometric W-based oxides from X-ray, electrons and neutron diffraction experiments [18][19][20][21][22] , and show a qualitative similarity to data for sub-stoichiometric W oxide 23 .
Those structural characteristics are more clearly observed in the total X-ray g R ( ) X and neutron g R ( ) N PDF defined according to , 2 where c α , c β are the concentrations of α, β atoms [c α,β = N α,β /N], b α , b β denotes the X-ray/neutron scattering length of species α, β, and αβ g R ( ) denotes the partial PDF according to with αβ d n R ( ) the ensemble average number of β atoms in a shell dR at a distance R of an α atom. Here ρ is the number density and αβ g R ( ) is the probability to find a β atom at a distance R from an α atom. The calculated g R ( ) X exhibits two peaks at R ≈  Fig. 3d]. In Fig. 3e-g, maxima in the par-   Table 1]. The low intensity shoulder at the left side of g W-W (R) X is due to W-W pairs with bond-length R ≈ 3.1 Å, in neighboring edge-sharing WO x structural units [inset Fig. 3f]. Those short W-W bonds induce small polaron formation upon insertion of oxygen-vacancies [O v ], or alkali metal impurity ions [Li + ] and charge-balancing electrons 10,24 . It has been suggested that electrochromism in aWO 3 arises from the optical absorption due to small polaron hopping associated to the formation of W 5+ states due to transfer of electrons from O v -sites and inserted Li + species 10, 24 . Short W-W bonds have been previously reported for ion exchange and sputtered aWO 3 solid thin film oxides 18,20 .
The RMC-EXAFS simulations on aWO 3 not only fit the experimental χ k k ( ) 3 and FT χ k k ( ) 3 spectra, but also the calculated X-ray D(R) = 4πρR · g(R) function [ Fig. 3h], accurately reproduces the relative amplitude, peak position and the line-shape of earlier D(R)'s reported from electron and X-ray diffraction experiments 13,[18][19][20][21][22] . Especially, the peak at R ≈ 7.3 Å in the D(R) of the RMC-EXAFS optimized structures of aWO 3 accurately reproduces the formation of WO 6 six-membered rings [ Fig. 3i], in agreement with experimental data 18,21,22 . Particularly, the distribution of distorted WO x octahedra-units in the structure of aWO 3 leads to the formation of distorted WO x -like chains analogous to those found in the Magnéli phases W x O z 25 , and a wide distribution of WO 4,5,6,7 -membered rings comprising distinct WO x units with large free volume forming spacious channels as shown in Fig. 3i. Analogous channels have been reported for hexagonal h-WO 3 and nanostructured WO 3 systems, which have WO 6 -three-and six-membered rings forming trigonal cavities with hexagonal-and four-coordinated square channels 26,27 . Those local structures provides large available sites for cation intercalation and superior charge densities, simplifying charge-ion injection/diffusion and providing enough free volume for ion storage, which reduce activation energies and electrode volume variation during ion insertion/extraction. These results support the fact that the disordered structure of aWO 3 enhances its electrochromic performance when comparing with its crystalline counterparts, as has been assumed in previous technical studies 10 . Formation of different distorted WO x -like chains and WO 4,5,6,7 -membered rings comprising distinct WO x units suggest that the atomic short-range order of aWO 3 should consist of a mixture of the different symmetries existing in the polymorphs and Magnéli phases of WO 3 , rather than comprising a single hexagonal or distorted ReO 3 -octahedra phase, as suggested previously 13,18,22 . This would also explain why aWO 3 turns into a mixture of the monoclinic, hexagonal and triclinic phases of crystalline WO 3 upon heating 28 . Difficulties with the assignment of a single phase to aWO 3 , arises because previous structural models were deduced by hand from direct comparison of the D(R)'s of crystalline and aWO 3 . Thus, such model results are unrealistic since they do not take into account the contribution of distinct atomic environments to the total D(R) of aWO 3 . Structural characterizations of aWO 3 in terms of crystalline phases are ambiguous because the monoclinic/hexagonal WO 3 , and the W x O z -like Magnéli phases exhibit very analogous D(R) functions 23 . Because of the local-structural reconstruction at nonequivalent atomic environments, amorphous materials should display a