## The dynamics of polarization switching can mathematically be described by the Kolmogorov-Avrami-Ishibashi

The dynamics of polarization switching can mathematically be described by the Kolmogorov-Avrami-Ishibashi (KAI) model,7-9 where polarization reversal starts with the statistical formation of a huge quantity of nucleation sites followed by homogeneous domain growth.8 The variation in polarization as a function of time, ?P(t), is expressed as:10-11
1/(2P_r ) ?P(t)=1-exp?(-?_i?S_i/S_0 )
where S0 is the area of the sample and Si reflects the area of growing sporadic domains. The change of polarization is normalized to 2Pr. This normalized value equals 1 for complete polarization reversal. The basic assumption of the KAI model is that a domain can expand unrestrictedly after successful nucleation. Therefore:
S~ (v.t)^n
where v is a constant domain-wall velocity and t is the time. n is the Avrami index, which depends on the dimensionality of the domains. The normalized variation of polarization upon switching can then be written as a compressed exponential function:
1/(2P_r ) ?P(t)=1-exp?(-(t/t_0 )^n )
According to the empirical Merz law,12 the switching time, t0, is related to the activation field, Eact, and can be written as:
t_0=t_? exp?(E_act/E)
where t? is the switching time at infinite applied field. The activation field inversely relates with the temperature: Eact ~ 1/T . Hence, it is straightforward to conclude that polarization reversal in ferroelectric thin films is faster for large electric fields and for higher temperatures. The latter implies a decrease of the activation field.

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