Iron oxides such as hematite (α-Fe₂O₃) play an important role in diverse fields ranging from biogeochemistry to photocatalysis. In the majority of these cases their electrical properties are key because they facilitate electron transfer reactions occurring at their interfaces. The iron oxides tend to be wide band gap semiconductors with a narrow d band. Consequently, electron and hole charge carriers localize by self-trapping to form polarons, carriers whose mobilities are tied to the lattice distortions they create. Site-to-site polaron transport is thermally activated and the rate depends on the energy to reorganize the local structure into a suitable transition state, as well as the strength of the electronic coupling in that transient configuration. In turn, these depend on factors that determine the atomic and electronic structure, including crystallographic direction.

Because the structure of polarons, the reorganization energy, and the electronic coupling matrix element are not readily experimentally accessible, quantum mechanical calculations have been extraordinarily useful. However, this also means that the accuracy of predicting polaronic charge carrier mobilities depends strongly on the computational method used. In the ~20 years since the first application of ab initio cluster model computations to this topic for hematite, both the methodological rigor and the supporting computational power have greatly advanced. After a brief historical overview, this talk will highlight current findings obtained with gap-optimized hybrid density functional theory with periodic boundary conditions performed on massively parallel supercomputers. Comparisons will be made to the original calculations for hematite published in 2003. The DFT-based predictions for other iron oxide phases will also be discussed.

Specifically, we will present calculations of both the electron and hole polaron structures and associated reorganization energies for the bulk of hematite, lepidocrocite (γ‑FeOOH), goethite (α‑FeOOH) and white rust (Fe(OH)₂).¹ Through the use of gap-optimized hybrid functionals and large supercells under periodic boundary conditions, we remove some of the complications and uncertainties present in earlier cluster model calculations. It is found that while the hole polaron in these materials generally localizes onto a single iron site, the electron polaron delocalizes across two iron sites of the same spin layer as a consequence of the lower reorganization energy for electrons compared to holes. An exception to these trends is the hole of goethite, which according to our calculations does not form a localized polaron.

For hematite,² we find that upon ionization the hole relaxes from a delocalized band state to a polaron localized on a single iron atom with localization induced by tetragonal distortion of the six surrounding iron-oxygen bonds. This distortion is responsible for sluggish hopping transport in the Fe-bilayer, characterized by an activation energy of 70 meV and a hole mobility of 0.031 cm²/(V s). By contrast, the excess electron induces a smaller distortion of the iron-oxygen bonds resulting in delocalization over two neighboring Fe units. We find that 2-site delocalization is advantageous for charge transport due to the larger spatial displacements per transfer step. As a result, the electron mobility is predicted to be a factor of three higher than the hole mobility, 0.098 cm²/(V s), in qualitative agreement with experimental observations.

The advances made on the theory and simulation front have provided increasing insight into the nature of polaronic transport in the iron oxides, and the methods have been widely duplicated successfully to a variety of materials. However, there is tremendous room for expansion of the work to treat the more challenging aspects governing iron oxide behavior in natural systems and in device application, including the role of interfaces, surface potential, and defects to name a few. Theoretical computations seem poised now more than ever to help unlock the fundamental insight into charge carrier transport in polaronic materials needed to understand and adapt their electrical transport properties.

References

1. Ahart C.S., Blumberger J., Rosso K.M. (2020) Polaronic structure of excess electrons and holes for a series of bulk iron oxides. Physical Chemistry Chemical Physics, 22, 10699-10709.
2. Ahart C.S., Rosso K.M., Blumberger J. (2022) Electron and hole mobilities in bulk hematite from spin-constrained density functional theory. Journal of the American Chemical Society, 144, 4623-4632.