Relating local electric field in a ferroelectric capacitor to externally measurable voltages

The field-dependent polarization model is a commonly proposed model to describe the hysteresis loop of a ferroelectric material. The electric field in the model refers to the local field which varies with position within the ferroelectric film. However, the parameters in the field-dependent polarization are usually extracted based on the macroscopic experimentally averaged field, i.e., applied voltage divided by film thickness. In this work physical quantities such as field, potential, electric displacement and polarization inside the ferroelectric capacitor are obtained by assuming that the capacitor is wholly depleted. The resultant electric field distribution demonstrates the significant difference between the local electric field and the macroscopic average field, indicates that extension of some modeling works is necessary. For the assumption of complete depletion to be valid, a constraint is required upon the film thickness at a given value of doping. A design curve is constructed that shows the maximum allowable film thickness for typical PZT film doping levels in order for the assumption of complete depletion to be valid. The technique presented in this work provides a simple and direct way of relating the macroscopic properties of the capacitor to the actual electric field and potential distribution in the ferroelectric film, furthermore, it will serve as a basis for the C-V modeling of ferroelectric capacitors.