Introduction

Polarons consist in the spatial localization of excess charge carriers in crystals, through their interaction with the surrounding atomic lattice and the subsequent deformation of the latter1. As such, polarons can be seen as quasiparticles consisting of an electron or a hole, dressed by a cloud of virtual phonons attached to and moving with it. Depending on the crystal and strength of the electron-phonon interaction, polarons can be large or small but will inevitably affect electronic properties, making the concept of polaron a central, timely, and ubiquitous topic in material science2. As such, it is nowadays at the heart of active research in photovoltaics3, photocatalysis4, electrochromism5 or ion batteries6.

Anticipated by Landau7, the concept of polaron was first coined by Pekar8, from a model describing the interaction of an electron with a continuous polarizable medium. Following the same line of thought, Fröhlich9,10 then provided a more rigorous quantum mechanical description, relying on long-range Coulombic electron-phonon interactions. The addition of an extra charge in a polar crystal lattice naturally induces dielectric screening from the activation of longitudinal optical polar phonons, yielding the formation of a polaron, the spatial extension of which depends on the dielectric constant of the medium. This typically applies to large polarons extending over multiple sites in ionic and polar crystals. Independently, Holstein11,12 provided an alternative and more atomistic description, relying on short-range electron-phonon interactions involving acoustic phonons. Local elastic deformations can create a small quantum dot trapping the charge and can explain the formation of small polarons, usually confining the charge on one single site. Later, the concept of Jahn–Teller polaron was also introduced by Hock, Nickisch, and Thomas13 and appeared as an alternative mechanism for small polaron formation in a system with Jahn–Teller active ions. In the presence of degenerated energy levels, the addition of an extra charge can activate a local Jahn–Teller distortion that splits the energy levels and stabilizes the charge on a given site. Such a concept of Jahn–Teller polaron is intimately linked to important phenomena such as colossal magnetoresistance in manganites14,15 or the high-Tc superconductivity in cuprates16. All these consecutive models are still the backbone of present polaron interpretations. They remain however more conceptual than real tools for quantitative predictions.

Polarons are nowadays also accessible to density functional theory (DFT) calculations, although their first-principles study remains challenging17. A first limitation in direct calculations is the size of simulation boxes, restricting practical investigations to small polarons18,19. This drawback has been recently overcome with the development of an elegant formalism requiring only the computation of quantities within the primitive cell20. Including all kinds of electron-phonon interactions, this approach provides moreover a unified formalism for addressing large and small polarons. A second difficulty is to properly reproduce charge localization using usual local exchange-correlation functionals, that include spurious self-interactions. However, more advanced self-interaction corrections21,22, DFT+U23,24, and hybrid functionals24,25 often reproduce polarons in fair agreement with experimental data. Thanks to this recent progress, there has been a renewed interest in polarons17, with the hope that DFT could unravel meaningful new insights in polaron formation and important related phenomena.

WO3 is often considered as a paradigmatic polaronic compound26,27. Although the unusual medium size and disk-like shape of its polarons were already revealed in the experimental work of Salje27, the origin of such polarons remains unclear. Here, relying on first-principles investigations, we show that classical polaron models do not strictly apply to WO3. Instead, we introduce the concept of anti-distortive polaron, in which charge localization arises from the undoing of some inherent distortive motions. We rationalize our findings with a simple generic model and further demonstrate the generality of the concept and its concrete practical implications.

Results

Structurally, WO3 can be seen as an ABO3 perovskite with missing A cation: its \(Pm\bar{3}m\) cubic reference structure consists in a network of corner-shared oxygen octahedra, with W atoms at their centers28. On cooling, it exhibits a complex sequence of structural phase transitions toward a monoclinic ground state, originally considered as of Pc symmetry26,29 but better assigned now as P21/c30,31. Electronically, it is a wide bandgap semiconductor32. In practice however, it is typically sub-stoichiometric (WO3−δ), which makes it intrinsically n-doped. Then, those extra electrons are known to form polarons and even bi-polarons, which are expected to be closely associated with the remarkable electrical, chromic and superconducting properties of WO35,33,34,35.

