Introduction

The small polaron is a mobile quasiparticle composed of an excess carrier dressed by a phonon cloud1,2,3,4. It is manifested by local structural deformations and flat bands near the Fermi level and is significant for many applications including photovoltaics5,6,7, rechargeable ion batteries8, surface reactivity9,10,11, high-Tc superconductivity12 and colossal magnetoresistance13. Coupling polarons with other degrees of freedom can generate new composite quasiparticles, such as magnetic14, Jahn-Teller (JT)15,16, ferroelectric17 and 2D polarons18, to name just a few. The main driving forces favoring polaron formation: phonon-active lattice, electronic correlation, and electron-phonon coupling, are realized in 3d TMOs, which represent a rich playground for polaron physics1,19,20. In 5d TMOs, instead, charge trapping is hindered by the large d-bandwidth and associated weak electronic correlation, making polaron formation in a 5d orbital an unlikely event21.

The recent discovery of SO coupled Mott insulators22, where the gap is opened by the cooperative action of strong SO coupling and appreciable electronic correlation, has paved the way for the disclosure of novel, exciting quantum states of matter23,24. The coexistence of SO coupling and electronic correlation in the same TMO raises the possibility of conceptualizing a SO polaron25,26, whose properties are determined by the complex physical scenario arising from strong SO and electron-phonon interactions. Both effects can favor or hinder polaron formation, depending on their relative strength, as shown in recent model Hamiltonian studies27,28,29. In real materials, SO polarons can be observed if a correlated relativistic background develops in a structurally flexible lattice, as in the case of the double perovskite Ba2NaOsO6 (BNOO)30. With a SO coupling strength λ of 0.3 eV, a large on-site Hubbard U of 3.4 eV and sizable JT vibration modes31,32,33,34, BNOO represents the ideal candidate for questing 5d spin-orbital polarons.

BNOO is a Mott insulator with a low temperature canted antiferromagnetic (cAFM) ordered phase below 7 K, where SO splits the effective l = 1t2g levels on the Os7+ d1 ion into a lower Jeff = 3/2 ground state and a doublet Jeff = 1/230,34,35 (see Fig. 1a). Injecting electrons in Ba2Na1−xCaxOsO6 by chemical substitution of monovalent Na with divalent Ca ions does not cause the collapse of the Mott gap, which remains open up to full doping, when all d1 sites are converted in d236. This indicates that excess charge carriers do not spread uniformly in the crystal, forming a metallic state, but rather should follow a different fate. Here we provide evidence that the addition of excess electrons at a local Jeff = 3/2 site produces the formation of SO/JT entangled Jeff = 2 bipolarons, which block the onset of a metallic phase.

Fig. 1: Spin-orbit d2 bipolaron in Ba2Na0.875Ca0.125OsO6.
figure 1

a DFT charge density isosurface of the occupied Os t2g bands, showing the formation of d2 a Jeff = 2 bipolaron coexisting with d1 \({\rm{J}}_{{{{{{{{\rm{eff}}}}}}}}}=\frac{3}{2}\) sites characteristic of pristine x = 0 BNOO. Blue and red lobes refers to the entangled bipolaronic PB1 and PB2 bands displayed in (b). The Jeff spin-orbital levels are obtained from DFT+HI. b Band structure and relative density of states of pristine BNOO (x = 0, left) characterized by a multiband manifold of d1 states, and bipolaronic Ba2Na0.875Ca0.125OsO6 (right) with a localized d2 bipolaronic level below the Fermi level composed by two entangled d1 bands (PB1 and PB2). The green line represents the occupied d states of a single Os site in the pristine phase.

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Results

Polaron formation and dynamics

To gain insights on the effect of electron doping in BNOO we have performed Density Functional Theory (DFT) calculations on a Ba2Na1−xCaxOsO6 supercell containing eight Os atoms at x = 0.125, corresponding to one extra electron per supercell. Figure 1a shows that the extra charge is trapped at a d1 Os site, leading to a local modification of the electronic configuration from d1 → d2. The surrounding OsO6 oxygen octahedron expands isotropically by a few 10−2 Å, and new nearly flat bands develop in the mid gap region, all hallmarks of small polaron formation. This is confirmed by the local polaronic charge displayed in Fig. 1a. The delocalized alternative metal phase, with the excess charge equally distributed over all Os sites37, is less stable than the small polaron phase by 134 meV.

