Equilibrium and Stability of Polarization in Ultrathin Ferroelectric Films with Ionic Surface Compensation

Abstract

Thermodynamic theory is developed for the ferroelectric phase transition of an ultrathin film in equilibrium with a chemical environment that supplies ionic species to compensate its surface. Equations of state and free energy expressions are developed based on Landau-Ginzburg-Devonshire theory, using electrochemical equilibria to provide ionic compensation boundary conditions. Calculations are presented for a monodomain PbTiO$_3$ (001) film coherently strained to SrTiO$_3$ with its exposed surface and its electronically conducting bottom electrode in equilibrium with a controlled oxygen partial pressure. The stability and metastability boundaries of phases of different polarization are determined as a function of temperature, oxygen partial pressure, and film thickness. Phase diagrams showing polarization and internal electric field are presented. At temperatures below a thickness-dependent Curie point, high or low oxygen partial pressure stabilizes positive or negative polarization, respectively. Results are compared to the standard cases of electronic compensation controlled by either an applied voltage or charge across two electrodes. Ionic surface compensation through chemical equilibrium with an environment introduces new features into the phase diagram. In ultrathin films, a stable non-polar phase can occur between the positive and negative polar phases when varying the external chemical potential at fixed temperature, under conditions where charged surface species are not present in sufficient concentration to stabilize a polar phase.