Accurate simulation of a self-trapped single polaron in WO3 from DFT has only been reported recently25,36,37. Those calculations properly captured the spatial extension of the polaron over a few unit cells, together with its unusual disk-like shape. Building on those seminal results, we start here with a careful analysis of the polaron structural distortion in order to unravel the microscopic mechanism behind its formation.

Our investigations are performed in the framework of DFT using a full hybrid-functional approach on large supercells (up to 576 atoms), which was previously shown to provide an accurate description of polarons in WO325. To study polaron formation, an extra electron is added to the system while charge neutrality is restored by the inclusion of a positive background (see method section). The atomic distortion is then analyzed from its projection on phonon eigendisplacements of the reference (\(Pm\bar{3}m\)) and ground-state (P21/c) phases.

The reference \(Pm\bar{3}m\) cubic phase of WO3 shows various phonon instabilities30,31, in line with its complex sequence of structural phase transitions with temperature. The largest instability is a polar mode (\({\Gamma }_{4}^{-}\)) at the Brillouin-zone center that could make WO3 ferroelectric. However, the ground state and room temperature phases are both non-polar and show instead P21/c and P21/n symmetries30. These phases can be described as small distortions \(\left\vert \Delta \right\rangle\) with respect to the cubic phase, that arises from the condensation of distinct unstable modes \(\left\vert {\eta }_{i}^{c}\right\rangle\) of this cubic phase (i.e., \(\left\vert \Delta \right\rangle={\sum }_{i}{Q}_{i}\left\vert {\eta }_{i}^{c}\right\rangle\)). In the ground-state P21/c phase30, the distortion \(\left\vert {\Delta }_{GS}\right\rangle\) is dominated by the condensation of (i) unstable antipolar motions of W against O atoms along the z axis, \(\left\vert {\eta }_{M}^{c}\right\rangle\) (\({M}_{3}^{-}\) mode lowering the symmetry from \(Pm\bar{3}m\) to P4nmm), and (ii) additional unstable antiphase rotations of the O octahedra, \(\left\vert {\eta }_{R}^{c}\right\rangle\) (\({R}_{4}^{+}\) mode), that further produce together the appearance of (iii) secondary antipolar motions in the xy plane, \(\left\vert {\eta }_{X}^{c}\right\rangle\) (\({X}_{5}^{-}\) mode), through improper coupling30. In the room temperature P21/n phase30, the distortion \(\left\vert {\Delta }_{RT}\right\rangle\) involves similar modes, but differently oriented, and also includes additional in-phase oxygen rotations (\({M}_{3}^{+}\) mode), which importantly reduce the \({X}_{5}^{-}\) in-plane antipolar motions along x direction. Our approach nicely reproduces the geometry of both phases (see ref. 31 using the same method) and describes the P21/c ground state as an insulator with a bandgap of 3.4 eV, in close agreement with the experimental value (Eg ≈ 3.25–3.4 eV32,38).

Adding an extra electron to the system, it should a priori occupy a state at the bottom of the conduction band, delocalized over the whole crystal (the left panel of Fig. 1a, b). However, and consistent with previous calculations25, the proper coupling of this original state with atomic relaxations leads to the formation of a polaron level below the conduction band edge, with the charge located mainly on a central W atom and spreading partly over the first and second W neighbors (the right panel of Fig. 1a, d). In line with experimental expectations by Salje27 (Fig. 1f), the polaron charge is confined in a single xy plane (Fig. 1g) showing a strongly anisotropic and characteristic 2D disk-like shape (radius Rp ≈ 7.5–8.5 Å). This unusual shape originates from the fact that, in the P21/c phase, the lowest conduction level giving rise to the polaron state is made of strongly directional dxy orbitals. Distinctly from the original report25, our more converged calculations in a bigger supercell locates the polaron state 520 meV below the delocalized state (see also Supplementary Note 1), supporting the spontaneous formation of polarons at low temperature in the P21/c phase of WO3.

