Zr concentration-dependent sub-lattice phase-field model of Hf1− <i>x</i> Zr <i>x</i> O2: Analysis of phase composition and polarization switching

Abstract

We develop a sub-lattice phase-field model of Hf1−xZrxO2 incorporating zirconium (Zr) concentration (x)-dependence. Our framework expands the time-dependent Ginzburg–Landau equation to the sub-lattice level and incorporates x-dependent interaction parameters and gradient coefficients. Our experimentally calibrated model captures the evolution of charge–voltage (Q–V) characteristics for x ranging from 0.5 to 1.0. The sub-lattice formulation explains the thermodynamic preference and kinetic transition barriers of competing orthorhombic phase (o-phase) and tetragonal phase (t-phase), while the phase-field framework enables spatially resolved analysis of polarization (P) and electric-field (E-field) profiles, allowing multi-domain (MD) P and mixed-phase states to emerge naturally. Our model reproduces the experimentally observed ferroelectric (FE)-to-anti-ferroelectric (AFE) transition as x increases from 0.5 to 1.0. At low Zr concentration (x = 0.5–0.6), the o-phase dominates, yielding distinct FE behavior. At high concentration (x ≥ 0.9), the t-phase is stabilized, leading to AFE transitions. A key finding of our work is the unique behavior at intermediate Zr concentrations (x = 0.7–0.8). Here, the o- and t-phase energies are comparable, making the system strongly influenced by local variations in the electric field (E-field), which arise from stray fields near the domain walls. This non-uniform field distribution results in a mixed-phase composition and spatially staggered P reversal, which manifests as a more gradual Q–V evolution (compared to other values of x). By linking energy landscapes to spatial field effects, the model provides insights into the FE-to-AFE crossover in Hf1−xZrxO2.