Electron trapping in TMOs generally occurs in an empty d0 manifold at the bottom of the conduction band, causing a d0 → d1 transition at the trapping site associated with one mid-gap flat band. In BNOO, where the 5d orbital is singly occupied and strongly hybridized with Oxygen p states36, we observe a conceptually different mechanism. In the undoped phase the fully occupied d1 states are grouped among the topmost valence bands (see Fig. 1b) and each site contributes equally to the density of state (DOS) (green line, bandwidth W ≈ 0.3 eV). The chemically injected excess electron goes to occupy an empty d band at the bottom of the conduction manifold, which is shifted into the gap and couples with the original d1 band at the same lattice site, forming a local d2 configuration (PB1 and PB2 bands in Fig. 1b) well separated by the remaining d1 bands. The resulting d2 dual-polaron complex can be assimilated to a bipolaron38, as evident from the charge isosurface shown in Fig. 1a, where the PB1 and PB2 orbitals are interwoven together.

Polaron formation is confirmed by 23Na nuclear magnetic resonance (NMR) and muon spin rotation (μSR) measurements on a BNOO sample having 12.5%Ca concentration shown in Fig. 2a, b, respectively. NMR shows an anomalous peak at TP,1 ≈ 130 K in the spin-lattice relaxation rate 1/T1 (squares), well above the temperature associated to the magnetic transition (6.8 K); correspondingly, a peak is observed in the spin-spin relaxation rate 1/T2 at TP,2 ≈ 50 K (triangles). Since the fast paramagnetic fluctuations are beyond the frequency window employed and no specific magnetic interaction is expected in the explored regime39, we attribute the NMR anomalous peaks to a charge-related thermally activated process, such as that associated with the small-polaron dynamics. This dynamical process drives electric field gradient (EFG) fluctuations which are probed by the quadrupolar interaction with the 23Na nuclear quadrupole. A peak is expected in 1/T1 when the frequency of the EFG fluctuations ν = 1/τc (being τc the fluctuation correlation time) matches the Larmor frequency (here ω0 ≈ 5 ⋅ 108 s−1), while a peak in 1/T2 is anticipated when τc is of the order of the experimental NMR echo delay time (here of the order of microseconds). In order to confirm that the origin of the observed peaks in the NMR rates can solely be associated with the small-polaron dynamics, we have performed the μSR measurements, which are only sensitive to magnetic fluctuations. The μSR results exhibit strong critical relaxation rates at the magnetic transition temperature, 7 K, but no further relaxation peak above, in agreement with the quadrupolar polaronic mechanism16, since the spin 1/2 muon is not coupled to EFGs. The anomalous peak in NMR 1/T1 temperature dependence is fitted using a Bloembergen-Purcell-Pound-like (BPP) model for quadrupolar spin-lattice relaxation40,41

$$\frac{1}{{T}_{1}}={\Delta }^{2}\left[\frac{{\tau }_{c}}{1+{({\omega }_{0}{\tau }_{c})}^{2}}+\frac{4{\tau }_{c}}{1+{(2{\omega }_{0}{\tau }_{c})}^{2}}\right],$$
(1)

where Δ2 is the second moment of the perturbing quadrupole-phonon coupling and the correlation time \({\tau }_{c}={\tau }_{0}\exp ({E}_{a}/kT)\) is expressed in terms of the activation energy Ea and the characteristic correlation time τ0 of the dynamical process. The resulting fitting curve is represented by the solid line in Fig. 2a and predicts a dynamical process with activation energy of Ea = 74(2) meV, τ0 = 0.7(0.2) ps and Δ2 = 2.75(6) × 1010 s−2.