Fig. 1: Polaron in the P21/c phase of WO3 (W atoms in green and O atoms in yellow).
figure 1

a Sketch of the atomic structure (center) and of the electronic bands, when adding an extra electron, in the free carrier (left) and polaron (right) cases. b Computed charge density map (in purple) of a fully delocalized extra electron in the xy plane of the P21/c phase. c Computed atomic distortion (red arrows) associated to the formation of a single polaron in the P21/c phase. d Computed charge density map (in purple) of a single polaron in the xy plane of the P21/c phase. e Comparison of the atomic environment of the polaron central W atom (within the dashed-line circle area in c) with that in the pristine P21/c and P4/nmm phases (interatomic distances in Å). f Experimental charge density map of a single polaron in the P21/c phase as reported by Salje (based on ref. 27). g Side view of the computed charge density (in purple) of a single polaron in the P21/c phase, highlighting its 2-dimensional character.

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While the polaron charge is rather isotropically distributed in the xy plane (flat disk-like shape, Fig. 1d), the related atomic distortion \(\vert {\Delta }_{pol}\rangle\), illustrated in Fig. 1c, is intriguingly asymmetric. In order to clarify the nature of \(\vert {\Delta }_{pol}\rangle\), we can project it on the phonon modes \(\left\vert {\eta }_{i}^{m}\right\rangle\) of the monoclinic P21/c phase. The dominant contributions are highlighted in Figs. 2d, e and are distributed all over the Brillouin zone. Amazingly, focusing first on the Brillouin-zone center (Γ point), we do not see any significant involvement of polar, Jahn–Teller, or acoustic modes as expected from usual Fröhlich, Jahn–Teller or Holstein models (see also Supplementary Note 2, Supplementary Table II). Instead, the major contribution at Γ comes intriguingly from non-polar Raman modes \(\vert {\eta }_{Ra}^{m}\rangle\). They are these modes that are mainly responsible for the atomic distortion pattern in the polaron core region, but which alone would reproduce this Raman distortion homogeneously to the whole crystal (see Supplementary Note 2, Supplementary Fig. 3). Then, additional non-Γ phonon modes are also activated: those yield a global distortion that cancels out the Raman-type distortion outside the core region (see Supplementary Note 2, Supplementary Fig. 3) and are so responsible for the local character of \(\vert {\Delta }_{pol}\rangle\). As a whole, \(\vert {\Delta }_{pol}\rangle\) can therefore be seen as a zone-center Raman-type distortion produced by \(\left\vert {\eta }_{Ra}^{m}\right\rangle\), but localized over only a few unit cells around the polaron core.

Fig. 2: Electronic and phonon properties of WO3 (W atoms in green and O atoms in yellow).
figure 2

a Comparison of the electronic bandgap amplitudes (in eV) and position of the valence band (VB) and conduction band (CB) edges in distinct phases of WO3. b Simple sketch of the valence and conduction bands in the P21/c (left) and P4/nmm (right) phases of WO3, with yellow and green area highlighting respectively the positions of O 2p and W 5d states. The change of hybridization between O 2p and W 5dxy orbitals, yielding the gap opening under atomic distortions from P4/nmm to P21/c, are illustrated from the evolution of wavefunctions valence and conduction states at Γ. c Evolution of the electronic band structure of the P4/nmm phase of WO3 when condensing the \({X}_{5}^{-}\) in-plane antipolar motion, and \({R}_{4}^{+}\) oxygen rotation motions, bringing the system toward the P21/c phase. d Contributions to the polaron atomic distortion of phonon modes at distinct (qx, qy, 0) points of first Brillouin zone of the P21/c phase (using a 6 × 6 × 1 supercell). e Contributions to the polaron atomic distortion of distinct phonon modes from Γ and Γ − C and Γ − Y lines of first Brillouin zone of the P21/c phase. Amplitudes of respective contributions are proportional to the red circles' radii.

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At a more atomistic level, the W environment is asymmetric in the pristine P21/c phase and, in practice, the polaron distortion acts in order to make this environment more symmetric (Fig. 1e). Projection of \(\left\vert {\eta }_{Ra}^{m}\right\rangle\) on the phonons of the cubic phase identifies it as a linear combination of \(\left\vert {\eta }_{R}^{c}\right\rangle\) and \(\left\vert {\eta }_{X}^{c}\right\rangle\) inherent to the P21/c phase distortion \(\left\vert {\Delta }_{GS}\right\rangle\) but oriented in order to cancel it (see Supplementary Note 2, Supplementary Table III). As such, the polaron formation here should not be seen as the activation of an additional virtual phonon cloud attempting to screen the extra charge as in traditional models but, instead, as the local undoing of the \(\left\vert {\eta }_{R}^{c}\right\rangle\) and \(\left\vert {\eta }_{X}^{c}\right\rangle\) distortive motions, originally present in the pristine P21/c phase, in order to drive the system closer to the more symmetric P4/nmm phase (QR = QX = 0). As such, and in analogy with anti-Jahn–Teller polarons39, we coin here the name of anti-distortive polaron.

