  An extended Kolmogorov-Avrami-Ishibashi (EKAI) model is proposed, which
represents dynamic characteristics in polycrystalline ferroelectric films
consisting of grains under time-dependent electric fields. The original
Kolmogorov-Avrami-Ishibashi (KAI) model described time-varying polarization
reversal in a single-crystal film under a constant electric field. The field
and the polarization were directed along the film normal. The polarization
reversal dynamics were represented by a time-evolution function, c(t),
regarding the volume fractions of the downward- and upward-polarization domains
under a constant electric field. In the EKAI model, a grain in a
polycrystalline ferroelectric film is indexed by l with an angle theta which is
the angle between the spontaneous polarization Ps and the film normal. The EKAI
model first assumes KAI-like polarization variations under a constant field
inside the grain whose angle is tilted by theta. The second assumption is
concerned with polarization variation under time-varying electric field. During
a time period delta_t from t_now to t_next (=t_now+delta_t), the volume
fractions of the downward- and upward-polarization domains changes according to
the function c(t) under an electric field at t_now. By combining the derived
polarization and an electrostatic potential equation across a gate stacked
structure at t_next, an electric field at t_next is derived. Since the electric
field and the volume fractions of the downward and upward domains are known at
t_next, the calculation at the next delta_t step is possible. This procedure is
repeated, and the EKAI model can simulate time-varying polarization reversal
phenomena. Good agreement of the numerical results with the experimental
indicates that the model is appropriate to simulate dynamic characteristics of
FeFETs with polycrystalline ferroelectric films under swept and pulsed
gate-voltages.
