Defect chemistry and transport in materials with three carriers

R. Zohourian, R. Merkle

Proton-conducting acceptor-doped perovskites have three carriers: oxygen vacancies  , protons , and electron holes . Their concentrations are determined by two reactions describing water and oxygen uptake:


Alternatively, proton uptake can occur by a redox reaction as expense of holes:


(this reaction is a linear combination of reactions (1) and (2), i.e. not independent). Under conditions leading to a high  and low  concentration (high pO2, and/or redox-active perovskites), this is the dominating mode of proton uptake.

Because in the three carrier system only all three carriers together have to fulfil the electroneutrality conditions, an increase of pH2O can lead to complex stoichiometry relaxation phenomena. At sufficiently high hole concentration, "two-fold" relaxation is observed, which also comprises a characteristic non-monotonic change of the hole concentration.

Qualitatively, in first approximation this can be viewed as fast uptake of protons at expense of holes, followed by a slower uptake of oxygen again compensated by hole migration. The overall reaction is essentially water uptake, but it proceeds via an over-reduced intermediate state of the sample. A derivation of exact analytical expressions and numerical examples are given on the publication below.

(a) Simulated evolution of space-resolved defect concentration profiles after an increase in pH2O for a material with sufficient hole concentration to decouple the proton and oxygen vacancy concentration changes (defect diffusivity ratios Dh. = 100 DOHo. = 10DVo.. ; colors indicate different times after the pH2O increase). (b) Integral defect concentrations plotted as a function of square root of time; this representation allows one to extract effective hydrogen and oxygen diffusivities DHeff > DOeff for the fast initial reduction and the slower reoxidation processes from the slopes.


  • D. Poetzsch, R. Merkle, and J. Maier
    Solution of the three-carrier problem: A natural explanation for surprising observations Annual Report MPI-FKF 2014
  • D. Poetzsch, R. Merkle, and J. Maier 
    Stoichiometry Variation in Materials with Three Mobile Carriers—Thermodynamics and Transport Kinetics Exemplified for Protons, Oxygen Vacancies, and Holes 
    Advanced Functional Materials 25(10), 1542–1557 (2015). DOI: 10.1002/adfm.201402212
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