Looking at the various metastable phases of WO330, it appears in Fig. 2a that the electronic bandgap is much larger in the P21/c (3.4 eV) and P21/n (2.9 eV) phases than in the P4/nmm phase (1.7 eV), which is in line with previous theoretical and experimental reports30,40 and can be rationalized as follows (see also Supplementary Note 3, Supplementary Fig. 4). The bandgap in WO3 is typically between valence and conduction states respectively of dominant O 2p and W 5d t2g character41. In the reference cubic phase, all three t2g states are degenerated at the bottom of the conduction band. In the P4/nmm phase in which all in-plane W-O bonds are equivalent, the condensation of \({M}_{3}^{-}\) antipolar motions along z has pushed up dxz and dyz states so that the lowest conduction states are of dxy character. Then, the appearance of \({X}_{5}^{-}\) antipolar in-plane motion, and to a lower extent \({R}_{4}^{+}\) oxygen rotation motions, create alternating short and long W-O bonds in-plane. This amplifies hybridizations between the occupied O 2p and empty W 5dxy states and significantly opens the bandgap by lowering, in the valence band, the occupied state localizing the charge in the short W-O bond and by moving up, in the conduction band, the empty state localizing the charge in the long W-O bond (Fig. 2b, c). Our conjecture is therefore that, in order to form the polaron, the system suppresses locally the \({X}_{5}^{-}\) and \({R}_{4}^{+}\) atomic distortions inherent to the pristine P21/c phase in order to lower the conduction state there and trap the extra charge at a lower energy. This mechanism is reminiscent of the phase transformations previously reported in WO3 over heavy doping42.

As a first proof of concept, we built a supercell of the P21/c phase, in which we artificially imposed QR = QX = 0 around the central unit cell (see Supplementary Note 4, Supplementary Fig. 9a) to bring the system back there to P4/nmm. Adding an extra electron to that frozen structure, it naturally localizes in the central region, showing features similar to the relaxed polaron (see Supplementary Note 4, Supplementary Fig. 9b). This confirms the possibility to achieve charge localization from the tuning of Raman mode distortions but it does not demonstrate that such a mechanism will spontaneously appear.

In order to go one step further, we developed a simple quantum-dot model43 (see Supplementary Note 5). Adding an extra electron to the system, it should a priori occupy the lowest conduction energy level and delocalize over the whole crystal (Fig. 1b), which defines our reference energy state. Starting from there, forming a polaron requires activating the atomic distortion \(\vert {\Delta }_{pol}\rangle\) in the dynamically stable P21/c phase. This costs an energy Elatt(x, R) which scales both with the amplitude x of the Raman-type atomic distortion \(\left\vert {\eta }_{Ra}^{m}\right\rangle\) associated to the polaron (with x normalized to 1 in the polaron structure relaxed in DFT) and the radius R of the region within which this distortion appears. Concomitantly, condensing a Raman distortion \(\left\vert {\eta }_{Ra}^{m}\right\rangle\) of amplitude x in the P21/c phase lowers the lowest conduction state containing the extra electron, producing then a lowering of electronic energy Eelec(x). The latter is an electronic band term, expected when condensing \(\left\vert {\eta }_{Ra}^{m}\right\rangle\) homogeneously over the whole crystal. Restricting the distortion over a spacial region of radius R will further localize the extra electron in this region and so produce an extra energy cost Econf (x, R) due to the confinement of the electron. In this context, the polaron formation energy can be written as

$${E}_{f}^{pol}={\min} _{(x,R)}\left\{{E}_{latt}(x,R)+{E}_{elec}(x)+{E}_{conf}(x,R)\right\}$$
(1)

and the polaron will form spontaneously as soon as the lowering of Eelec produced by the appearance of the Raman distortion compensates the energy costs produced by Elatt and Econf.

