DOI: 10.1002/adma.202004490 charge transfer at surfaces [2] or by potentially increasing catalytic selectivity by the localization of minority carriers, [3] and have been predicted to form better transparent conducting oxides (TCOs) than materials that conduct through band transport. [4] But polaron transport is sluggish, and the polaron-induced limitations to charge carrier mobility are well known to the applied science community. [3,5,6] These limitations are especially prevalent in transition metal oxides, which are prone to form smallpolarons (tightly bound) due to the strong electron correlation within the valence d-orbitals. [7] For instance, next-generation p-type TCOs like CuAlO 2 and ZnRh 2 O 4 , which are essential to realizing low-power transparent electronics, have limited performance due to the low mobilities of smallpolarons. [8] The transport of small-polarons in energy-conversion materials have necessitated decreasing diffusion path dimensions to the nanometer length scale [3] and smallpolarons can establish a fundamental limit for the Fermi level in materials like Fe 2 O 3 . [9] Similarly, in small-polaron oxides used as active materials in Li-ion batteries, the low mobility of polarons has been observed to slow down Li diffusion, leading to accumulation of Li-ions at the electrode interface and low battery capacities. [2] The small-polaron hopping model has been used for six decades to rationalize electronic charge transport in oxides. The model was developed for binary oxides, and, despite its significance, its accuracy has not been rigorously tested for higher-order oxides. Here, the small-polaron transport model is tested by using a spinel system with mixed cation oxidation states (Mn x Fe 3-x O 4 ). Using molecular-beam epitaxy (MBE), a series of single crystal Mn x Fe 3-x O 4 thin films with controlled stoichiometry, 0 ≤ x ≤ 2.3, and lattice strain are grown, and the cation site-occupation is determined through X-ray emission spectroscopy (XES). Density functional theory + U analysis shows that charge transport occurs only between like-cations (Fe/Fe or Mn/Mn). The site-occupation data and percolation models show that there are limited stoichiometric ranges for transport along Fe and Mn pathways. Furthermore, due to asymmetric hopping barriers and formation energies, the Oh Mn + + 2 2 polaron is energetically preferred to the Oh Fe + + 2 2 polaron, resulting in an asymmetric contribution of Mn/Mn pathways. All of these findings are not contained in the conventional small-polaron hopping model, highlighting its inadequacy. To correct the model, new parameters in the nearestneighbor hopping equation are introduced to account for percolation, cross-hopping, and polaron-distribution, and it is found that a near-perfect correlation can be made between experiment and theory for the electronic conductivity.

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.202004490.

Despite its importance, there is a gap between empirical studies and theoretical models because the common expression used to model small-polaron transport has not been meticulously tested for ternary and more complex oxides. For instance, ignoring the configurational disorder that can occur in the cation site locations may lead to overestimation of the theoretical mobility values. [10] To quantify small-polaron transport researchers rely on an equation that was developed more than 60 years ago by Heikes and Ure [11] and derived from the Einstein diffusion model and the random-walk model (see Supporting Information for derivation). This conventional model was developed for binaries, for example, spinels of the type M 3 O 4 (M = metal cation), and considers only two critical steps for charge transport to occur: 1) the existence of charge carrier donor/acceptor pairs (or hopping pairs of M n+ /M (n+1)+ and 2) the ability to surmount the activation barrier between hopping pairs. While accurate for binaries, applying this crude model to higher order oxides (e.g., ternaries) with multiple cations that could contribute to donor/acceptor transport might not be valid. For instance, in the binary spinel Fe 3 O 4 , all of the charge-conducting octahedral (O h ) sites are occupied by Fe cations, and charge transport occurs along pathways having an alternating arrangement of Fe 2+ /Fe 3+ (Figure 1a). On replacing an Fe 2+ cation with a Mn 2+ cation, although the donor/acceptor pair arrangement is still present, the charge transport may be affected by the differences introduced by the hopping barriers or different spinstates between the Fe 3+ /Mn 2+ cation pairs. Additionally, charge transport might only occur between like-cation pairs (i.e., Fe 2+ / Fe 3+ and Mn 2+ /Mn 3+ only), necessitating the separate consideration of donor/acceptor pair concentrations for each species, and for the relative concentrations of these pairs to form conductive pathways. These pathway and configurational parameters are not considered in the conventional polaron hopping model. Furthermore, the conventional model has never been rigorously tested for complex oxides lacking long-range order because the characterization methods required to precisely determine cation site-occupation have only recently matured. [12] For this work, we create a well-controlled experimental testbed (Figure 1b) to investigate the conventional polaron hopping model, determine its accuracy, and develop a new /Fe 3+ cations leads to charge flow. Blindly applying this model to ternary spinels might not be valid; the presence of another cation-type at O h sites (for example, replacing Fe 2+ with Mn 2+ ), may affect the charge transport pathway since the charge carrier may or may not cross-hop between Fe/Mn cations. This condition is not considered in the conventional polaron hopping model. b) Schematic illustration of the experimental plan to determine the role of cation site occupation and charge transport pathways in controlling transport properties in ternary spinels. Mn x Fe 3-x O 4 thin films are grown using molecular-beam epitaxy. The films are characterized using various X-ray techniques followed by temperature-dependent conductivity measurements. DFT+U analysis is carried out to correlate the composition, electronic, and structural properties to the observed transport behavior.

