Metal–insulator transition in composition-tuned nickel oxide films

Thin films of the solid solution Nd 1−x La x NiO3 are grown in order to study the expected 0 K phase transitions at a specific composition. We experimentally map out the structural, electronic and magnetic properties as a function of x and a discontinuous, possibly first order, insulator–metal transition is observed at low temperature when x = 0.2. Raman spectroscopy and scanning transmission electron microscopy show that this is not associated with a correspondingly discontinuous global structural change. On the other hand, results from density functional theory (DFT) and combined DFT and dynamical mean field theory calculations produce a 0 K first order transition at around this composition. We further estimate the temperature-dependence of the transition from thermodynamic considerations and find that a discontinuous insulator–metal transition can be reproduced theoretically and implies a narrow insulator–metal phase coexistence with x. Finally, muon spin rotation (µSR) measurements suggest that there are non-static magnetic moments in the system that may be understood in the context of the first order nature of the 0 K transition and its associated phase coexistence regime.

Thin films of the solid solution Nd 1−x La x NiO 3 are grown in order to study the expected 0 K phase transitions at a specific composition. We experimentally map out the structural, electronic and magnetic properties as a function of x and a discontinuous, possibly first order, insulator-metal transition is observed at low temperature when x = 0.2. Raman spectroscopy and scanning transmission electron microscopy show that this is not associated with a correspondingly discontinuous global structural change. On the other hand, results from density functional theory (DFT) and combined DFT and dynamical mean field theory calculations produce a 0 K first order transition at around this composition. We further estimate the temperature-dependence of the transition from thermodynamic considerations and find that a discontinuous insulator-metal transition can be reproduced theoretically and implies a narrow insulator-metal phase coexistence with x. Finally, muon spin rotation (µSR) measurements suggest that there are non-static magnetic moments in the system that may be understood in the context of the first order nature of the 0 K transition and its associated phase coexistence regime. * Author to whom any correspondence should be addressed.
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Introduction
A fundamental and very widely used tool of materials engineering is tofine-tune the chemical composition to achieve a specific property. Beyond this, families of materials with phase transitions are particularly interesting for applications as well as fundamental physics and the chemical composition often provides a control knob to set the transition at a desired condition, e.g. temperature or pressure.
The rare earth nickelates, RNiO 3 , where R represents a rare earth or lanthanide element, comprise a family of perovskite oxides well-known for their metal-to-insulator transition (MIT). The fundamental structural motif is the cornershared NiO 6 octahedron. The important Ni electronic states are the e g (d 3z 2 −r 2 /x 2 −y 2 ) doublet of the 3d shell, and formal valence considerations imply each Ni is in the 3 + ionisation state with a d 7 electron configuration (one electron in the e g multiplet). In fact, strong hybridisation and close level alignment of Ni-d and O-p states means that the Ni state is better approximated as d 8L , corresponding to two electrons in the Ni e g states and a hole shared amongst the nearby oxygens [1]. At high temperature the materials are metallic with a Pnma crystal structure in which all the Ni sites are equivalent (up to a rotation). As the temperature is decreased all of the nickelates except LaNiO 3 undergo a MIT [2,3]. The insulating state is characterised by a P2 1 /n structure involving a two sublattice breathing distortion in which the mean Ni-O bond length alternates between a short and a long value. The lattice distortion is accompanied by a charge disproportionation which, from the point of view of fundamental many-body electronic structure theory should be treated as a hybridisation or site-selective order involving an alternating pattern of d 8 /d 8L2 [4][5][6] but for most purposes may be thought of as a d 8 /d 6 'charge order' corresponding to e 2 g and e 0 g respectively [7]. At low enough temperatures the insulating state is antiferromagnetic with a striped ordering pattern defined by a wavevector of [ 1 4 , 1 4 , 1 4 ]. Because LaNiO 3 in bulk form remains metallic and nonmagnetic with no two-sublattice breathing distortion down to lowest temperature, while NdNiO 3 has an insulating, latticedistorted antiferromagnetic ground state, we expect that the solid solution Nd 1−x La x NiO 3 will exhibit zero temperature insulator-metal, antiferromagnetic-paramagnetic and structural transitions for x between 0 and 1. If continuous (second order), a 'quantum' (zero temperature) phase transition may be expected to give rise to exotic behaviour [8]. In addition, LaNiO 3 forms in the R3c symmetry, distinct from the Pnma symmetry of NdNiO 3 , so a composition-driven lattice symmetry transition must occur at a specific value of x that may depend on temperature. This transition has been found to occur at around x = 0.5 at room temperature [9].
In this paper we present a comprehensive study of the composition-driven phase transitions occurring in Nd 1−x La x NiO 3 as x is varied from 0 to 1. Our study uses a suite of films with a very dense set of x values allowing us to locate the transition with precision and determine the evolution of properties as the transition is approached. We perform transport, magnetic, light scattering and structural measurements as well as theoretical calculations. We find a zero temperature first order transition at a x ≈ 0.2 between a low x P2 1 /n -symmetry (presumably lattice-distorted and charge-ordered) magnetic insulator and an intermediate x Pnma symmetry paramagnetic metallic state.