Fig. 2: Polaron hopping: experiment and DFT.
figure 2

a NMR spin-lattice (square) and spin-spin (triangles) relaxation rates showing an anomalous peak at 130 K and 50 K due to a dynamical process. The 1/T1 anomalous peak is detailed in the right plot. The curve fit (red solid line) to a thermally activated BPP model provides an activation energy Ea = 74(2) meV. b μSR data showing only a peak due to the magnetic transition but no high temperature anomalous feature corresponding to the NMR one. c Evolution of the density of states around the Fermi level for selected snapshots across the hopping path displayed in (d), projected onto the initial (I, dark blue) and final (F, orange) Os sites. The five plots correspond to reaction coordinate equal to 0.0 (I), 0.3 (IT), 0.5 (T), 0.7 (TF) and 1.0 (F). The d2 bipolaron charge is gradually transferred from the initial and final hosting sites. At the transition state (T, at 0.5) the charge is equally distributed between both Os sites giving rise to an adiabatic weakly metallic transient state (brown). d Potential energy for a bipolaron migrating from I to F with the charge density projected on the two neighboring Os atoms, using a color gradient from blue (bipolaron fully localized in I) to orange (bipolaron fully localized in F). The insets show the charge density isosurface decomposed over the bipolaron bands PB1 (blue) and PB2 (red). The resulting hopping barrier, 66 meV, is in excellent agreement with the experimentally-derived activation energy. e band structure around the Fermi level at the initial (I), transition (T) and final (F) point of the hopping process.

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The energy barrier extracted from 1/T1 is in good agreement with the activation energy predicted by DFT for a thermally activated adiabatic hopping, 66 meV, estimated at the same doping level in the framework of the Marcus-Emin-Holstein-Austin-Mott (MEHAM) theory42,43. The energy path between two energetically equivalent initial (I) and final (F) polaron sites Os1 and Os2 along the [\(\bar{1}\)10] direction, constructed with a linear interpolation scheme (LIS)44, is shown in Fig. 2d. The hopping is a complex mechanism involving a three electron process: at the initial stage (I) the DOS (see Fig. 2c) is characterized by the d2 polaron peak at the Os1 site (blue) and the unperturbed d1 and d0 bands at the final Os2 site (orange lines); When the hopping process starts, the Os1-d2 and Os2-d0 bands get progressively closer, and a fraction of the polaronic charge in Os1 transfers to the empty band in Os2. At the transition (T) state the polaron charge is equally distributed between Os1 and Os2 resulting in a local weakly metallic state (see Fig. 2c, e), as expected from an adiabatic hopping process44. At this point a reverse mechanism begins: the original Os2-d1 and (now filled) Os2-d0 merges to form a d2 polaron in Os2, whereas the original Os1-d2 is depleted by one electron and generates a d1 band below the polaron peak. As a result, the DOS at the final point F is symmetrical to I (compare panels I and F in Fig. 2c and corresponding polaron charge isosurfaces in the insets of Fig. 2d).

Spin-orbital polaron structure and coupling with Jahn-Teller modes

Next we investigate the nature of the polaron, unraveling an intermingled action of SO and JT-distortions in determining the energy levels and degree of stability of the polaron45. As single-particle approach, DFT is bound to predict jj-coupled levels46, where the total angular momentum J is the vector addition of the single-electron angular momentum j. Indeed, the d2 polaron occupation matrix computed by projecting the Kohn-Sham energy levels onto the d2 polaronic subspace using spinorial projected localised orbitals47 shows that the trapped electrons occupy two single-particle Jeff = 3/2 levels (see Supplementary Tab. 1 and Supplementary Fig. 1) corresponding to the PB1 and PB2 bands reported in Fig. 1b. To compute the d2 two-electron levels we employed Dynamical Mean-Field Theory (DMFT) within the Hubbard-I (HI) approximation applied to the DFT lattice structure relaxed with the polaronic site. This many-body approach finds that the two electrons forming the d2 polaron occupy Jeff = 2 LS-coupled levels (see Supplementary Note 2), separated from the excited Jeff = 1 triplet by a SO gap of about 0.4 eV, as schematized in Fig. 1a. The non-polaronic d1 sites (gray isosurfaces in Fig. 1a) preserves a Jeff = 3/2 ground state, as in the pristine material34. Regardless the specific type of coupling, jj or LS, both DFT and DMFT predict spin-orbital Jeff states, clearly indicating that the polaron is integrated into the SO-Mott background, it exhibits an individual spin-orbital state, and does not break the preexisting Jeff = 3/2 state at the other Os sites. This leads to the coexistence of two distinct SO-Mott quantum states in the same material, a hitherto unreported physical scenario.

