Triggered ferroelectricity in HfO$_2$ from hybrid phonons and higher-order dynamical charges

TL;DR

The paper addresses the unclear origin of polarization in HfO$_2$ and challenges the conventional proper/improper ferroelectric frameworks by proposing a hybrid-triggered mechanism in which polarization emerges through cooperative trilinear and quadlinear couplings among stable modes. It combines symmetry-guided Landau-Ginzburg-Devonshire theory with first-principles DFT to map abstract order parameters to real phonon modes, revealing a significant electronic polarization contribution from higher-order, nonpolar modes (approximately 41% of the total). The study shows that a unified trigger lowers the polarization threshold and maps a complete coherent switching pathway under epitaxial strain, with the Aeaa nonpolar phase playing a key role. These insights provide a foundational framework for understanding ferroelectricity and antiferroelectricity in fluorite-related structures and offer design principles for next-generation ferroelectric materials and devices.

Abstract

Ferroelectric HfO$_2$ has emerged as a highly promising material for high-density nonvolatile memory and nanoscale transistor applications. However, the uncertain origin of polarization in HfO$_2$ limits our ability to fully understand and control its ferroelectricity. Ferroelectricity, the emergence of a spontaneous and switchable polarization in solids, is conventionally understood to be governed by unstable structural modes (phonons), arising either directly from an unstable polar phonon or indirectly through coupling of unstable nonpolar phonons with a polar mode. While these 'proper' and 'improper' mechanisms successfully explain ferroelectricity for most systems, they do not encompass all possible phenomena. Here, we present a novel mechanism of 'hybrid-triggered' ferroelectricity, where a polar order emerges through trilinear coupling without any structural instabilities. Our group theoretical analysis starting from a high-symmetry reference structure shows that this mechanism is realized in intensely-debated ferroelectric HfO$_2$, along with quantitative confirmation from first-principles calculations. We also show that dynamical charges in this material are highly unconventional, and a significant contribution to the total polarization arises solely from high-order couplings of nonpolar phonons. These findings underline that even simple crystal structures can host surprisingly complicated interplay between different structural orders, elucidate the origin of ferroelectricity and antiferroelectricity in fluorite-related structures, and provide foundational understanding for designing superior ferroelectric materials.

Figures (15)

• Figure 1: Modes and unique couplings in fluorite HfO$_2$. (a) Eight modes observed in the ferroelectric Pca2$_1$ phase of HfO$_2$. Blue arrows and signs represent ionic displacements in the front half of the unit cell ($x=0.75$ in direct coordinates for oxygen), and the pink arrows represent those in the rear half ($x=0.25$). (b) Unique 2nd, 3rd, and 4th-order couplings among modes. Modes that can condense in the dielectric phase are highlighted in green, whereas modes that condense in the triggered phase are shown in red.
• Figure 2: LGD model and DFT validation of hybrid-triggered ferroelectricity. (a) Solution of the theoretical LGD model for hybrid-triggered ferroelectricity described by Equation (\ref{['simple']}), formulated in terms of hypothetical order parameters $p_0$, $q_1$, and $q_2$. (b) Same as part (a), but for Equation (\ref{['eq:multicoupling']}). The inclusion of an additional trilinear coupling term, $q_1 q_3 q_4$, significantly reduces the critical polarization required to induce the triggered ferroelectric transition. (c) DFT-calculated energies of HfO$_2$ as functions of the structural order parameters $P_0$, $Q_1$, and $Q_5$, providing first-principles validation of hybrid-triggered ferroelectricity. The amplitudes are constrained such that $Q_1/Q_5$ is 4.4 (Figure S4). Results are shown for cases where all other order parameters are constrained to zero (left) and where they are fully relaxed (right), revealing a substantial reduction of the critical polarization $P_{\mathrm{0,c}}$ from 0.49 Å to 0.17 Å.
• Figure 3: Coherent switching barrier of hybrid-triggered and hybrid improper ferroelectrics. Example of an energy contour diagram from LGD model of (a) Hybrid-triggered ferroelectrics from Equation (\ref{['simple']}) and (b) Hybrid improper ferroelectrics. The former switch though a barrier nonparallel to $p_0$ axis, while the latter encounter an intrinsic energy barrier parallel to the $p_0$ axis associated with the switching of either one of the nonpolar modes, which does not directly couple with voltage.
• Figure 4: Higher-order dynamical charge arising from nonpolar modes in HfO$_2$. Interpolation between the high-symmetry reference structure and the polar ground state configuration, decomposed into polar and nonpolar mode contributions. HfO$_2$ exhibits a substantial higher-order polarization component originating from nonpolar modes (accounting for approximately --41% of the bulk response), in sharp contrast to the proper ferroelectric LiNbO$_3$, where such contributions are symmetrically forbidden (0.0%), and to hybrid improper ferroelectric Ca$_3$Ti$_2$O$_7$, where they are allowed but negligible (0.4%).
• Figure 5: The effect of polar hybrid modes on triggered ferroelectricity. (a) Plots for a hybrid-triggered ferroelectric material with polar hybrid mode, including energy-polarization (top left), polarization-voltage (bottom left), order parameters-polarization (top right), and two-dimensional energy-$p_0$,$P$ (bottom right) modeled with additional hybrid mode contribution to polarization to Equation (\ref{['simple']}). The triggered ferroelectric phase is separated from the dielectric phase in the two-dimensional energy plot, introducing an additional layer of hysteresis and a disruptive phase transition. (b) Coherent switching pathway of HfO$_2$ via the hybrid-triggered mechanism. Five order parameters that remain zero in the dielectric phase condense together at $P_\mathrm{c} = 0.30$ C m$^{-2}$ disruptively.