Thermodynamics of electromechanically coupled mixed ionic-electronic conductors: Deformation potential, Vegard strains, and flexoelectric effect

Strong coupling among external voltage, electrochemical potentials, concentrations of electronic and ionic species, and strains is a ubiquitous feature of solid statemixed ionic-electronic conductors (MIECs), the materials of choice in devices ranging from electroresistive and memristive elements to ion batteries and fuel cells. Here, we analyze in detail the electromechanical coupling mechanisms and derive generalized bias-concentration-strain equations for MIECs including contributions of concentration-driven chemical expansion, deformation potential, and flexoelectric effect. This analysis is extended toward the bias-induced strains in the uniform and scanning-probe-microscopy-like geometries. Notably, the contribution of the electron-phonon and flexoelectric coupling to the local surface displacement of the mixed ionic-electronic conductor caused by the electric field scanning probe microscope tip has not been considered previously. The developed thermodynamic approach allows evolving the theoretical description of mechanical phenomena induced by the electric fields (electromechanical response) in solid state ionics toward analytical theory and phase-field modeling of the MIECs in different geometries and under varying electrical, chemical, and mechanical boundary conditions.