Sample preparation
The fully epitaxially strained Nd 1−x La x NiO 3 films were grown by intermittent radio-frequency off-axis magnetron sputtering as described in section 7.
The films are, within our experimental resolution, homogenous solid solutions. The possibility that the solid solution is segregated into nanocomposites of NdNiO 3 and LaNiO 3 can be excluded. Indeed, figures 1(a) and (b) are compositional maps from electron energy loss spectroscopy (EELS) and energy dispersive x-ray spectroscopy (EDXS) showing that lanthanum and neodymium are homogeneously distributed down to the atomic scale.

Electronic and structural properties
The inset of figure 1(c) displays representative resistivity curves as a function of temperature for a set of 20 unit cell (u.c.) (=7.6 nm) thick Nd 1−x La x NiO 3 with x = (0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1) on LSAT substrate. The films fall into four groups. As expected, the x = 0 (pure NdNiO 3 ) sample displays a clear MIT at around 150 K separating a high temperature, relatively poorly conducting, metallic state from a low temperature state with a very high resistance that grows exponentially as the temperature is decreased. The transition exhibits a hysteresis loop. The x = 0.1 and 0.2 Nd 1−x La x NiO 3 have metallic-state resistivities that are significantly lower than for x = 0 but similar between the two samples and higher than the ones with greater La content. They also show first order MITs albeit at lower temperatures than the x = 0 sample. The x = 0.1 film has an insulating state very similar to that of the x = 0 film, but with a lower resistivity. The low temperature state of the x = 0.2 film is classified here as insulating because the resistance exhibits a minimum at a non-zero temperature, but the low temperature resistivity is much lower than in the x = 0.0, 0.1 films and increases much less rapidly as T → 0. Indeed at low temperature the conductivity of the x = 0.2 sample is ≈5000 Ω −1 cm −1 which, using a Ni-Ni spacing of ≈0.4 nm is not much larger than the Mott 'minimum metallic conductivity' [10], suggesting that at this composition the material is only weakly insulating despite the relatively high temperature of the resistance minimum. We suggest that the low temperature resistivity of the x = 0.2 sample arises from a mixture of insulating and metallic domains, with the metallic domains not quite percolating. All the further samples are metallic in the sense of having a resistivity that increases with increasing temperature at all temperatures, but the metallic films fall into two groups, with x = 0.3, 0.4 having very similar resistivities that are noticeably higher than the resistivities of the larger-x films.
The main panel of figure 1(c) presents, as a function of x, the temperature at which the resistivity of each film (recorded on warming) is minimal for all the measured compositions on the three substrates. On the low x side of the plot, the points correspond to the MIT and the temperature of the transition decreases gradually with increasing La content. On the high x side the resistivity is clearly metallic with a resistance that is minimised at T = 0. We may therefore approximately interpret this panel as a metal-insulator phase diagram, separating a low-T and low-x insulating regime from a high-T/high-x metallic regime defined by dρ/dT > 0. However we must bear in mind that first order transitions are generically characterised by regions of phase coexistence, corresponding here to domains of insulating and metallic behaviour, so that some of the region with T MI = 0 may correspond to a region of phase coexistence with percolating metallic domains while, as seen for the x = 0.2 sample, some of the regime with T MI ̸ = 0 may correspond to a phase coexistence regime with non-percolating metallic regions. Thus, phase coexistence stemming from the first order nature of the transition with x provides an explanation for both the absolute resistivities (e.g. the weakly insulating behaviour for x = 0.2) and the discontinuous behaviour of the temperature of the minimum of resistivity.
The precise location of the microscopically defined phase boundary is therefore subject to some uncertainty. Comparison of the main panel of figure 1(c) with data previously reported for bulk solid solutions [11] suggests that the MIT in these films occurs at a somewhat lower x and lower T with respect to bulk, similar to films under compressive strain [12]. As an example, bulk Nd 1−x La x NiO 3 with x = 0.3 is still insulating at low temperature with T MI = 110 K [11].
Given the strong correlation between the lattice and electronic degrees of freedom in the rare earth nickelates, the MIT is expected to be strongly influenced by the structure. The difference in symmetry between the Pnma structure of room temperature NdNiO 3 and the R3c structure of LaNiO 3 is a natural candidate for the source of the first order transition. However, our structural measurements rule out this scenario. Figures 1(d) and (e) show room temperature high-angle annular dark field (HAADF) images from scanning transmission electron microscopy for x = 0.1 and x = 0.3 Nd 1−x La x NiO 3 prepared in a planar view. There is no noticeable change in the microstructure between the composition that becomes insulating (d) versus the one that remains metallic (e). Moreover, the fast Fourier transform patterns generated from these images, and displayed as insets, show that both films are in the Pnma symmetry by the presence of additional half-integer reflections, highlighted by yellow circles. Thus the change in symmetry from Pnma to R3c with increasing x must occur at an x that is larger than x = 0.3. Indeed, previous work found that the room temperature crossover in symmetry from Pnma to R3c with increasing x in the Nd 1−x La x NiO 3 phase diagram occurs at approximately x = 0.5 [9].
The known strong association of lattice distortions and the MIT in the nickelate materials means that we can use the temperature-dependent Raman scattering spectra shown in figures 2(a) and (b) to bound the region of phase coexistence and therefore the x at which the MIT occurs. Panel (a) shows spectra taken on a Nd 1−x La x NiO 3 with x = 0.1, which undergoes the MIT and, therefore, is assumed to also undergo the Pnma-P2 1 /n symmetry lowering. The peak at ≈300 cm −1 arises from an A g -symmetry phonon that is related to rigid antiphase rotations of corner-connected oxygen octahedra [13,14]. This peak undergoes a clear frequency shift as the temperature is tuned across the temperature of the resistance minimum for this sample (≈100 K). There is no evidence of the low-frequency peak in the high-T data and no evidence of the high-frequency peak in the low-T data, strongly suggesting that at this x the sample completely transforms symmetry without significant phase coexistence. Similarly, for x = 0.3 (b), which remains metallic, only one peak is evident at all temperatures, again suggesting a single-phase sample. Figure 2(c) shows the Raman spectra at 10 K for different compositions. The A g mode frequency is marked with stars and then plotted as a function of composition x in panel (d) with error bars corresponding to one standard deviation from a Lorentzian fit. A change of slope is observed at around x = 0.2, reinforcing the conclusion that there is a change of space group at the same composition as the insulator-metal transition.
This sudden suppression of the MIT, which has been resolved in large part due to the high number of samples analysed, obscures the potentially-tunable quantum phase transition that was expected and cannot be explained by chemical or electronic phase separation.
For instance, a discontinuous change of bandwidth upon undergoing the symmetry crossover to R3c may occur at higher x but cannot account for the rapid suppression of the MIT at lower lanthanum content, this phenomenon manifesting within the Pnma phase only.

