Two-lattice-vibration theory for proton transfer in cubic perovskites: Barium zirconate versus potassium tantalate

Abstract

Proton transfer (PT) in proton-conducting oxides is an intrinsically quantum phenomenon, due to the very strong quantization of the OH stretching vibration (ℏω~ 0.4 eV). Up to high temperatures, the proton is frozen in the ground state associated with the OH stretching motion, and thus does not undergo the thermal agitation for this vibration. Therefore, these are the thermal fluctuations of the (heavier) lattice atoms which make PT possible, at least above ~ half the Debye temperature of the lattice. These fluctuations may occasionally and randomly produce specific lattice configurations in which the quantum protonic ground levels in the two wells are equalized (coincidence), making PT possible, however with a certain probability. An analytical expression of this transfer probability may be obtained as the solution of a curve-crossing quantum-mechanical problem, and thus described by the Landau-Zener (LZ) formula. Two lattice vibrations play a fundamental role in intra-octahedral PT: (i) a reorganization of the lattice, that sends the protonated system from its initial self-trapped configuration up to the coincidence manifold, and (ii) a reduction of the (donor) oxygen - (acceptor) oxygen distance Q, which facilitates PT by leading the system to coincidence configurations with smaller proton barrier, and thus larger LZ probability. In cubic perovskites, the set of the coincidence configurations plays the role of the transition state for PT. The implementation of this theory of proton transfer [G. Geneste, Solid State Ionics 323, 172 (2018)] is here strongly improved on several points: description of the coincidence configurations, proton zero-point energies and proton potential at coincidence. It is then re-applied to barium zirconate (BZO), and also to potassium tantalate (KTO), with parameters derived from density-functional theory (DFT) calculations. The theory confirms the adiabatic PT regime in BZO, common to most proton conductors, with a negligible contribution of non-adiabatic tunneling transfers to the transfer rate. In KTO, by contrast, the relative contribution of non-adiabatic tunneling transfers is larger than in BZO, especially below ~ 250–300 K. The present work helps to characterize intrinsic features common to most proton conductors, at least from the point of view of proton mobility within the lattice. The activations energies for proton transfer at high temperature are predicted at 0.11 eV (BZO) and 0.25 eV (KTO).