model that incorporates parameters that play a critical role in controlling charge transport. To produce a tightly defined sample set we employ molecular-beam epitaxy (MBE) to grow a stoichiometric series of epitaxial thin films of the spinel Mn x Fe 3-x O 4 . We choose this ternary spinel system for two reasons. First, despite the large miscibility gap in the bulk phase diagram, [13] it is possible to synthesize this system as a single phase over a wide range of "x" under similar conditions using MBE. Maintaining similar growth conditions is important because the spinels have been shown to have processing-condition-dependent cation siteoccupation and electronic and electrochemical properties. [14][15][16] MBE-grown films overcome consistency challenges in sample synthesis that have obfuscated past results in this area. These challenges include accessing metastable phases of alloys not accessible through bulk techniques, phase-purity, achieving single-crystals since the presence of grain boundaries impede the charge flow and make it difficult to study the intrinsic material properties, and strain engineering to prevent Jahn-Teller distortions which gives rise to an abrupt transition of the unit cell from cubic to tetragonal with increasing Mn content, [14,17,18] see Supporting Information for more details. Second, from our previous work [19] we learned that 3d ternary spinels can have active and inactive cation species at the spinel octahedral sites, which are the location of the donor-acceptor transport pairs. Both Fe and Mn cations have mixed oxidation states at O h sites, allowing them each to be active in the transport process, and enabling us to directly test the veracity of the conventional polaron hopping model. To precisely determine the cation-site-occupation and oxidation-state concentrations in our Mn x Fe 3-x O 4 thin films, we use our recently developed method [12] of K β X-ray emission spectroscopy (XES). XES data gives complete information for each species at each site and can be coupled with simpler techniques such as X-ray photoemission spectroscopy (XPS) to verify the results and generalize the technique. Finally, using density functional theory + U (DFT+U), we calculate the electronic structures and theoretical activation barriers for polaron transport. From our results we find that the conventional model is inaccurate for our ternary spinel system: charge transport only occurs through like-element pathways (decoupled pathways), without cross-hopping between different elements. We further show that asymmetric hopping barriers are present between cross-hopping pairs and that the

polaron, indicating that charge transport occurs preferentially along the Mn/Mn pathways over the Fe/Fe pathways. To account for these observations, we introduce a percolation parameter, a polaron distribution parameter, and a cross-hopping parameter to the conventional electronic conductivity equation that correct the model for higher order spinels.

Our new model provides a platform for optimizing mobility and charge transport, and hence opens new avenues for designing high-performance TCOs and batteries.

The Mn x Fe 3-x O 4 thin films are grown on MgO (001) substrates using MBE. Since XPS probes only the surface (top ≈5-10 nm), [20] XES, with a probe depth of few micrometers, is employed to characterize the average composition through the complete depth of the films (≈50-60 nm, determined from X-ray reflectivity scans). Only minor deviation with a mean squared error ≈2% is observed in the "x" values between the two techniques (Figure S3c, Supporting Information) indicating that the films have a homogeneous composition (i.e., no composition gradient). For all further analyses, the "x" values obtained using XPS are used. The high quality of our thin films is confirmed using a suite of characterization techniques. In situ reflection high-energy electron diffraction (RHEED) pattern shows sharp diffraction streaks and Kikuchi lines for x ≤ 2.30 samples, indicating epitaxial growth and a smooth surface (Figure 2a and Figure S4, Supporting Information). [21,22] Well-defined Kiessig fringes observed in the X-ray reflectivity measurements further confirm a smooth surface morphology and an abrupt film-substrate interface (Figure S5a, Supporting Information). Coupled θ-2θ X-ray diffraction (XRD) patterns (Figure 2b) show only a 004 reflection from the samples that shifts to lower 2θ values with increasing Mn content, similar to peak position trends in bulk Mn x Fe 3-x O 4 (Figure S5b, Supporting Information). No peaks corresponding to monoxides are observed, confirming the films to be phase pure. For x ≤ 2.30, distinct film peaks and thickness fringes indicate the films are single phase, and crystalline with a well-defined substrate-film interface. For x ≥ 2.52 (not shown), broad peaks are observed indicating poor crystallinity, and are thus excluded from further analysis. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) reveals the thin films are epitaxial and show characteristic spinel structure (overlaid with a schematic of the spinel unit cell) (Figure 2c and Figure S6, Supporting Information). At the interface, a few monolayers of pseudomorphic rock-salt growth is observed. Figure 2c insets show two different spinel domains separated by an antiphase boundary (APB) that is associated with the mismatch of symmetry between the spinel films (space group Fd3m) and the rock-salt MgO substrate (space group Fm3m). [20] In particular, the blue inset shows the characteristic spinel structure, while the green inset shows a pattern resulting from a spinel structure with APBs present along the zone axis. An accumulation of Fe in the film near the Mn x Fe 3-x O 4 /MgO interface is detected from elemental maps from electron energy loss spectroscopy, extending ≈2 nm into the film, beyond which Mn and Fe cations show uniform distributions in the lattice (Figure S6, Supporting Information).

