A method for the modeling of P-E hysteresis curves of the ferroelectric thin film capacitor was developed. Local electric field, local polarization and local electric potential in the ferroelectric film can be calculated by using the model. With the present modeling method, P-E hysteresis curves can be obtained from the ferroelectric film with any distributions of inhomogeneous charge density N(x) and local polarization parameters ($P_s(x)$, $P_r(x)$, $E_c(x)$ and $ε_f(x)$), and the separate or combined effect of the parameters on the P-E hysteresis curve can be investigated. Thus the model is useful for the investigation of the degradation mechanism and the asymmetric behavior of the ferroelectric thin film capacitors. In this study, the effects of inhomogeneity of charged defect density N(x) and polarization parameters on the hysteresis curves of ferroelectric capacitors were simulated through mathematical determination of the local electric field in ferroelectric films.
Microscopic variation in charge density of the ferroelectric film induced asymmetry and voltage shift of the hysteresis curve. The distribution and the sign of the charged defect determined the direction and the magnitude of the voltage shift of the hysteresis curves. Inhomogeneities in $P_s(x)$, $P_r(x)$, $E_c(x)$ and $ε_f(x)$ did not result in the voltage shift of the hysteresis curves. However, when they combined with the inhomogeneity of the charged defect density, the voltage shift induced by the inhomogeneous charged defect is generally enhanced. The apparent Ps and Pr values measured from the calculated hysteresis curves were decreased with inferior N(x) and $E_c(x)$ and with decrease of $P_s(x)$ and $P_r(x)$ parameters. The magnitudes of +Ps and +Pr may differ from those of $-P_s$ and $-P_r$, depending on the distribution and the sign of the charged defect.
The hysteresis curves measured by Sawyer-Tower method with capacitance of sense capacitor was simulated. As the capacitance of sense capacitor was decreased, the apparent Ps and Pr were decreased. Apparent Ec did not depend on the capacitance of sense capacitor. The direction of the voltage shift in the hysteresis curve measured by Sawyer-Tower method was opposite to that measured by virtual ground method, which satisfied the condition LEFT |$+P_s$|=|$-P_s$|. The direction of the voltage shift of the C-V curve was the same with that of the P-E hysteresis cure measured by virtual ground method.
Considering the screening length $l_s$ and $ε_r$ of the electrode, the P-E hysteresis curves were calculated. As the screening was increased, the apparent $P_s$ and $P_r$ were decreased, since the more applied voltage was across the electrode.