Limitations of the uniform effective field approximation due to doping of ferroelectric thin-film capacitors

The electric-field distribution in a ferroelectric capacitor often is treated as a uniform effective field for circuit-level modeling. By solving Poisson's equation and treating the ferroelectric capacitor as a back-to-back Schottky-barrier system, the nonuniform electric-field distribution is calculated inside a ferroelectric thin film, assuming that the thin-film capacitor is completely depleted and has a constant doping concentration. It is found that the departure of the local field from the uniform effective field increases with an increase in the doping concentration of the film. Within this model, the uniform field approach to extraction of microscopic ferroelectric parameters is inaccurate for doping levels great enough that the surface field exceeds the coercive field even at zero bias. Based on this criterion, the critical doping concentration for parameter extraction using the uniform field approximation is found to be about 5×10¹⁷ cm^-3. That is, according to the assumed model, for the high doping concentrations reported for typical lead-zirconate-titanate thin films (10¹⁸-10¹⁹ cm^-3), the extraction of microscopic film properties based on a uniform electric field approximation is inaccurate.