The film thicknesses are optimized to study intrinsic transport behaviors without strain relaxation (Figure S7a, Supporting Information). The out-of-plane 004 Bragg peaks, fitted with a Gaussian function to extract the lattice parameter, increases linearly with increasing Mn content (Figure S5b, Supporting Information). The samples' peak positions deviate from bulk reference standard peaks, implying that the lattice is strained. To further confirm the commensurately strained growth and extract the lattice parameters, reciprocal space maps (RSM) are collected around the 22 6 reflections of the Mn x Fe 3-x O 4 thin films (Figure 2d and Figure S8, Supporting Information). RSM shows that the in-plane (IP) peak position (Q ∥ ) overlaps with the MgO substrate (highlighted with a dashed line in Figure 2d), whereas the out-of-plane (OOP) peak position (Q ⊥ ) deviates away from the substrate peak (Δ = 0.17 and 0.32 Å for x = 1.14 and x = 2.30, respectively), indicating that the film has the same in-plane lattice parameter as the substrate but an increasing out-of-plane lattice parameter with increasing "x". The in-plane lattice parameter (Figure 2e) extracted from the RSM remains constant at ≈8.43 Å and as expected, it is twice as large as the underlying MgO substrate (4.217 Å). The out-of-plane lattice parameter varies linearly from 8.3 to 8.7 Å x" ranges between 0 and 2.30 and is determined from XPS (verified using XES). The intense peak at ≈43° (indicated by *) corresponds to the 002 Bragg peak of the MgO substrate. The 004 Bragg peak is from the Mn x Fe 3-x O 4 samples. c) Cross-sectional HAADF-STEM image of x = 1.62 thin film showing the characteristic spinel structure. The green and blue insets show two different spinel domains separated by anti-phase boundaries resulting from the structural inequivalence between the film and the substrate. The insets are overlaid with spinel unit cell schematics, where cations (Fe/Mn) are highlighted as yellow spheres and the oxygen atoms as red spheres. d) RSM around the 2 2 6 reflection of the x = 1.14 (top) and x = 2.30 (bottom) thin films grown on MgO substrate. The in-plane (IP) peak position overlaps with the MgO substrate, whereas the OOP peak position (y-axis) deviates away from the substrate peak with increasing "x". e) IP and the OOP lattice parameters calculated from 2 2 6 reflections in RSMs. The OOP lattice parameter (black) increases linearly with "x", whereas the IP lattice parameter (blue) does not change, indicating that the films are coherently strained to the underlying MgO (001) substrates. f) Comparison of unit cell anisotropy extracted from RSM (black) and STEM (red) data. The unit cell anisotropy is calculated as the ratio of OOP and IP lattice parameters (a oop /a ip ). g) Tensile strain mapping of x = 1.62 thin film calculated by the gradient of the phase of the OOP 004 and IP 400 Bragg peaks. The clear distinction between the film and substrate present in the OOP maps, but not the IP maps confirms that the film lattice is tetragonal. The dashed line highlights the film-substrate interface. h) Site-occupation data for Mn cations (top) and Fe cations (bottom), extracted from the Mn-edge and Fe-edge K β-1,3 (3p→1s) XES spectra. For Mn x Fe 3-x O 4 thin films, the degree of inversion "i" equals the fraction of T d sites occupied by Fe 3+ (dark green).

(Figure 2e), and it deviates from the lattice parameters of the bulk relaxed unit cell. [15] Tensile strain maps (Figure 2g and Figure S9, Supporting Information) are calculated from HAADF-STEM data by determining the gradient of the phase for the out-of-plane 004 and in-plane 400 Bragg peaks. The out-of-plane tensile strain maps show a clear distinction between the film and substrates but not in the in-plane direction. This indicates the lattice in the films is tetragonally distorted. Additionally, the increase in average tensile strain with increasing "x" indicates that the magnitude of tetragonal distortion increases with the increased Mn content. The unit cell anisotropy quantified by calculating the ratio of outof-plane and in-plane lattice parameters (a oop /a ip ) increases in the range of 0.995 to 1.036 with increasing Mn content and is consistent with the values extracted using RSM (Figure 2f).

Using XES, we determine that the cation site-occupation changes dramatically with stoichiometry, inverting the spinel structure and affecting the types of donor/acceptor pairs (Figure 2h and Figure S10, Supporting Information). To extract the cation site-occupation, we employ the K β XES because of its sensitivity to the oxidation state, spin state, and ligand geometry of the cations. [12,23,24] We have shown that XES is more accurate than conventional techniques such as X-ray absorption spectroscopy. [12,19] The site-occupation of Mn x Fe 3-x O 4 can be summarized as

, where () and [] represent tetrahedral (T d ) and octahedral (O h ) sites, respectively, a + b + c + d = 2, and 0 ≤ i ≤ 1. From x = 0 to x ≈ 1.20 the Mn x Fe 3-x O 4 films have partially inverse spinel structure, where the degree of inversion "i" is defined as the fraction of M 3+ cations at T d sites (where M = Fe, Mn). [25] Since only Fe cations are present as 3+ at T d sites, "i" simply equals the fraction of T d sites occupied by

T + Fe 3 d cations. The Fe 3 O 4 sample is entirely inverse as all T d sites are occupied by Fe 3+ cations. On increasing the Mn content, the Mn 2+ cations partially replace the Fe 3+ cations at T d sites, resulting in a decrease in the degree of inversion toward a normal spinel arrangement (Figure 2h, dark green line). At O h sites, on increasing the Mn content, the + Fe O 2 h and + Fe O 3 h species are replaced by + Mn O 2 h and + Mn O 3 h . More importantly, we confirm the presence of multiple types of potential electron donor/ acceptor pairs at O h sites: Fe 2+ /Fe 3+ , Mn 2+ /Mn 3+ , and crosshopping pairs of Mn/Fe. The extracted site-occupation data provides the number of donor/acceptor pairs of each type that is used in the analysis that follows to directly test the polaron hoping model. Since the O h sites are primarily considered as the pathways for polaron hopping, [17,19,26] we are particularly interested in the number of hopping pairs at O h sites. Previously, the charge transport in bulk Mn x Fe 3-x O 4 synthesized at high temperature has been associated with Mn 3+ /Mn 4+ pairs, [14] however, since spinels show synthesis temperature-dependent site-occupation, we do not observe Mn 4+ in our samples (possibly due to low oxidizing power of molecular O 2 at low pressure in MBE growth), hence we exclude the possibility of Mn 3+ / Mn 4+ pairs contributing to charge transport.