distribution of interatomic distances, and lower average atomic coordination. Here, energetically and structurally pre-converged MD structural trajectories of aWO 3 were used. This ensures that effects of static disorder due to the fluctuation of interatomic distances, atomic coordination and bond-angles, were intrinsically taken into account in the simulations. Multiple-scattering processes were included by self-consistent calculations considering atoms beyond the first coordination shell. Thus, pre-converged amplitudes and phase-shifts for different scattering processes around the photoabsorbing W atoms, allow the explicit treatment of three-body correlations. The use of bond-distance and coordination constraints prevent the aWO 3 simulated structures getting away from the atomic short-range order defined in the measured χ k k ( ) 3 and FT χ k k ( ) 3 spectra. Thus, since the structure was optimized at each atomic displacement, until it reached an accurate one-to-one matching with the experimental χ k k ( ) 3 and FT χ k k ( ) 3 spectra, we expect our RMC-EXAFS-based simulation to properly describe the atomic short-range order of aWO 3 . 3 . We now use the RMC-EXAFS optimized structures of aWO 3 to assess the correlations between atomic short-range order and the electronic properties by detailed calculations of the electronic structure by hybrid density functional theory [DFT] 29 , and electronic transitions associated to the XANES spectra by ab-initio finite difference methods [FDM] 30  The VB tail-states shows slightly more localization than those electronic orbitals at the CB tail states. The mobility band gap defined as the energy gap between extended VB and CB states, is estimated to be ≈3.22 eV. The extent of localization of the VB and CB tail states in aWO 3 depends on the charge density contribution due to the atomic short-range order of distinct WO x -units. the VB and CB edges are strongly dependent on the type of ligands and on the atomic short-range order around the O and W sites in the first and second coordination shells. Figure 5a,b display the experimental and ab-initio FDM-RMC computed W-L 3 -edge XANES spectra of aWO 3 . The normalized spectra exhibit a strong and broad white-line absorption maximum above the absorption edge-energy centered at ≈10210.8 eV. The relative intensity, energy position, line shape and the electronic transitions in the experimental spectrum are well reproduced by the FDM-XANES function calculated from the RMC-EXAFS optimized structures of aWO 3 . According to the dipole selection rules, the absorption W-L 3 -edge is due to allowed electronic dipole transitions of the photoelectron from the initial W-[2p 3 Since the electronic correlations due to the interaction between the core-hole and the excited electron are small at the O-K-edge, the projected DoS, which reflects the electronic ground state, provides a consistent interpretation of the O-K-edge XANES spectra of aWO 3 . Figure 5d shows the ab-initio FDM calculated O-K-edge XANES spectrum of aWO 3 as calculated from the RMC-EXAFS optimized structures. The calculated O-K-edge FDM-XANES spectrum properly reproduces the relative intensity, energy position and shape of earlier reported spectra for aWO 3 34 . In the framework of the dipole selection rules, the spectrum is due to electronic transitions from the O-[1s] core-level into the unoccupied O-[2p] orbitals. Due to the electronic hybridization between the W-[5d] and O-[2p] orbitals, the O-K-edge also provides the features of the density of W-[5d] states. The O-K-edge XANES spectrum in Fig. 5d shows a main peak at γ ≈ 530.2 eV. From the projected DoS in Fig. 5e interactions. The relative intensity of the τ-ζ-peaks depends on the atomic short-range order of the WO x -units present in the structure of aWO 3 , and thus, the ζ-peak exhibits lower intensity when comparing with crystalline WO 3 34 . The CB states of the O projected DoS in Fig. 5e, suggests that the O-K-edge XANES spectrum of aWO 3 , which gives the unoccupied final states located above the Fermi level, emerges mainly from contributions due to the W-[5d]-O-[2p] hybridized states. The W-[s, p] states do not contribute significantly to the absorption at low energies, but they contribute to the ζ-peak, and also contribute in some extent to the broad feature at ε ≈ 565.1 eV, observed in the O-K-edge XANES spectrum of aWO 3 .

