Density functional theory based calculation of small-polaron mobility in hematite

Nicole Adelstein, Jeffrey B. Neaton, Mark Asta, and Lutgard C. De Jonghe

Phys. Rev. B 89, 245115 – Published 11 June 2014

Abstract

The mobility of electron small polarons in hematite, $α$ - ${Fe}_{2}$ $O_{3}$ , is calculated by density functional theory within the generalized gradient approximation including Hubbard $U$ corrections. Our work goes beyond previous computational investigations of this system by computing both the prefactor and activation energies for adiabatic polaron transport. The results obtained using a Hubbard $U$ value of 4.3 eV yield a calculated value of the room-temperature basal plane mobility of 0.009 S* ${cm}^{2}$ /s, which compares to within an order of magnitude with experimental measurements. Further, the values of the electronic-coupling parameter in the Marcus theory for small-polaron transport are estimated from DFT $+ U$ calculations of the defect energy levels in the stable and saddle-point configurations. Our results predict an adiabatic polaron transfer, in good agreement with previous wave function based calculations.

Received 16 October 2013

DOI:https://doi.org/10.1103/PhysRevB.89.245115

Authors & Affiliations

Nicole Adelstein^1,*, Jeffrey B. Neaton^2,3,4, Mark Asta^5,6, and Lutgard C. De Jonghe^5,6,†

¹Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA
²Molecular Foundry, Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
³Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
⁴Kavli Energy NanoSciences Institute at Berkeley, Berkeley, California 94720, USA
⁵Materials Science and Engineering Department, University of California, Berkeley, Berkeley, California 94720, USA
⁶Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

^*adelstein1@llnl.gov
^†Current address: PolyPlus Battery Company, Berkeley, California 94710, USA.

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Vol. 89, Iss. 24 — 15 June 2014

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Images

Figure 1
Plot showing the results for adiabatic and nonadiabatic energy barriers for polaron hopping in $α$ - ${Fe}_{2}$ $O_{3}$ , as calculated by the DFT $+ U$ method and the approach described in the text. The dashed lines show harmonic energy curves with the calculated nonadiabatic energy barrier ( $Δ E_{dia}$ ), given by the intersection of the curves, corresponding to the value calculated in this work. The red line shows the adiabatic energies along a reaction path defined by linearly interpolating the atomic positions between the initial and final polaron states. The black symbol gives a refined value for the adiabatic activation energy barrier ( $Δ E_{ad}$ ) defined by relaxing the approximate saddle-point geometry determined by linear interpolation. The difference between $Δ E_{dia}$ and $Δ E_{ad}$ corresponds to the magnitude of the electron coupling matrix element, $V_{A B}$ . The images from the linear interpolation and the relaxed saddle point are calculated for the 30-atom unit cell with $U = 4.3$ eV.
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Figure 2
(a) c-axis projection of the $2 \times 2 \times 1$ supercell of $α$ - ${Fe}_{2}$ $O_{3}$ shows the initial ground state of the polaron and its three nearest neighbors in the basal plane: Fe1, Fe2, and Fe3 (Fe ions in gold) along with the associated bond lengths and local magnetic moments. The charge density of the ${Fe}^{2 +}$ small polaron is illustrated by the purple isosurface. The black lines give a sense of the a and b lattice vectors and the red oxygen ions show hexagonal close packing. (b) The same supercell viewed along the a axis at the saddle point for a hop between nearest neighbors shows the polaron is delocalized over the initial and final Fe ions and a few oxygen ions. (These images were made with VESTA [50].)
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Figure 3
(a) Calculated density of states for a $1 \times 1 \times 1$ unit cell containing an electron small polaron in both the ground-state and saddle-point configurations for nearest-neighbor hopping shows the narrow band due to the polaron in the ground state near the Fermi energy ( $E_{f}$ ) in green/dotted lines. At the saddle point, the “bonding” and “antibonding” orbitals are both in the spin-down channel and show appreciable bandwidth (in blue/solid lines). (b) The peaks due to the polaron sharpen in the calculated DOS results obtained with the $2 \times 2 \times 1$ supercell compared to the $1 \times 1 \times 1$ cell for both the linear interpolation (light blue/dashed) and the saddle point (black/solid). (c) Calculated density of states for a saddle-point configuration for hopping to next-nearest neighbors calculated in a $2 \times 2 \times 1$ supercell. The bonding and antibonding defect states are very close together, indicating small electronic coupling, in contrast to the results shown for the nearest-neighbor saddle point.
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