Temperature-dependent electronic measurements show nearest-neighbor small-polaron hopping as the charge transport mechanism and indicate a transition in active species with increasing Mn content. The electronic conductivity "σ" at 325 K (Figure 3a) shows a gradual decrease up to x ≈ 1.20, beyond which a significant drop is observed. Overall, a dramatic 10-order of magnitude change in electronic conductivity is observed between the conductive Fe 3 O 4 phase and the insulating Mn 3 O 4 phase. The electronic charge transport in semiconducting materials is typically driven by a thermally activated small-polaron hopping mechanism, [27] derived from the general charge transport equation: [11] T T

where σ 0 is the pre-exponential term and corresponds to the concentration of donor/acceptor pairs discussed in detail later, T is temperature, E A is the activation energy, k B is Boltzmann's constant, and α and p are constants. The type of charge hopping is determined from the power-law dependence (p) in the www.advmat.de www.advancedsciencenews.com electronic conductivity equation, with p = α = 1 for nearestneighbor hopping (NNH) and p α = = 2 0.5 (or 0.25) for variable range hopping (VRH). [28] To determine the mechanism of electronic charge transport, the transport is written in terms of conductivity as a function of inverse temperature,

and p is extracted by fitting the Arrhenius curves (Figure 3b). Based on the least-squares fitting of the Arrhenius curves, the transport mechanism is NNH (i.e., p = α = 1) throughout the entire "x" range at the measured temperature ranges. The preexponential term shows an inflection point with a local minimum at x = 1.20 (Figure 3c, red). The activation energy varies in the range of 0.07 to 1.40 eV (Figure 3c, blue), consistent with the range observed previously for bulk Mn x Fe 3-x O 4 samples. [14] The activation energy increases gradually in the range of x = 0 to 1.20, beyond which it increases sharply. The increase in activation energy in Mn-containing spinels has been previously associated with the tetragonal distortion of the unit cell (see Supporting Information for details). [14,18] Small-polaron charge transport can be affected by the presence of oxygen vacancies, however, since the oxygen vacancy concentration is found to have a negligible change with varying stoichiometry (see Supporting Information for details), we conclude that the oxygen vacancies do not affect charge transport as a function of cation composition. Since σ 0 and E A are a measure of the concentration of donor/acceptor pairs in the system (discussed in detail later) and the barrier required for the charge carrier to hop between these pairs, respectively, the stoichiometry-dependent trends of σ 0 and E A provide insight on the active species participating in the electronic charge transport. The various possibilities of donor/acceptor pairs include Fe 2+ /Fe 3+ , Mn 2+ /Mn 3+ , Fe 2+ /Mn 3+ , and Mn 2+ /Fe 3+ . Both the σ 0 and E A trends (Figure 3c) show a transition at x ≈ 1.20 indicating this stoichiometry is a crossover point between different types of transport behavior. For low "x" samples the O h sites are predominantly occupied by Fe cations, and for high "x" samples they are predominantly occupied by Mn cations. This leads to our hypothesis that the charge transport transitions from predominantly Fe 2+ /Fe 3+ couples to predominantly Mn 2+ /Mn 3+ couples as the Mn content in the sample increases (x increases). This simplistic analysis does not, however, address the contribution of cross-element hopping (Mn to Fe) versus like-element hopping (Fe to Fe or Mn to Mn, only), or, in the case of like-element hopping, insight into which elemental path is active. This information is especially important in the mid "x" range where both Mn and Fe cations are equally distributed at O h sites.

Using DFT+U, we calculate the total energies and oxidation states in epitaxially-strained normal and inverse Mn x Fe 3-x O 4 and find that the 2+ cations' presence in the samples results in an n-type behavior, that is, electrons as the charge carrier type (see Figure S12 and Table S1, Supporting Information, and accompanying discussion). Analysis of the spin-resolved projected density of states (PDOS) of the strained Mn x Fe 3-x O 4 cells, calculated using DFT+U, show that the unoccupied tetrahedral Mn and Fe states, in the normal and inverse spinel, respectively, are located significantly above the conduction band minimum, indicating their lack of contribution for electronic conduction. This justifies the assumption that only O h sites participate in polaron hopping, in agreement with previous reports. [17,19,26] Further analysis of the PDOS indicates that conduction occurs predominantly between like-cations (e.g., Fe to Fe) and not as cross-element hopping (see Figures S13 and S14, Supporting Information, and accompanying discussion). The charge transport being restricted to only like-elemental pathways has major implications when applying the conventional polaron hopping model. The conventional model was developed for binary spinels, and since all O h sites in binary spinels are occupied by the same cation-type, the condition necessary for hopping to occur is the formation of a donor/acceptor pair. Based on this model, replacing one Fe 2+ with Mn 2+ cation still meets the criteria for charge transport, and charge flow should be unchanged from the pure Fe 3 O 4 case. As shown by our analysis of Mn x Fe 3-x O 4 , however, charge cannot hop between Mn and Fe cations. This creates a requirement for a contiguous elemental path and leads to an additional condition for charge transport to occur: separate, decoupled percolation networks need to be formed by both Fe and Mn cations, and within each elemental percolation network a M n+ /M (n+1)+ arrangement needs to be present. The charge transport behavior in Mn x Fe 3-x O 4 is summarized in the schematic shown in Figure 4a: at low Mn content, Fe/Fe percolation pathways impart the charge transport, but some of the Fe percolation pathways are broken by Mn cations and they become inactive. On increasing the Mn content, however, there are sufficient Mn cations at O h sites to form separate percolation pathways and charge can conduct through the Mn/Mn pathways as well.

