Thickness Dependence of Coercive Field in Ferroelectric Doped-Hafnium Oxide

Ferroelectric hafnium oxide (${HfO_2}$) exhibits a thickness-dependent coercive field $(E_c)$ behavior that deviates from the trends observed in perovskites and the predictions of Janovec-Kay-Dunn (JKD) theory. Experiments reveal that, in thinner $HfO_2$ films ($<100\,nm$), $E_c$ increases with decreasing thickness but at a slower rate than predicted by the JKD theory. In thicker films, $E_c$ saturates and is independent of thickness. Prior studies attributed the thick film saturation to the thickness-independent grain size, which limits the domain growth. However, the reduced dependence in thinner films is poorly understood. In this work, we expound the reduced thickness dependence of $E_c$, attributing it to the anisotropic crystal structure of the polar orthorhombic (o) phase of $HfO_2$. This phase consists of continuous polar layers (CPL) along one in-plane direction and alternating polar and spacer layers (APSL) along the orthogonal direction. The spacer layers decouple adjacent polar layers along APSL, increasing the energy barrier for domain growth compared to CPL direction. As a result, the growth of nucleated domains is confined to a single polar plane in $HfO_2$, forming half-prolate elliptical cylindrical geometry rather than half-prolate spheroid geometry observed in perovskites. By modeling the nucleation and growth energetics of these confined domains, we derive a modified scaling law of $E_c \propto d^{-1/2}$ for $HfO_2$ that deviates from the classical JKD dependence of $E_c \propto d^{-2/3}$. The proposed scaling agrees well with the experimental trends in coercive field across various ferroelectric $HfO_2$ samples.