Theory
In this section we present a theoretical analysis. Extensive previous work has documented an intimate interplay between lattice/structural and electronic properties across the broad family of rare earth nickelates and confirmed in particular the importance of the Ni-O-Ni bond angle [15][16][17][18][19][20][21]. As observed in earlier work [9], the A g -like modes of Nd 1−x La x NiO 3 both soften as the La content is increased, demonstrating that the oxygen octahedral rotations decrease in magnitude. With the reduction of octahedral rotations comes a straightening of the Ni-O-Ni bond angle. We use the combination of density functional theory (DFT) and dynamical mean field theory (DMFT) to investigate the structural evolution and its impact on the electronic behaviour.
We perform bulk calculations (as described in [22]) and analyse-as described in [18]-DFT and DFT+DMFT calculations for NdNiO 3 and x = 0.25 Nd 1−x La x NiO 3 , in order to compare with the parameters obtained for NdNiO 3 previously benchmarked for these references. We use the NdNiO 3 DFT-relaxed structure and parameters from said previous references and approximate the randomly alloyed Nd 1−x La x NiO 3 x = 0.25 compound by employing a 20-atom unit cell with three Nd and one La (note that any location for the novel La atom is equivalent by symmetry, therefore a single structure need only be considered). As before, the functional used is the Perdew-Burke-Ernzerhof [23], as it leads to structures with nearly identical bond angles to the experimental ones, with our DMFT calculations very sensitive to these results. We then generate structurally disproportionate structures, as previously shown in [22], to extract the parameters g and k, which are the electron-lattice coupling and lattice stiffness respectively. After DFT relaxation, the average bond angle for the DFT-relaxed NdNiO 3 is 156.3 • in-plane, and 157.1 • out-ofplane. For the x = 0.25 structure, the average bond angle is 157.5 • in-plane and 157.8 • out-of-plane. This slight straightening is qualitatively consistent with our experimental observations and also with our expectations given the RNiO 3 bulk phase diagram.
We then select interaction parameters following [22] and, then, following [18] perform DFT plus DMFT calculations of the electronic charge disproportionation ∆N for a series of structures that interpolate between the DFT-relaxed Pnma structure and fully distorted P2 1 /n structure. We fit the results to the free energy functional: with: where Q denotes the amplitude of the lattice distortion that breaks the Pnma symmetry and the k and g are obtained from the DFT calculations as described in [22] and the parameters in F el come from the DMFT calculations. Representative results are shown in table 1. Note that following the conventions of Georgescu et al [22] the free energy is per two Ni (i.e. per unit cell of the disproportionate structure). We observe that all of the parameters change modestly between x = 0 and x = 0.25.
To study other x values we linearly interpolate. Figure 3 shows the calculated free energy landscape in the plane of Q and ∆N for x = 0.0 (panel (a)) and x = 0.25 (panel (b)). The landscape is characterised by two minima, one at the Pnma structure Q = ∆N = 0 and one at a relatively strongly distorted P2 1 /n structure Q ≈ 0.05 Å and ∆N ≈ 1, connected by a sharply defined saddle point path. In this calculation, as x is changed from x = 0.0 to 0.25 the global minimum switches from the P2 1 /n structure to the Pnma structure, confirming a first order composition-driven transition at an x between 0 and 0.25.
For a different perspective on the calculation, we present in figures 3(c) and (d) cuts of the free energy along the line connecting the origin to the distorted minimum for a series of x. The first order nature of the transition is clearly evident.
Comparison of the two panels shows that the precise location of the transition with x is sensitive to details (shown here are the results obtained for two values of the on-site exchange J), but the important features are robust.
We now consider the temperature dependence of the phase boundary, on the assumption that the insulating state is characterised by a large electronic gap and a low-temperature entropy arising mainly from spin waves and acoustic phonons, both of which give a free energy contribution ∼ T 4 at low T. On the other hand the metallic state has an entropic contribution from the Fermi surface = AT 2 where A is of the order of the inverse Fermi temperature determined by the effective mass. Neglecting the entropy of the insulating state, the metal-insulator free energy difference is then with E ′ = ∂E/∂x the first derivative of the ground state energy with respect to x so that at very low T the phase boundary is given by The data in figure 3 yield E ′ ≈ 20 meV uc −1 where the low temperature unit cell contains two Ni ions. To estimate A we note that DMFT calculations for the undistorted paramagnetic metallic state yield a mass enhancement ≲ 2 while the bare density of states (summed over orbitals, spins and the two Ni sites in the unit cell) is about 12 eV −1 . The low temperature electronic specific heat coefficient is then given by π 2 /3 times the density of states times the mass enhancement implying A ≈ 26 eV -1 uc −1 so that we may estimate Thus the DMFT calculation suggests that the   figure 1, which is in reasonable agreement with the experimental data.

