Decoding the Complexity of Ferroelectric Orthorhombic HfO2: A Unified Mode Expansion Approach

Abstract

The ferroelectricity in $\mathrm{HfO}_2$ thin films is widely attributed to the formation of a polar orthorhombic phase named OIII phase. However, the complexity of OIII phase originated from its low symmetry becomes an obstacle for studying ferroelectric properties of $\mathrm{HfO}_2$. Here, we developed a unified framework based on phonon mode expansion for studying ferroelectric $\mathrm{HfO}_2$. In this framework, phase structures, domain walls and switching paths of orthogonal crystal system can be studied from the same basis of mode analysis. The OIII phase and other orthogonal phases can be represented by the high-symmetry cubic phase with the excitation of cubic phonon modes, into which the complexity of orthogonal phases is faithfully coded. To present the capability of this mode expansion approach, we clarified the origin of orthorhombic stability from the energy functional of modes; enumerated inequivalent domain walls and calculated their stable criteria; and summarized all possible switching mechanisms. This unified framework can be used to simplify the study of domain wall structures and transition paths. Furthermore, it can provide a new perspective for ferroelectricity in $\mathrm{HfO}_2$ from phonon mode analysis.

Figures (7)

• Figure 1: Group-subgroup relation of $\mathrm{HfO}_2$ phases. The low symmetry phase can be seen as the high symmetry phase superimposed by phonon modes and strained by strain tensor.
• Figure 2: Phonon spectrum of $\mathrm{HfO}_2$ using conventional cell. (a) The Brillouin zone of C phase primitive cell in blue and that of conventional cell in green. Due to band folding, $\Gamma$, X, Y, Z points of primitive cell are folded into the $\Gamma$ point of conventional cell. (b) Phonon spectrum of C phase conventional cell. The $\Gamma$, $X$ symbols on the wavy line indicate the k points of these phonons in reciprocal space of primitive cells.
• Figure 3: An unified framework to study ferroelectric properties of $\mathrm{HfO}_2$ using mode expansion. The phases, superstructures such as domain walls and transition paths can be generated from cubic phase by adding phonon modes, and their symmetry groups are related to the symmetry group of parent cubic phase.
• Figure 4: Effects of domain phonon modes on the stability of domain walls. (a) Domain wall model comprised of two domains, A and B. (b) The inequivalent dipole directions of two domains comprising the domain wall. (c) Dependence of stability of domain walls on pseudo-chirality numbers of two domains, A and B. The x-axis and y-axis of each stability map are the pseudo-chirality numbers of domain A and B in subfigure (a). The figure title of each stability map indicates the dipole directions of domain A and B.
• Figure 5: All possible switching paths of OIII phase and the evolution of modes along one of the paths. (a) Five switching mechanisms of OIII phase. The IDs of switching paths are the same as these in Table \ref{['tab:OIII-phase-switching']}. (b-d) The evolution of selected modes $Q_1$, $Q_4$ and the non-zero triplet of $Q_6$ along the $90^\circ$ transition path from OIII $a0$ unitcell to OIII $b2$ unitcell. The third dimension is visualized using colorbar.