Percolation can only occur when enough components align in a contiguous pathway. We, therefore, determine the percolation threshold limit, that is, the minimum fraction of a cation-type to be present at O h sites to form a separate percolation pathway. Based on the percolation theory developed for 2-phase composites, [29] the fractional occupancy of a phase in the composite (referred to as the percolation value "P") should be higher than the threshold percolation value to form a contiguous pathway. McLachlan et al. determined that in a 2-phase composite with each site having six nearest-neighbors, a 3D percolation network is formed for either phase when its fractional occupancy is greater than the threshold value of 0.311. [29] An analogous case can be considered for the nearest-neighbor polaron hopping in a spinel structure where a charge carrier localized at the central O h site can hop to one of the six surrounding edge-sharing O h sites (Figure S7b, Supporting Information). Therefore, in Mn x Fe 3-x O 4 , a 3D percolation network for charge transport is formed when the percolation value P j ≥ 0.311, that is, when at least 31.1% of the O h sites are occupied by cation type "j" (where j = Mn, Fe). Here,

and P Fe = 1 -P Mn . The percolation values P Fe and P Mn are determined for the Mn x Fe 3-x O 4 system from the XES site-occupation data and plotted as a function of "x" (Figure 4b). The Fe and Mn active ranges (i.e., P Fe, Mn ≥ 0.311) are 0 ≤ x ≤ 1.91 and 1.35 ≤ x ≤ 3.00, respectively. Based on this percolation model, the stoichiometry range ("x" range) is divided into 3 regions: I)

35, where only Fe 2+ /Fe 3+ hopping pairs are active; II) 1.35 ≤ x ≤ 1.91, where both Fe 2+ /Fe 3+ and Mn 2+ /Mn 3+ hopping pairs are active; and III) 1.91 ≤ x ≤ 3.00, where only Mn 2+ / Mn 3+ hopping pairs are active. Since the conventional polaron hopping model does not consider any percolation limitations, Fe 2+ /Fe 3+ and Mn 2+ /Mn 3+ hopping pairs are both treated as active over the full stoichiometry ("x") range, which is obviously incorrect to ternary spinels, highlighting the inadequacy of this widely-used model. Using DFT+U to generate polaron structures, we find that the + Mn O 2 h polaron is energetically preferred to the + Fe O 2 h polaron across the stoichiometric range tested. Based on the a) Since the charge carriers cannot hop between Mn/Fe O h sites (see DFT+U analysis discussed in main text), only Fe/Fe percolation paths are present for charge transfer in Fe-rich samples (left). The presence of multiple cation-types at O h sites breaks some of the percolation paths (center). On increasing Mn cations at O h sites beyond the percolation threshold, enough Mn cations are present at O h sites to form a separate percolation path. b) The percolation value plotted as a function of "x". In percolation theory, for sites having six neighboring hopping locations, a 3D network for charge transport can only form when percolation values P j ≥ 0.311. Here, P j (where j = Mn, Fe) is defined as the relative concentration of cation-type "j" at O h sites. Based on the percolation model, the "x" range is divided into three regions: I) 0 ≤ x ≤ 1.35, where only Fe 2+ /Fe 3+ hopping pairs are active; II) 1.35 ≤ x ≤ 1.91, where both Fe 2+ /Fe 3+ and Mn 2+ /Mn 3+ hopping pairs are active; and III) 1.91 ≤ x ≤ 3.00, where only Mn 2+ /Mn 3+ hopping pairs are active (right). c) Octahedrally coordinated Fe 3+ (top) and Fe 2+ (bottom) in Mn 1.75 Fe 1.25 O 4 . The additional electron in Fe 2+ localizes in a 3d state, shown as an isosurface of the charge difference between the Fe 2+ and Fe 3+ charge densities around the cation. The addition of an electron leads to an elongation of cation-anion bond lengths and breaks the site's local symmetry. d) DFT+U calculation results used to generate Marcus theory curves for NNH between the hopping pairs in Mn 1.75 Fe 1.25 O 4 . Calculations of Fe/Fe hopping result in non-adiabatic curves, while for Mn/Mn hopping, adiabatic curves are obtained. For comparison between the Fe and Mn results, the non-adiabatic lines (dashed) were extrapolated for Mn/Mn transport from the adiabatic curves. e) Experimental (black) and theoretical (blue and green) activation energy values. The theoretical activation barriers for Fe/Mn and Mn/Fe hopping pairs are excluded from the plot, due to the lower likelihood of such events. HSE06 is used to calculate the Mn/Mn barrier for x ≥ 2 samples. The experimental activation energy for 0 ≤ x ≤ 1.35 is consistent with the theoretical Fe/Fe hopping barrier. For x > 1.35, a sharp increase in experimental activation energy is consistent with the theoretical Mn/Mn hopping barrier trends. Note: cation size is enlarged in (a,c) to highlight the charge transfer pathways.