Electronic properties of aWO
Finally, we remark that our scheme could offer a consistent route to experimentally and theoretically unveil the atomic short-range order of aTMOs, and how local disorder affects their underlying electronic properties. The approach provides a one-to-one matching of experimental data and corresponding model structure from which electronic properties can be directly calculated in agreement with the electronic transitions giving rise to the XANES spectrum of aTMOs. Average coordination constraints were set and their weighting was gradually reduced at each RMC-EXAFS cycle. This leads to mean coordination constraints N W-O ≈ 4-6 and N W-W ≈ 4-6, which were found to be the most suitable steady-state conditions to decrease the residual and reach spectral convergence. Hence, we are using structural constraints to avoid the exploration of configurations far from the already energetically and structurally pre-converged structures. RMC-EXAFS structures of aWO 3 were then optimized allowing a 3-5% of atoms to undergo displacements of ≈0.08 Å, at every RMC-EXAFS cycle. Each atomic movement is evaluated according to the degree of consistency R 2 between the experimental and the refined spectral-data-points. Thus, if the atomic movement increases the consistency R 2 it is accepted. If instead, it lowers the consistency R 2 it is accepted with probability ℘ = R R . When the experimental and refined data-points are statistically the same then the value of R 2 is less than the number of degrees of freedom 38 . Total χ k k ( ) 3 RMC and FT χ k k ( ) 3 RMC functions; equal to the averaged single spectrum of each photoabsorbing W atom were re-calculated at each RMC-EXAFS cycle. Reaching of convergence to a minimum residual of ≈1-5 × 10 −3 was attained by running ≈8 × 10 5 RMC-EXAFS cycles. After refinements, all twelve RMC-EXAFS optimized structures of aWO 3 attained similar atomic-bonding distributions and displayed small variation in the atomic coordination and bond-angle distributions. This is of course, due to the use of already pre-converged MD trajectories and structural constraints applied in simulations, which force the input structures to achieve similar structural order to fit the EXAFS spectra. Here the reported structural correlations and physical quantities correspond to the average over those twelve RMC-EXAFS optimized structures.

Hybrid density functional theory [DFT]. Electronic properties of the RMC-EXAFS optimized structures
of aWO 3 were studied by ab-initio hybrid DFT 39 40 , as implemented in VASP 29 . A maximal force criterion convergence of 0.01 eV/Å was used and an energy cut-off of 700 eV was used to expand the Kohn-Sham orbitals in the plane wave basis set. A 1 × 1 × 1 Monkhorst-Pack mesh 41 centered at the Γ-point was used for k-sampling and the non-local range separated screened hybrid functional HSE06 [10% HF; 90% PBE; ω = 0.2 Å −1 ] 42 was used.
Ab-initio calculations of RMC-XANES spectra. Ab-initio calculations of the XANES spectra for RMC-EXAFS optimized structures of aWO 3 were carried out in order to assess the electronic transitions associated to unoccupied states in aWO 3 . The W-L 3 and O-K edge RMC-XANES spectra were calculated self-consistently by ab-initio FDM, as implemented in the near-edge structure FDMNES code 30 . The full potential FDM method does not approximate the potential's form, providing a precise description of occupied and unoccupied electronic states. The energy dependent exchange-correlation potentials by Hedin-Lundqvist and Von Barth were used and evaluated using relativistic DFT. Electronic correlations and spin-orbit coupling were approached by Fock-Dirac schemes. Single W-L 3 and O-K edge XANES signals were calculated on grids of 7 Å centered at each photoabsorbing W or O atom and then averaged to a total FDM-RMC-XANES function.