Although the d2 polaron possesses an intrinsic spin-orbital nature, SO coupling does not play in favor of polaron formation, as inferred from the progressive increase of the polaron energy Epol as a function of the SO strength shown in Fig. 3a: the inclusion of SO destabilizes the polaron by about 80 meV. This behavior is linked to the effect of SO on the JT distortions recently elaborated by Streltsov and Khomskii, which suggests that for a d2 configuration, SO suppresses JT distortions45,48. To shed light on this complex cross-coupling we have studied JT and polarons properties as a function of the effective SO coupling strength \(\tilde{\lambda }={c}_{\lambda }\lambda\) from cλ = 0.1 to cλ = 1 (full SO). The resulting data are collected in Fig. 3 and explained in the following.

Fig. 3: Role of Jahn-Teller and SOC on polaron stability.
figure 3

a Polaron energy Epol as a function of the SO coupling scaling factor cλ (For cλ = 0 SO is completely suppressed, whereas cλ=1 refers to the full SO regime). Epol is progressively less negative for increasing SO coupling. b JT tetragonal distortion amplitude Q3 as a function of cλ. The color scale and circle’s size indicate the variation of Epol relative to the cλ = 1 case: SO coupling rapidly quenches Q3 and reduces the polaron stability. c JT energy EJT at the polaron trapping site as a function of cλ for all three JT modes. Q3 is the only mode influenced by SO coupling. d–f Geometrical interpretation of the JT non-zero modes Q2, Q3 and Qxy respectively.

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Pristine d1 BNOO exhibits a cooperative JT ordering involving the Eg modes Q2 and Q334 as well as the trigonal Qxy mode (see Supplementary Table 4); these modes are graphically displayed in Fig. 3d–f and defined in Supplementary Table 3. Upon charge trapping, the electrostatic potential of the OsO6 octahedron increases due to the additional excess charge. To counterbalance this energy cost, the oxygen cage expands according to the isotropic (A1g) breathing-out mode Q1. This expansion favors charge localization, and contributes 67% of the total Epol (Supplementary Fig. 5), making Q1 the major lattice contribution to polaron stability. However, Q1 is an isotropic deformation that does not break any local symmetry and therefore it is not related to the JT effect. Moreover, Q1 is insensitive to SO and cannot play any role in the strong decrease of Epol with increasing cλ (see Supplementary Fig. 6). According to our data, only the tetragonal elongation along the [001] axis Q3 is strongly dependent on cλ (see Supplementary Fig. 6). In particular, Fig. 3b shows that increasing cλ yields a continuous decrease of Q3, in agreement with the analysis of ref. 45. Moreover, this SO-induced suppression of JT distortions is reflected on the JT energy (EJT), estimated from the potential energy surface at different values of cλ and displayed in Fig. 3c. Therefore, Q3 appears to be the key JT mode explaining the coupling between SO and polaron stability, correlating the SO-induced decrease of Epol with the progressive quenching of Q3 and associated reduction of EJT.

This analysis provides a new conceptual framework to interpret the complex concerted interaction between JT, SO and polaron stability. Without SO the isotropic expansion Q1 and the JT modes (Q2, Q3 and Qxy) help polaron stabilization providing an energy gain EJT which depends on the distortion amplitude. SO dampens the JT distortion Q3 leading to a reduction of EJT and consequentially a progressive increase of Epol (less stable polaron) with increasing SO coupling strength. This entangled spin-orbital Jahn-Teller bipolaron develops in a relativistic background, is described by a spin-orbital Jeff = 2 state and its stability is weakened by the SO-induced reduction of JT effects.

Polaron-mediated robustness of the Mott gap at high doping

Finally, we generalize our analysis to all doping concentrations disclosing the critical role of bipolarons in preserving the Mott state and elucidating the doping-induced modulation of the polaron phonon field as measured by NMR. It is well established that a critical amount of carrier doping drives a metal-insulator transition (MIT)49. Formation of small polarons can delay the MIT, but at a critical polaron density, coalescence into a Fermi liquid prevails, leading to a metallic (or superconducting) phase50,51,52. Notably, electron doped BNOO represents an exception: the insulating gap remains open at any concentration as shown in Fig. 4a (the corresponding DOS are collected in Supplementary Fig. 7). This unique behavior is explained by the absence of coherent hybridization between the bipolarons, facilitated by the large Os-Os distance of ≈ 5.9 Å in the double perovskite lattice. As illustrated in Fig. 4(b), NMR shows the bipolaronic peak at TP,1 ≈ 130 K at any Ca concentration with a virtually unchanged activation energy (see Supplementary Table 5), confirming the DFT results which indicate a linear increase of number of d2 bipolarons with increasing doping (see Fig. 4a).