experimental observations of + Mn O 2 h and + Fe O 2 h site-occupations in the O h sublattice, electron polaron geometries are formed by localizing an excess electron on octahedral sites in the strained Mn x Fe 3-x O 4 cells. The excess electron localizes in a 3d state on the metal, leading to an elongation of MO (M = Fe/Mn) bond lengths and altering the site's local symmetry, as depicted in Figure 4c for a + Fe O 2 h polaron in Mn 1.75 Fe 1.25 O 4 . The localized charges exhibit changes in magnetization and effective charge that are consistent with a change of oxidation state from 3+ to 2+ (Table S2, Supporting Information). The resulting elongations of FeO and MnO bond lengths for the polaron structures show increased elongation as the Mn content is raised and the cells become more tetragonally distorted, with the + Mn O 2 h polarons showing a more pronounced elongation than the + Fe O 2 h polarons (Table S3, Supporting Information). The total free energies of cells with a + Mn O 2 h polaron are consistently lower than for cells with a + Fe O 2 h polaron (Table S3, Supporting Information), implying that the preferred charge localization is on the Mn cation. The preferred localization on + Mn O 2 h suggests that transport from a + Mn O 2 h polaron to a + Fe O 2 h

polaron requires a large driving force. Indeed the difference in free energy between the states is large: on average 0.497 eV for 1.25 ≤ x ≤ 2.5. This large energy barrier that prevents asymmetric cross-elemental hopping supports our hypothesis on separation of hopping pathways between elements.

By examining the role of the stoichiometry on the activation barrier using DFT+U and hybrid functional calculations using the Heyd-Scuseria-Ernzerhof (HSE06) functional (Figure S15, Supporting Information), we find that Fe/Fe barriers are nearly constant (≈0.2-0.3 eV) for all stoichiometries tested, while Mn/Mn barriers show a general increase with stoichiometry (≈0.3-0.6 eV). Fe 2+ /Fe 3+ transport calculations show characteristic non-adiabatic (i.e., tunneling) Marcus theory [30,31] curves, where the charge remains localized at one of the endpoints throughout the transport process, and the barrier is defined as the intersection of the two quadratic energy curves (Figure 4d, bottom, blue). For Mn 2+ /Mn 3+ , the generated curves are adiabatic (i.e., hopping), with the charge being spread out over both endpoints during the transport process, resulting in a smooth energy surface and an activation barrier that lies below the non-adiabatic barrier (Figure 4d, bottom, green). Due to the different nature of the barriers for Fe 2+ /Fe 3+ and Mn 2+ /Mn 3+ , a direct comparison is not possible. In order to facilitate a comparison between the Fe 2+ /Fe 3+ and Mn 2+ /Mn 3+ barriers, we extrapolate the non-adiabatic barriers for Mn 2+ /Mn 3+ transport, as detailed in the Supporting Information (Figure 4d, bottom, green dashed line). We find that the Fe 2+ /Fe 3+ barriers are consistently smaller than the extrapolated Mn 2+ /Mn 3+ barriers for all stoichiometries examined. Transport between mixed pairs shows a low activation barrier for Fe 2+ /Mn 3+ , but a markedly high barrier for Mn 2+ /Fe 3+ (Figure 4d, top; Figures S16 and S17a, Supporting Information), due to the difference in the free energies of the Fe 2+ and Mn 2+ polarons states. The implication of this asymmetric barrier is that charges transported through a mixed-elemental pathway could get "trapped," with the Fe/Mn pair acting as a one-way valve (Fe to Mn only). In region III (x ≥ 2), the theoretical activation barriers calculated using DFT+U underestimate the experimental values (Figure S17b, Supporting Information), possibly due to an inadequate description by DFT+U of charge localization effects in cells with large Mn content. Previous theoretical studies have shown that hybrid functionals are better suited for generating the electronic structure of Mn 3 O 4 . [32,33] When HSE06 hybrid functional calculations are used the resulting activation barriers (Figure S15, Supporting Information) perform better than the barriers obtained using DFT+U in predicting the experimental trends (Figure 4e).

We find that charge transport preferentially occurs along the Mn/Mn pathways over the Fe/Fe pathways in the region where both pathway-types are active (region II) for the following reasons. First, since Mn 2+ polaron states are more energetically favorable than Fe 2+ polarons, by estimating the rate constants for the charge transfer reaction (i.e., Fe/Mn transport) at thermal equilibrium we find that the Mn polaron population exceeds Fe polarons by at least five orders of magnitude (see Supporting Information). Second, if a charge transfers from the Fe/Fe path to the Mn/Mn path, it is likely to be get "trapped" on the Mn/ Mn path, since there is a large energy barrier preventing it from returning to an Fe polaron (Figure 4d, top and Figure S16, Supporting Information). Third, the PDOS plots (Figures S13 and S14, Supporting Information) show that Mn states are preferred energetically for charge carriers because the majority (or main peak) of Mn states in the conduction band is lower than the main peak of Fe states. This means, simply, that the Mn states are more readily available for conduction electrons. Fourth, the correspondence between the calculated and measured activation energies (Figure 4e, green and black points) indicates that the activation energy from the Mn/Mn pairs dominate the trends in region II, which implies that although Fe percolation pathways are present in this region, the charge transport predominantly occurs through the Mn pathways.