Magnetic properties
Antiferromagnetism and the MIT are strongly linked in NdNiO 3 but antiferromagnetism is difficult to probe in thin films. In this section we report results of resonant soft x-ray scattering (REXS) and µSR probes of magnetism in our films. REXS reveals coherent long-ranged antiferromagnetic ordering via emergence of a Bragg scattering peak at the ordering wavevector. µSR reveals short-ranged magnetic ordering, fluctuating moments, and can give information on the magnetic volume fraction.

Resonant x-ray scattering.
REXS is a probe of longrange order. REXS [24] was carried out at the pseudocubic [ 1 4 , 1 4 , 1 4 ] diffraction condition while the beam energy was set to resonate with the Ni L 3 edge at 853 eV. At low temperatures, a peak is observed that has been ascribed to the antiferromagnetic order of the nickel sublattice [25]. Here, this same peak was recorded in a series of symmetric θ-2θ scans from 40 K up to room temperature with the incident beam π-polarised. The maximal intensity of each of these scans for a 100 u.c. (= 38 nm) sample of Nd 1−x La x NiO 3 with x = 0.1 on LSAT substrate is plotted as orange squares in figure 4(a) normalised to the low temperature value. As can be seen, the intensity of this peak drops sharply at the same temperature as the sample becomes metallic (resistivity plotted in black).
The full width at half-maximum of the resonant diffraction peak at low temperature is 2.34 • , which, at this Bragg angle and x-ray wavelength, corresponds to a (0 0 1) coherence length of 100 u.c., precisely in agreement with the thickness of the film. This confirms that, at low temperature, the magnetic order is coherent across the entire film with no extra broadening from in-plane grains.