Fig. 4: Polaron dynamics in Ba2Na1−xCaxOsO6: DFT+NMR.
figure 4

a DFT energy gap (black diamonds) and number of bipolarons (red filled squares) as a function of doping. Chemically doped BNOO remains insulating for any doping concentration. The number of d2 bipolaron sites grows linearly with doping. At full doping (x = 1, corresponding to BCOO) all Os sites are doubly occupied (d2) and polaron formation is completely quenched (empty square). b Polaronic 1/T1 anomalous peak data and fitting curves calculated with the quadrupole relaxation model of Eq. (1). For all investigated doping concentration doped BNOO exhibits a polaronic peak at approximately the same temperature T. c Second moment Δ2 of the fluctuating field as extracted from the experimental data (black squares) compared with our predicted data obtained from the DFT polaron phonon field (red circles). The dashed lines connect to pristine BNOO (x = 0) and BCOO (x = 1) where the absence of polaron leads to Δ2 = 0.

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The second moment of the fluctuating field Δ2 (see Eq. (1)) as a function of doping (Fig. 4c) exhibits a dome shape, characterized by a progressive increase until ≈ 75% Ca concentration, followed by a rapid decrease towards the full doping limit (x = 1, BCOO) with all Os sites converted into a non-polaron d2 configuration in a undistorted and JT-quenched cubic lattice. To interpret these NMR measurements we have derived a model linking the polaron-induced spin-lattice relaxation rate 1/T1 with the oscillation of the polaron modes \(\Delta {Q}_{\xi }={ \langle {Q}_{\xi } \rangle }_{{d}^{2}}-{ \langle {Q}_{\xi } \rangle }_{{d}^{1}}\) (with ξ running over the dominant modes Q1 and Q3). In particular, the average is taken over the distortions at the d2 and d1 sites obtained from DFT calculations. The resulting compact formula reads (see the Methods section for a full derivation)

$${\Delta }^{2}(x)=\frac{54{(eQ{q}_{ox})}^{2}}{5{\hslash }^{2}{R}_{0}^{8}(x)}\left[-\Delta {Q}_{1}^{2}(x)+2\Delta {Q}_{3}^{2}(x)\right]$$
(2)

where Q is the quadrupole moment of the 23Na nucleus, qox is the charge of the oxygen ions (as obtained by DFT, 1.78e), and R0 is the average Na-O bond length. The obtained numerical data summarized in Fig. 4c reproduce the experimental trend and indicate that, upon doping in Ba2Na1−xCaxOsO6, the main phonon contributions to polaron dynamics are encoded in the modulation of the breathing-out mode Q1 and the tetragonal distortion Q3 as a function of doping. In this regard, Fig. 4c provides a transparent unprecedented microscopic interpretation of the polaron-driven spin-lattice relaxation rate 1/T1, here demonstrated for the quadrupolar polaron mechanism.

Discussions

Summarizing, our study discloses a new type of polaron quasiparticle which is responsible for blocking the MIT even at ultrahigh doping and enables the coexistence of different spin-orbital Jeff states in the same compound. This mixed-state can be interpreted as a precursor state towards the formation of the homogeneous Jeff = 2 state at full Na → Ca substitution (BCOO53), and the polaron is the main driving force of this transition. In perspective, this work provides the conceptual means to explore polaron physics in quantum materials with strong spin-orbit coupling including topological54, Rashba55 and 2D materials56, and pave the way for polaron spintronics57, polaron heavy-elements catalysis58 and polaron multipolar magnetism34,59,60,61.