The explicit (x, R) dependence and parameters of all three energy terms can be independently estimated from DFT calculations (see Supplementary Note 5, Supplementary Equation 6). Then, determining \({E}_{f}^{pol}\) from the minimization in Eq. (1) will not only tell us if the polaron will naturally appear (i.e., \({E}_{f}^{pol} < 0\)) but also what is the spatial extension R and distortion amplitude x at the minimum. As discussed in Supplementary Note 5, the model calculations reproduce the formation of a polaron over a relatively broad range of effective masses entering into Econf(x, R). Moreover, a formation energy (\({E}_{f}^{pol}\approx -100\) meV), polaron radius (Rp ≈ 8 Å) and distortion amplitude (x ≈ 1) comparable to DFT data are obtained consistently for realistic values of the parameters.

The ability of this simple quantum dot model to reproduce, with realistic parameters, the results achieved from the fully self-consistent DFT atomic relaxation, provides strong support to our initial conjecture that polaron formation in the P21/c phase of WO3 arises from local lowering of the bandgap achieved from the partial undoing of the atomic distortions inherent to the pristine phase. According to this picture, the distinct phases of WO3 should not be similarly prone to form polarons. Consistently with that, in the room-temperature P21/n phase, which shows smaller QX along x and a lower bandgap, polarons become much more delocalized in our calculations (see Supplementary Note 6, Supplementary Fig. 12), as also confirmed experimentally27. Then, in the P4/nmm phase (QR = QX = 0), we were unable to stabilize any similar polaronic state.

Discussion

The above results highlight the natural formation of stable polarons in WO3, arising from the local undoing of non-polar distortive motions (inherent to the pristine phase), which tunes orbital hybidizations, locally lowers the conduction state and drives the charge localization. This was not anticipated in previous studies25,36,37,44 and we coin the name of anti-distortive polaron for this alternative mechanism of lattice-mediated charge trapping.

It is worth contrasting this anti-distortive polaron with traditional conceptual models that were better expected to apply to WO3. In WO3, the strong polar character of the compound (giant Born effective charges Z*45, soft polar mode in the cubic phase) was naturally pointing in the direction of an efficient dielectric screening of the extra charge through the activation of longitudinal optical polar modes in line with Landau–Pekar–Fröhlich models or of a ferroelectric polaron. However, our DFT calculations show that no significant polar motion is involved in the polaron distortion. Alternatively, WO3 might also be prone to form Jahn–Teller polarons due to the Jahn–Teller active character of the W ion. However, the polaron distortion in the P21/c phase does not overlap with Jahn–Teller motions either. The present anti-distortive polaron of WO3 shows closer similarities with small Holstein polarons in the sense that charge trapping relates to the formation of a quantum dot. However, while the conventional Holstein polaron is typically linked to the activation of a local elastic response induced by the extra charge, here the mechanism involves changes of hybridizations driven by the undoing of soft atomic motions inherent to the displacive nature of the compound. The unusual medium size of the polaron in WO3 supports that both mechanisms are not necessarily strictly equivalent.

The case of WO3 could be a bit specific since the absence of the A cation gives rise to large distortions and the 5d character of W makes the d states particularly delocalized, but the mechanism of anti-distortive polaron is a priori totally generic and should more broadly apply to various displacive compounds like some related ABO3 perovskites. The latter are typically mixed ionic-covalent compounds and their giant Z* testify of the strong sensitivity of orbital hybridization to atomic displacements46. Depending on their Goldschmidt tolerance factor47, ABO3 compounds are prone to develop different kinds of structural distortions48, which are known to tune their bandgap, and could get similarly involved in polaron formation.

As a guiding rule, inspection of the electronic bandgap and its evolution from phase to phase might be a good indicator to track the eventual propensity of alternative compounds to develop analogous anti-distortive polarons. A quick search in materials databases for compounds showing large increase of bandgap from a high-symmetry to a lower-symmetry phase, provides a long list of perovskite and even non-perovskite potential candidates, including WO3 (see Supplementary Note 7). All of them might not be relevant since other aspects need to be considered, such as the nature of the bandgap. Nevertheless, considering for instance P21/c MoO3 (isostructural to WO3) or Pnma YAlO3 and P2/c YTaO4 as representative members of the perovskite R(Al/Ga)O3 and fergusonite R(Ta/Nb)O4 families of compounds (with R = Y or a rare-earth) and adding an extra electron, we can stabilize in those compounds an anti-distortive electronic polaron (see Supplementary Note 7, and Supplementary Figs. 13, 15, 16) arising from the partial undoing of distortive motions, exactly as in WO3. This illustrates concretely the generality of the reported mechanism that might also be relevant to many other families of displacive compounds.