Low energy µSR.
µSR is a well-demonstrated probe of local magnetic moment [26]. By decreasing the muon beam energy down to 4 keV, the mean stopping depth is reduced to around 20 nm for materials such as Nd 1−x La x NiO 3 with a density of around 7 gcm −3 according to Monte Carlo simulations [27]. Using low energy muons, therefore, maximises the sensitivity to the thin film. Low energy µSR [28] was carried out on a series of 100 u.c. (≈38 nm) thick Nd 1−x La x NiO 3 , on LSAT substrate. Each measurement, from 10 K up to room temperature, consists of at least two million events and was performed in a weak magnetic field of 10 mT applied transverse to the initial muon spin polarisation. The muon acts as a local magnetic sensor. There are two extremes for the case of an applied transverse magnetic field: 1. The muon spin ensemble experiences dominantly the applied field because it is much stronger than any internal field. In this case the muon spin precession is around the applied field with the maximal asymmetry amplitude. 2. The local static magnetic field distribution at the muon site is comparable to or stronger than the applied field: in this case the muon spin ensemble dephases very rapidly. Therefore, the asymmetry amplitude of the resulting muon spin precession is taken to be proportional to the non-magnetically-ordered volume fraction of the sample. Figure 4(a) shows different temperature-dependent behaviour of the same Nd 1−x La x NiO 3 x = 0.1 sample when comparing the REXS intensity to the magnetic volume fraction determined from µSR. The REXS shows a sharp transition at the same temperature at which the sample undergoes the MIT. On the other hand, µSR shows a gradually increasing magnetic volume fraction that begins when the sample is still in the metallic phase and does not reach 1.0 even at 10 K. Figure 4(b) shows that a similar broad onset of the magnetic volume fraction is observed in other compositions of Nd 1−x La x NiO 3 with x ̸ = 0. These differences may be reconciled by considering that muon relaxation samples the system on a microsecond time scale while x-ray scattering occurs within picoseconds. In particular, either there are local magnetic moments that increase in density towards lower temperatures or the spin-spin correlation function increases, which decreases the fluctuation rate of the electronic moments into the MHz regime, quasi-static for the muons. Super-MHz fluctuation of local moments at higher temperatures would thus give rise to this behaviour.

Discussion of magnetic properties
Further evidence for fluctuating local moments is provided by figure 5(a). In the case of a static magnetic order, the µSR asymmetry should decay rapidly to a flat tail. The slow ZF-µSR decay observed here is typical of fluctuating systems.  The origin of such fluctuations in Nd 1−x La x NiO 3 is unknown. La 3+ replaces Nd 3+ so a background of cations with spin 1 2 is gradually being diluted by cations with spin 0. Such a substitution may increase disorder in the insulating, antiferromagnetic state through fluctuating local moments. Previous µSR work on bulk (powder) Nd 1−x La x NiO 3 shows a sharp transition in magnetic volume fraction that closely tracks the MIT [29]. Our samples may have amplified fluctuations due to the thin film nature of these crystals. Finally, it is possible that fluctuations at higher x are related to small inclusions of insulating phase.