Methods

Density functional theory

The electronic structure, structural deformations, and polaron hopping were studied using the fully relativistic version of VASP, employing the Perdew-Burke-Ernzerhof approximation for the exchange-correlation functional62,63. All DFT calculations were performed with the magnetic moments’ directions fixed to those of the low-temperature cAFM phase of BNOO34. In addition, Dudarev’s correction of DFT+U was applied to account for strong electronic correlation effects, using a value of U = 3.4 eV, which stabilizes the cAFM ordering in the pristine material. The computational unit cell is a \(\sqrt{2}a\times \sqrt{2}a\times a\) supercell containing eight formula units, with a = 8.27 Å referring to the lattice constant of the standard double perovskite unit cell, which contains four formula units. The sampling of the reciprocal space was done with a k-mesh of 4 × 4 × 6, and an energy cutoff of 580 eV was selected for the plane wave expansion. The SO contribution to the DFT energy functional could be manually controlled through a scaling parameter cλ using an in-house modified version of VASP. The analysis of the JT effect was conducted using vibration modes defined by Bersuker64, with some minor modifications, such as neglecting the rigid translation of the octahedra and including rigid rotations (see Supplementary Note 3). Electron doping was achieved by manually increasing the number of electrons in pristine BNOO. Charge neutrality is restored by adding a homogeneous background. To extract polaronic energy levels and wavefunctions, a Wannier-like projection of Kohn-Sham wavefunctions was employed, using VASP non-collinear projected localized orbitals calculated on the polaronic Os site with the TRIQS’s converter library47 (see Supplementary Note 1). DFT calculations for comparison with NMR data were performed on relaxed chemically doped \(\sqrt{2}a\times \sqrt{2}a\times a\) supercells. The same number of k-points, energy cutoff and U were used as for the previous doping method.

Dynamical mean field theory

For the analysis of the spin-orbital structure of the polaron levels we performed charge-self-consistent DFT+DMFT calculations within the Hubbard-I approximation65,66, using WIEN-2K67 and the TRIQS library68,69. By using a \(\sqrt{2}a\times \sqrt{2}a\times a\) supercell where one Na atom was substituted by one Ca (12.5% Ca concentration), we first calculated the polaronic ground state lattice structure, as explained in the previous section Density Functional Theory, using VASP. In DFT+HI calculations, the Wannier functions representing Os 5d states are constructed from the Kohn-Sham bands within the energy range [ − 1, 5] eV around the Kohn-Sham Fermi energy, which contains the Os t2g and most of the eg levels. The fully-localized-limit double counting term on the polaronic Os site is set for the nominal d2 occupancy, as is appropriate for the quasi-atomic Hubbard-I approximation70, whereas for the rest of Os sites it is calculated for nominal d1. The on-site interaction vertex for the full 5d shell is specified by the parameters U = 3.5 eV and JH = 0.5 eV, in agreement with the previous studies of pristine BNOO34.

Sample preparation

Powder samples were prepared by the solid state method; stoichiometric amounts of BaO (Sigma-Alrich, 99.99% trace metals basis), CaO (Sigma-Alrich, 99.9% trace metals basis), Na2O2 (Alfa Aesar, 95%), and Os powder (Sigma-Alrich, 99.9% trace metals basis) were ground in a mortar and pestle, transferred to an alumina tube, and sealed in a quartz tube under vacuum. A separate alumina cap containing PbO2 was also included in the sealed quartz tube as the decomposition of PbO2 into PbO and O2 at 600 °C provided the oxygen source to oxidize Os metal. Because highly toxic OsO4 can form from the reaction of Os metal and O2 at or above 400 °C, this reaction was carried out inside an evacuated silica tube and the furnace was positioned in a fume hood in case the silica tube ruptured. To ensure the full oxidation of the osmium, the amount of PbO2 was chosen to generate an excess of 1/4 mol of oxygen for every mol of the desired product. The reaction vessel was heated at 1000 °C for 24 h. For several samples an additional step of grinding and heating for an additional 12 h at 1000 °C was necessary to form a homogeneous perovskite phase. See also ref. 36.

Nuclear magnetic resonance and muon spin rotation

We exploited the nuclear spin I = 3/2 of 23Na nuclei in order to perform NMR spectroscopy on powder samples of Ba2Na1−xCaxOsO6, with x = 0.0%, 12.5%, 25.0%, 37.5%, 50.0%, 75.0% and 90.0%. In particular, 23Na nuclei have a sizeable quadrupolar moment that allows to probe both magnetic and charge related dynamics. We report spin-lattice (1/T1) and spin-spin (1/T2) relaxation rates as a function of temperature measured using an applied field of 7 T (details are reported in Supplementary Note 6). We further analysed the anomalous peaks observed in these data by implanting a beam of polarised muons spin antiparallel to their momentum into the sample and applying a magnetic field of 10 mT and 100 mT parallel to the initial muon spin polarisation in order to measure the longitudinal muon relaxation rate \({\lambda }_{\mu }\equiv 1/{T}_{1}^{\mu }\) (see Supplementary Note 7).