We test the accuracy of the conventional model's pre-factor and find a poor fit to experimental data. In the conventional NNH polaron model (Equation ( 2)), the pre-exponential term in the electronic conductivity, σ 0 is [26] gNc c e a k σ ν ( )

where g is the geometric factor, N is the total density of charge carriers at O h sites, a is the hopping distance between donor/ acceptor sites, e is the electron charge, ν is the acoustic phonon frequency, and c(1c) is the probability of forming a donor/ acceptor pair. In the conventional model, the probability of forming a hopping pair is solely determined based on the total concentration of 2+ and 3+ cations at O h sites. Hence, the mixed donor/acceptor pairs include Fe 2+ /Fe 3+ , Mn 2+ /Mn 3+ , Fe 2+ /Mn 3+ , and Mn 2+ /Fe 3+ and c = + + + +

The hopping frequency ν for Mn 2+ /Mn 3+ pairs and for Fe 2+ / Fe 3+ pairs is 50 × 10 13 Hz and 8 × 10 13 Hz, respectively. [14,34,35] For comparison purposes we can assume the parameters g, N, and a are independent or have insignificant change with "x", thus the only parameters affected by stoichiometry for the preexponential term are the donor/acceptor concentrations. For the mixed-elemental hopping, donor/acceptor concentrations are scaled with the hopping frequency where the frequency is assumed to be limited by the lower value, that is, 8 × 10 13 Hz.

The pre-exponential terms, ln

show stoichiometric-independent behavior for low Mn content, followed by a sharp decrease (Figure 5a, red). Comparing this σ 0 to our experimentally determined pre-exponential factor, σ 0,exp , (see Figure 3c and Equation ( 2)) we find an extremely poor correlation (Figure 5a), indicating that the conventional polaron hopping model is insufficient for describing transport in this ternary spinel. We have previously found the conventional polaron hopping model is inaccurate for the Co-Mn-O ternary spinel, [19] and we believe this is generally the case for most ternary spinel systems. On considering the hypothetical case of like-element-only hopping, that is, decoupled pathways but without the percolation limits in the conventional model, the overlap of σ 0 with σ 0,exp improves slightly, however, the correlation is still poor (Figure S18, Supporting Information).

To correct the model, we consider like-element-only hopping, a polaron distribution factor, and a percolation parameter. The percolation parameter (γ ) considers a network to be "active" only when a percolation pathway is formed (P ≥ P threshold ). Therefore, γ = 1 when P ≥ P threshold and γ = 0 when P < P threshold , where P is determined using XES; and the new pre-exponential term σ ′ 0 is now defined as gNc c e a k σ γ σ γ ν ( )

Based on this model, and because we know there is no crosshopping between Fe and Mn, the Fe 2+ /Fe 3+ hopping pairs (Figure 4b, blue) are only active (γ = 1) in regions I and II, and Mn 2+ /Mn 3+ hopping pairs (Figure 4b, green) are only active in regions II and III. This expression (Equation ( 4)) is significantly different from the conventional model in which both Fe and Mn are considered active throughout the entire "x" range (γ = 1 always) and all mixed-hopping is allowed (both likeelemental and cross-elemental hopping). The Fe 2+ /Fe 3+ hopping pair concentration (Figure 5a, blue) shows a gradual decrease through "x" while the Mn 2+ /Mn 3+ hopping pair concentration (Figure 5a, green) shows a gradual increase with increasing Mn content. Overall, a strong correlation is seen between the concentration of hopping pairs (σ ′ 0 ) and σ 0,exp (Figure 5a), confirming our hypotheses on the prohibition of cross-elemental hopping, the importance of percolation networks for like-elementonly transport, and the change in these networks with varying stoichiometry. From the Pearson correlation function, used to quantify the monotonic correlation between datasets, [36] (a coefficient of ±1 and 0 implies ideal correlation and no correlation, respectively) we find the conventional transport model shows a low Pearson correlation coefficient of -0.11 for cross-elemental hopping (versus 0.95 for the new model), which confirms the limitation of the conventional model and the precision of our augmented model for ternary or higher-order spinels.

Based on the new model, we present a generalized equation of conductivity for nearest-neighbor small-polaron hopping and find a strong correlation between the model-predicted and experimental electronic conductivity values. The new general NNH transport equation, which considers the cases of mixedhopping and like-elemental-only hopping can be written as

where χ = 1 or 0; χ = 1(0) indicates mixed-element (like-elementonly) hopping is active. γ M = 1 or 0; γ M = 1(0) indicates a percolation network formed (not formed) by cation-type "M". In addition,

, where

, where M = Fe, Mn. Here, γ M is the percolation term, χ is the mixed-hopping term, N M is the charge carrier density along a pathway-type, and M is cation element type. The mixed-elemental (Fe/Fe, Fe/Mn, Mn/Fe, and Mn/Mn) hopping case (χ = 1) reduces this equation to the conventional hopping equation which considers: 1) a critical hopping barrier E A,mixed , and 2) a percolation path being always present since the hopping can occur between any donor/acceptor pair. On the contrary, when like-element-only (Fe/Fe and Mn/Mn) hopping is active (χ = 0), conduction only occurs along decoupled pathways, and since the Fe/Fe and Mn/Mn hopping barriers are different, the total conductivity can be written as the sum of conductivities along each pathway type. Additionally, as discussed above, the decoupled pathways are active only when P ≥ 0.311, hence the percolation parameter "γ " is included in the like-element-only hopping case. For the case where multiple pathway-types are active, the charge carrier density is distributed along the different pathways. This distribution of carriers can be excessively asymmetric, as evidenced in our Mn x Fe 3-x O 4 films where the Mn polaron population exceeds the Fe polaron population by five orders of magnitude. This asymmetry has substantial implications: the charge conduction is dominated by Mn-pathways. To account for this distribution, a polaron distribution parameter (f M ) is included in the pre-exponential term and the charge carrier density along a pathway-type (N M ) is given as:

Here, f M is the fraction of carriers along a pathway

and can be calculated computationally by estimating the rate constants for hopping (see Supporting Information). We test the accuracy of the new generalized NNH transport equation for the case of our Mn x Fe 3-x O 4 thin films by comparing the electronic conductivity calculated using the new model (ln σ) to the experimentally determined electronic conductivity (ln σ exp ) at a fixed temperature (325 K) (Figure 5b). For the case of our Mn x Fe 3-x O 4 thin films, we have like-element-only hopping (χ = 0) over all regions, and γ Fe = 1 in region I and II, while γ Mn = 1 in regions II and III. Since the + Mn O 2 h polaron is overwhelmingly preferred energetically, f Mn = 1 in region II (see Table S4, Supporting Information for all parameter values). To calculate the electronic conductivity, the donor/acceptor pairs determined from XES (Figure 5a) are combined with activation energy values determined from the four-point-probe measurements (Figure 3c, blue). An excellent overlap with the experimental electronic conductivity (Figure 5b, black) and a Pearson correlation coefficient of 0.97 substantiates the accuracy of the new generalized NNH transport equation in modeling the charge transport in ternary oxides. Equally, the theoretically determined E A values (Figure 4e) can be used to calculate the electronic conductivity (using Equation ( 5)) leading to a similarly excellent overlap with the experimental values (Figure S18b, Supporting Information). In contrast, the conventional model provides no information on the source of the activation barrier, and if the energy barriers were calculated from first principles it is not clear how to apply them to the conventional model. Intuitively, on assuming that charge conduction occurs along the pathways with the lowest activation barrier (E A,Fe/Fe for x < 1.5 and E A,Fe/Mn for x ≥ 1.5, Figure S17a, Supporting Information), an extremely poor overlap is observed between the conventional model and experimental electronic conductivity (Figure S18c, Supporting Information).

Based on the findings in this work, we can recommend design principles for incorporation of spinels in devices. We hypothesize that by identifying two suitable parent binary oxides that exhibit good band alignment (unlike the case for Mn 3 O 4 and Fe 3 O 4 ) it should be possible to design a ternary spinel with higher mobility through symmetric cross-element hopping. In practice, determination of band edge alignments can be performed theoretically or experimentally. Theoretically, the valence band maximum (VBM) and conduction band minimum can be estimated by finding the band gap center and subtracting (or adding) half of the materials bandgap. [37] Experimentally, the valence band edge alignments can be obtained using XPS, [38] by comparing the oxide's core level and VBM spectra. The experimental approach is widely used for determining the band edge alignments for interfaces and junctions. [39] For applications, since supercapacitors involve redox reactions which require movement of electrons through the active material, increased conductivity can lead to faster extraction of electrons and hence enhanced power density. In electrocatalysts at low overpotentials, only the first few monolayers in direct contact of the conductive supports have been observed to be active due to the electronic conductivity limitations. [40] Increased conductivity could thus enhance catalytic activity. Similarly, in batteries, low specific capacity has been associated with the limited charge transport through the active material to the electrolyte interface. This limitation can define a critical film thickness, beyond which the active material does not contribute to charge storage. [41] Conversely, the lack of cross-hopping could also be advantageous for applications. By identifying parent oxides with weak band alignment, systems with decoupled pathways can be leveraged to incorporate mixed properties in a system. For instance, magnetoelectric properties of a material can be enhanced by optimizing the dopant type and concentration in a base matrix of Fe 3 O 4 . The properties of the dopant can be leveraged such that the magnetic properties are controlled by the dopant while the transport occurs through highly conductive Fe pathways.

In summary, using Mn x Fe 3-x O 4 as a model system, we systematically highlight the shortcomings of the conventional polaron hopping model and present a new model for charge transport in higher order oxides. We overcome the major constraints restricting the study of ternary spinels by using MBE: we grow single-crystal, compositionally-uniform, and phasepure Mn x Fe 3-x O 4 thin films on MgO (001) under similar growth conditions and over a wide range of "x". Charge transport measurements show a stoichiometry-dependent behavior which cannot be explained based on the conventional nearest-neighbor polaron hopping model. Our investigations reveal that in Mn x Fe 3-x O 4 , the polaron hopping only occurs between like-elements (Fe to Fe or Mn to Mn), and as a result, the charge transport occurs along decoupled pathways. Using the site-occupation data and a percolation model, we determine that the transport along Fe and Mn pathways is limited in the range 0 ≤ x ≤ 1.91 and 1.35 ≤ x ≤ 3.00, respectively. DFT+U calculations show that Mn polaron formation is more favorable energetically and that charge transport between mixed cation pairs has a large activation energy, reinforcing the notion of the dominance of the Mn/Mn pathway as the Mn content increases. The transport calculation results are further supported by the energetic separation of Mn and Fe bands in the projected density of states. Based on these findings, we introduce a new percolation parameter, a cross-hopping parameter, and a polaron distribution parameter in the nearest-neighbor hopping equation. These parameters are not considered in the conventional model, as emphasized by the poor correlation between the predicted and the experimentally determined electronic conductivity trends. In contrast, our new model shows an excellent overlap with the experimental trends, confirming the role of percolation pathways and cross-hopping in describing the charge transport in ternary spinels. Our findings provide a pragmatic tool for designing non-equilibrium structures of spinel oxides with optimized parameters to incorporate spinels in various devices such as batteries and supercapacitors. It can also be used to design TCOs with enhanced or diminished conductivities. This analysis provides a non-contact predictive method: combining cation site-occupation information with DFT calculations, and using the new equation, the electronic conductivity of a smallpolaron material can be accurately predicted.