While anti-distortive polarons emerge from the undoing of distortive motions that had opened the electronic bandgap in the pristine phase, we could imagine alternatively distortive polarons activating instead a distortion lowering the conduction states. This points out some interesting connections with (anti)Jahn–Teller polarons16,39,49. But while in Jahn–Teller polarons the tuning of the gap arises from on-site splitting of degenerated states from the Jahn–Teller effect, in anti-distortive polarons, it is linked to inter-site covalency effects and the tuning of bonding-antibonding states. It is worth noticing that the present discussion also links to the metal-insulator transition in rare-earth nickelates in which the activation of distortive breathing oxygen motions lower energy states to localize the charge at some sites and is sometimes interpreted as a polaron condensation50.

In conclusion, the present concept of anti-distortive polaron is not only of academic interest. Appearing relevant to various compounds, it naturally connects to various fields including electrochromism5, high-Tc superconductivity16,35, colossal magneto-resistance14,15,35, photovoltaics3, photocatalysis4 or ion batteries6. Moreover, the distinct behaviors highlighted in the P21/c, P21/n and P4/nmm phases of WO3 (see Supplementary Note 6) demonstrate explicitly the strong sensitivity of the polaronic state with the amplitude of distortive motions. In WO351, related perovskites48 and other displacive compounds, phase stability and distortive motions can nowadays be efficiently monitored by strain52 and interface53,54 engineerings, electric fields55 or optical pulses56. So, linking explicitly here polarons in some compounds to tunable distortive motions, we open de facto concrete perspectives to achieve in those compounds rational static or dynamical control of the polaronic state and of its related properties by external fields.

Methods

Calculations were performed in the framework of DFT, using the B1-WC hybrid functional57 – updated with 20% of exact exchange – as implemented in the CRYSTAL17 code58. We use all-electron double-ζ basis sets for oxygen whereas, for the heavy tungsten atom, we use the effective core pseudopotential (ECP) technique – as implemented in the small core Detlev-Figgen pseudopotentials – associated with the correlation-consistent polarized valence triple-ζ basis set (cc-pVDZ)59 for the explicit treatment of valence electrons.

We have simulated the polaron by adding an extra electron into the system, with charge neutrality being ensured by a compensating positively charged background. Such a free-carrier charge trapping method25,44 avoid the complexities of considering the presence of donor cations and structural disorders, enabling us to solely investigate the effects of the extra charge rather than other factors. The use of this method is also justified by the experimental observation that self-trapped electrons in WO3 are disconnected from their defect origin27.

Calculations were carried out on distinct supercells to check convergency of the results. The reported results correspond to a supercell including 576 atoms and consisting in a 6 × 6 × 1 repetition of the P21/c unit cell (one P21/c unit cell contains sixteen atoms: four W and twelve O). Integrations over the Brillouin zone have been performed on a 1 × 1 × 4 grid of k-points for the 6 × 6 × 1 supercell.

Structural atomic relaxations of the doped system were carried using lattice parameters fixed to those of the pristine P21/c ground-state phase. The relaxation was stopped when the root mean square of the gradient and displacements were smaller than 5 × 10−5 Ha/Bohr and 10−3 Bohr, respectively.

The calculations were spin-polarized and the overall magnetic moment was constrained to be 1 μB during the first steps of the calculations, to stabilize the ferromagnetic (FM) solution and the polaron formation.

AGATE software60 was used to project the atomic distortions onto the phonon modes of the cubic and monoclinic phases (mode-by-mode decomposition analysis).

The above computational approach was checked to provide results equivalent to what was previously reported in ref. 31, demonstrating an excellent description of the structural, electronic, and phonon properties of the different phases of WO3. The present results show some similarities with the polaronic state previously reported in ref. 25 but the present calculations are better converged, highlighting then some key differences and yielding distinct analysis and interpretation, as pointed out in the main text.