Electron-Phonon Renormalization in the Proton-Conducting Electrolyte <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mrow><mml:mi>Ba</mml:mi><mml:mi>Zr</mml:mi><mml:mi mathvariant="normal">O</mml:mi></

Electrical leakage is an inherent problem in solid-oxide fuel and electrolyzer cells, limiting their energy-conversion efficiency. High concentrations of electrons or holes can exacerbate this issue. However, it is largely unclear how the typical high operating temperatures of the systems influence their charge-carrier concentrations. In this work, we use first-principles calculations to examine how lattice vibrations impact electrical conductivity in conventional electrolytes, using <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mrow><a:mi>Ba</a:mi><a:mi>Zr</a:mi><a:mi mathvariant="normal">O</a:mi></a:mrow></a:math><e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:msub><e:mi/><e:mn>3</e:mn></e:msub></e:math> as a representative material. Our analysis shows that phonon-induced shifts in the band gap and band edges lead to a dramatic increase in <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:mi>p</h:mi></h:math>-type carrier concentrations at temperatures above 600 K, compared to models that neglect temperature effects. Additionally, we reveal the importance of oxygen-ion motion on band-edge positions, which makes valence-band-edge shift dominate the band-gap change. Our study provides a protocol for calculating phonon-induced changes in similar oxides, paving the way for interrogating electrical leakage in electrolytes for high-temperature operation.