Spin-lattice relaxation model

The interaction of the nuclear quadrupole moment with an EFG Vαβ can be written using spherical tensor operators \({T}_{2}^{q}\) as71

$$H=\frac{eQ}{2I(2I-1)}\mathop{\sum }\limits_{q=-2}^{2}{(-1)}^{q}{V}_{q}{T}_{2}^{-q}$$
(3)

where Q is the quadrupole moment of the nucleus and I = 3/2 is the nuclear spin. To calculate the spherical component Vq of the EFG we adopted a point-charge model of the NaO6 octahedron. The explicit expressions of the Vq’s are given in Supplementary Note 8.

The perturbation induced by fluctuations of the Na-O bonds resulting from polaron hopping is obtained from Eq. (3) by expanding Vq in terms of the bond variations ΔRiα(t)

$${H}^{{\prime} }(t)=\frac{eQ}{6}\mathop{\sum }\limits_{i=1}^{6}\mathop{\sum }\limits_{\alpha }^{x,y,z}\mathop{\sum}\limits_{q{q}^{{\prime} }}{{{{{{{{\mathcal{D}}}}}}}}}_{{q}^{{\prime} }q}^{2}({{{{{{{\mathcal{R}}}}}}}}){T}_{2}^{-{q}^{{\prime} }}{w}_{q}^{i\alpha }\Delta {R}_{i\alpha }(t)$$
(4)

where \({w}_{q}^{i\alpha }\) are the derivatives of the spherical components of the EFG with respect to the α-th component of the i-th oxygen ion. In Eq. (4) we have introduced the Wigner D-matrix \({{{{{{{{\mathcal{D}}}}}}}}}_{{q}^{{\prime} }q}^{2}({{{{{{{\mathcal{R}}}}}}}})\) to give account for the random orientation of the EFG reference frame with respect to the external magnetic field in powder samples.

The transition rate between two Zeeman levels m and \({m}^{{\prime} }\) averaged over all possible directions is given by

$${W}_{m{m}^{{\prime} }}=\frac{{(eQ{q}_{ox})}^{2}}{6{\hslash }^{2}}\mathop{\sum }\limits _{q}{\left| \left\langle m\left| {T}_{2}^{-q}\right| {m}^{{\prime} }\right\rangle \right| }^{2}\mathop{\sum}\limits_{ij,\alpha \beta }{M}_{\alpha \beta }^{(ij)}\int\nolimits_{-\infty }^{\infty }{{{{{{{\rm{d}}}}}}}}t\,\overline{\Delta {R}_{i\alpha }(t)\Delta {R}_{j\beta }(0)}\,{e}^{-i{\omega }_{m{m}^{{\prime} }}t}$$
(5)

where qox is the oxygen ion charge in the point-charge model, \({\omega }_{m{m}^{{\prime} }}=| {\omega }_{m}-{\omega }_{{m}^{{\prime} }}|\) is the energy separation between the Zeeman levels, the matrices \({M}_{\alpha \beta }^{(ij)}\) are defined in Supplementary Note 8 and \(\overline{\Delta {R}_{i\alpha }(t)\Delta {R}_{j\beta }(0)}\) is the correlation function of the α-th component of the i-th bond with the β-th of the j-th one. To simplify the transition rate formula, some considerations on the crystal structure of Ba2Na1−xCaxOsO6 and small polaron dynamics are necessary.

First, we notice that NaO6 and OsO6 octahedra are corner sharing in Ba2Na1−xCaxOsO6. Therefore, the fluctuations of the i-th oxygen in the NaO6 octahedron can be defined using the distortion modes Qξ of the OsO6 octahedron sharing the i-th oxygen with the NaO6 one (see Supplementary Fig. 4). In this way the coordinate correlation functions can be expressed as correlation functions of the distortion modes Qi,ξ of the i-th OsO6 octahedron.

