Concentration Dependent Diffusion of H+ in TiO2: Analysis of Electronic Effects in Ionic Diffusion
Diffusion of charged impurities or defects in a non-metallic crystal generally results in changes in the electron Fermi level relative to the lattice energy bands and impurity levels, thus producing substantial internal electric fields. These fields in turn exert a force on the diffusing ion, hence modifying the diffusion process. A self-consistent treatment describing such diffusion processes in materials with band gap 1.5 eV will be developed. Our analysis shows that diffusion proceeds locally according to Fick’s laws, but with an effective diffusion coefficient, D(eff), which may be larger than the diffusion coefficient which would be observed in the absence of internal fields by a factor of 104 or more. D(eff) depends on the concentration of diffusing ions, and hence, changes with position and time. The problem of interdiffusion of two similar ions (as in an isotope exchange experiment) is also analyzed; it is shown that such an experiment virtually eliminates the effects of internal fields if the two species are more or less chemically equivalent, making it possible to determine accurately the field-free diffusion parameters. Detailed calculations for both single ion diffusion and interdiffusion of H and D in TiO2 will be presented; the overall rate for single diffusion typically shews an enhancement by a factor of 10 to 30, and a very distinctive concentration profile is predicted. Experiiments leading to accurate determination of the field-free diffusion parameters for H in TiO2, both by the isotope exchange technique and by ionic conductivity measurements, will be reported. In addition, single diffusion measurements have been carried out, which agree in quantitative detail with the calculations described above, both with respect to overall diffusion rate and concentration profile.
Criteria for applicability of our analysis will be discussed, as well as the relationship of this analysis to previous, more restricted treatments of the internal field problem.
KeywordsApparent Diffusion Coefficient Charge Neutrality Effective Diffusion Coefficient Internal Field Internal Electric Field
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