Jahn-Teller distortion driven magnetic polarons in magnetite

The first known magnetic mineral, magnetite, has unusual properties, which have fascinated mankind for centuries; it undergoes the Verwey transition around 120 K with an abrupt change in structure and electrical conductivity. The mechanism of the Verwey transition, however, remains contentious. Here we use resonant inelastic X-ray scattering over a wide temperature range across the Verwey transition to identify and separate out the magnetic excitations derived from nominal Fe2+ and Fe3+ states. Comparison of the experimental results with crystal-field multiplet calculations shows that the spin–orbital dd excitons of the Fe2+ sites arise from a tetragonal Jahn-Teller active polaronic distortion of the Fe2+O6 octahedra. These low-energy excitations, which get weakened for temperatures above 350 K but persist at least up to 550 K, are distinct from optical excitations and are best explained as magnetic polarons.

. S10       K and 550 K. All spectra were recorded with the incident X-ray energy set to 706 eV.
The RIXS data comprise an average of four runs of experimental results.

RIXS measurements
Using the AGM-AGS spectrometer at beamline 05A1 of the National Synchrotron Radiation Research Center (NSRRC), Taiwan 1 , we measured RIXS of Fe 3 O 4 excited at selected incident photon energies and polarisations. The polarisation of incident X-ray was in the scattering plane or perpendicular to the scattering plane, i.e., π and σ polarisations, respectively. Supplementary Fig. 2(a) illustrates the scattering geometry in which the scattering angle, i.e. the angle between the incident and the scattered X-rays, is denoted as φ, and the incident angle from the ab plane is θ. The photon energy was set to specific energies about the L 3 -edge (2p 3/2 → 3d) x-ray absorption of Fe to distinguish dominantly  Fig. 5(a)). If an effective molecular field H ex = 90 meV is included without the tetragonal distortion, these excitations are split further with the excitation energy centroid at 132 meV, but still the 200 meV feature is not obtained ( Supplementary Fig. 5(b)). We, therefore, need to either increase the effective molecular field to nearly 200 meV, or include the effect of the tetragonal distortion of FeO 6 octahedra. It is, however, unreasonable to use an molecular field much larger than the spin wave energy or the molecular field of Fe 3+ , 90 meV. Hence, we included a tetragonal distortion for calculating the RIXS spectrum of Fe 2+ in magnetite.
We performed a series of RIXS calculations for Fe 2+ with H ex = 90 meV and as a function of tetragonal distortions as shown in Supplementary Fig. 6. A positive ∆ t 2g , i.e. an elongated distortion along the local Jahn-Teller axis and contracted Fe-O bonds in the xy plane, does not yield correct energies of the excitations. For a small tetragonal distortion with −21 meV < ∆ t 2g < 0, the effective exchange field dominates the low-energy excitations and results in an energy centroid ∼ 120 meV. If the ∆ t 2g strength of the tetragonal compression is beyond −21 meV, the excitation profile is broadened dramatically with two major features; their excitation energies and the separation between them gradually increase with the increase of the ∆ t 2g strength. We found that H ex = 90 meV and ∆ t 2g about −24 meV most satisfactorily explain the measured RIXS spectra resulting from Fe 2 + states. The negative value of ∆ t 2g signifies that the energy of d xy is lower than that of d yz /d zx .
With the parameters obtained from RIXS measurements and calculations, we verified if these parameters are consistent with the XAS spectrum. Supplementary Figure 1 presents a comparison between measured XAS spectrum and the calculated spectrum obtained by using the parameters from RIXS results: the crystal field 10Dq = 1.13 eV for octahedral Fe 2+ and Fe 3+ , 10Dq = −0.6 eV for tetrahedral Fe 3+ , and ∆ t 2g = −24 meV.
The calculated XAS agrees will the measured XAS after correction for self absorption.
Supplementary Note 3

Band structure calculations
We performed band structure calculations 5 for the low-temperature P 2/c structure of magnetite using the accurate frozen-core full potential projector augmented wave method, as implemented in the VASP package. The calculations are based on the generalized gradient approximation (GGA). Supplementary Figure 7 shows the calculated band structure to highlight the splitting of the t 2g bands at the Γ point due to the tetragonal distortion in the P 2/c structure. We found that the splitting between the d xy and d yz/zx is 52 meV.

Supplementary Note 4 Single-crystal Synthesis and Characterisation
Single-crystal growth of magnetite was carried out in an infrared image furnace (NEC