Moving to polaron dynamics, if the i-th and the j-th octahedra are involved in an adiabatic hopping event within the time interval − τc ≲ t ≲ τc, in the LIS we have Qi,ξ(t) = Qj,ξ( − t). Thus, assuming that independent modes at the same site are uncorrelated, we can write all the correlation functions appearing in Eq. (5) as autocorrelation function of the independent distortion mode at each OsO6 octahedron \(\overline{{Q}_{\xi }(t){Q}_{\xi }(0)}\). To calculate these quantities, we recall that the MEHAM theory of adiabatic small polaron hopping relies on the classical treatment of phonon modes in the description of the site-jump process72. In this limit, one can describe fluctuations using a phenomenological Langevin equation73. Within this model, if the characteristic time of the fluctuations is much smaller than that of the hopping process (τc), we speak of overdamped regime and the fluctuations’ autocorrelations are given by

$$\overline{{Q}_{\xi }(t){Q}_{\xi }(0)}=\Delta {Q}_{\xi }^{2}\,{e}^{-| t| /{\tau }_{c}}$$
(6)

where ΔQξ is the amplitude of the fluctuations of the Qξ mode. To validate this assumption, we evaluated the vibration frequencies ωQ of the Q1 and Q3 oscillators from the potential energy curves obtained in the analysis of the Jahn-Teller modes (reported in Supplementary Note 3). We found them to be in the order of ωQ ~ 1013 s−1, as also reported in phonon spectra calculated by Voleti et al.60. On the other hand, the correlation times extracted from NMR measurements are in the order of τc ~ 10−10 s, therefore ωQτc ≫ 1, which corresponds to the overdamped regime73. Moreover, we notice that the behavior of the autocorrelations expressed in Eq. (6) is commonly assumed in the description of spin-lattice relaxation processes40,71,74.

By neglecting correlation functions between opposite sites (Os-Os distance ≈ 8.29 Å), i.e. assuming only nearest-neighbor hopping (Os-Os distance ≈ 5.86 Å), the transition rate becomes

$${W}_{m{m}^{{\prime} }}=\frac{3{(eQ{q}_{ox})}^{2}}{2{\hslash }^{2}{R}_{0}^{8}}\mathop{\sum}\limits_{q}{ \left | {\left \langle m {\left | {T}_{2}^{q}\right |} {m}^{{\prime} }\right \rangle} \right| }^{2}\left(-\Delta {Q}_{1}^{2}+2\Delta {Q}_{3}^{2}\right)\frac{{\tau }_{c}}{1+{({\omega }_{m{m}^{{\prime} }}{\tau }_{c})}^{2}}$$
(7)

where we have only considered fluctuations in the breathing-out mode Q1 and the tetragonal mode Q3 to be relevant, as deduced from the considerations expressed in the main text.

The spin-lattice relaxation time T1 for a I = 3/2 nucleus with quadrupolar interactions is given by71

$$\frac{1}{{T}_{1}}=\frac{12}{5}({W}_{1}+4{W}_{2})$$
(8)

where W1 and W2 are the transition rates for relaxations with selection rules Δm = ± 1 and Δm = ± 2 respectively. By combining the latter Eq. (8) with the transition rate formula in Eq. (7), we obtain the relation

$$\frac{1}{{T}_{1}}={\Delta }^{2}\left[\frac{{\tau }_{c}}{1+{({\omega }_{0}{\tau }_{c})}^{2}}+\frac{4{\tau }_{c}}{1+{(2{\omega }_{0}{\tau }_{c})}^{2}}\right]$$
(9)

which has been used to fit the NMR anomalous peak 1/T1(T) with \({\tau }_{c}={\tau }_{0}\exp ({T}_{a}/T)\), while Δ2 is given by

$${\Delta }^{2}=\frac{54{(eQ{q}_{ox})}^{2}}{5{\hslash }^{2}{R}_{0}^{8}}\left(-\Delta {Q}_{1}^{2}+2\Delta {Q}_{3}^{2}\right)$$
(10)

which corresponds to Eq. (2) of the main text. Based on the above analysis and in analogy with the standard BPP model, Δ2 corresponds to the second moment of the fluctuating perturbation \(H^{\prime} (t)\) which is expressed in terms of the amplitude of the distortion modes.