Conclusions
High quality epitaxial thin films of Nd 1−x La x NiO 3 are synthesised. They are found to be chemically homogeneous. Both experimental evidence and theoretical results suggest a discontinuous (likely first order) transition between a low-x, low-T P2 1 /n-symmetry insulating phase and a high-x high-T Pnma-symmetry metallic phase. Raman spectroscopy and STEM analyses, show that this transport anomaly cannot be attributed to any x-dependent structural change in the material, consistent with previous work on bulk solid solutions. In particular, the transition between the Pnma symmetry of the high temperature phase of the x = 0 material and the R3c symmetry of the x = 1 end-member occurs at x ≈ 0.5, far outside of the transition region [9]. This independence is also reflected in the effect of biaxial strain, which would be expected to renormalise any structural trend, but here does not have any influence on the behaviour with x.
The MIT, being first order, is likely to involve a region of phase coexistence near the transition point, driven e.g. by random variations in strain or local chemistry, for instance oxygen stoichiometry. The phase coexistence implies that the transport data do not necessarily provide the location of the thermodynamic phase transition, because once the metallic domains percolate, the resistivity is simply metallic. For example, the transport data at x = 0.2 reveal a resistivity with a minimum at ∼70 K indicating a transition to an insulating state, but a low T conductivity not much larger than the Mott minimum metallic conductivity consistent with large but not percolating metallic domains. The Raman data however clearly indicate that the x = 0.3 sample has a negligible volume fraction of insulating phase at any T. We therefore suggest that the MIT occurs at a concentration x between 0.2 and 0.3 with the precise location masked by phase coexistence effects.
The magnetic data reported in section 5 are broadly consistent with this interpretation. The REXS and muon data on the x = 0.1 sample reveal an almost complete coherent magnetic volume fraction with the hysteresis characteristic of a first order transition while at x = 0.3 the muon data indicate a small volume fraction without any hysteresis, very similar to the observations at x = 0 and T > T MIT .
Theory also supports this picture. While the precise location of the critical x depends sensitively on parameters, the existence of a first order transition and the variation of the T = 0 insulator-metal energy difference with x are very robust features, insensitive to details. A simple analysis of the thermodynamics of the first order transition, based on the assumptions that the entropy of the insulating state is negligible and the metallic phase is only moderately correlated, produces a temperature-dependent phase boundary reasonably consistent with experiment.
Important directions for future research include direct detection of the coexisting phases for x ≈ 0.2, determining the origin of the fluctuating magnetism observed even at large x, and refining the theory so that a more quantitative comparison to experiment can be made.

Methods
Films of Nd 1−x La x NiO 3 were grown by intermittent radio frequency off-axis magnetron sputtering. The growth atmosphere was 7:2 Ar:O 2 at 0.18 Torr and the temperature was 450 • C.
For electrical resistivity measurements, the films were contacted by Al bonds onto sputtered Pt pads in a van der Pauw geometry [30] or on films etched into a Hall bar. The measurements were taken in a 4 He dipping station.
The STEM specimens were prepared by conventional methods, following mechanical tripod polishing and argon ion beam milling until electron transparency. Stacks of planar view HAADF images were acquired in a double-aberrationcorrected FEI (Thermo Fisher Scientific) Titan Themis 60-300 microscope, located at the Interdisciplinary Centre for Electron Microscopy (CIME) inÈcole Polytechnique Fédérale de Lausanne (EPFL), operated at 300 kV using a convergence semi-angle of 20 mrad. The images of each stack were aligned with the SmartAlign plug-in for Digital Micrograph (GMS3.2), which corrects the image drift and the linear and non-linear scan distortions [31].
STEM-EELS spectrum image datasets were acquired with a Gatan GIF Quantum ERS spectrometer, using a collection semi-angle of ≃47 mrad. EDXS maps were acquired on the same microscope, using a four quadrant Super-X windowless silicon drift detector system and beam currents of around 100-250 pA. The maps were recorded and analysed using Thermo Fisher Scientiic Velox software.
The Raman spectra were acquired using an InVia Renishaw micro-Raman spectrometer, coupled with an Oxford HiRes microstat cooled with liquid He. Measurements were performed using the 633 nm line of a He-Ne laser. The spot size was around 1.3 µm in diameter. Polarised Raman data were acquired using a polariser, and no analyser after the sample due to the very low intensity of the signal. This results in the potential observation of all Raman-active symmetries in our spectra.

Data availability statement
Source data from DFT+DMFT is available at: https://github. com/alexandrub53/EnergyLandscapes. All muon spin measurements are available at musruser.psi.ch and can be found using the index in the main source data file.
The data that support the findings of this study are openly available at the following URL/DOI: https://purl.stanford.edu/ hc677dr8144. Grant (FNR/P12/4853155/Kreisel). The Flatiron Institute is a division of the Simons Foundation. The muon experiments were performed at the Swiss Muon Source SµS, Paul Scherrer Institute, Villigen, Switzerland. B M and D T L A acknowledge access to the electron microscopy facilities at the Interdisciplinary Centre for Electron Microscopy (CIME), Ecole Polytechnique Fédérale de Lausanne. REXS measurements were performed at the SEXTANTS beamline of the SOLEIL synchrotron (St. Aubin, France). The authors gratefully acknowledge fruitful discussions with Mael Guennou and Giacomo Mazza.