A family tree for hafnia

Abstract

Several candidate reference phases have been proposed to discuss phase transitions and ferroelectricity in hafnia in recent years. Although these proposals comply with crystallographic requirements, a physically compelling rationale connecting parent and daughter phases is often lacking. This problem is aggravated by the absence of clearly dominant polarization switching pathways, making it difficult to formulate a robust physical criterion for identifying the relevant high-symmetry states. Here we use first-principles calculations to show that pressure provides a simple and robust criterion for establishing physically meaningful family relationships among many hafnia polymorphs, including the monoclinic ground state and the technologically relevant ferroelectric phase, and their respective parent structures. Our simulations also reveal several previously unreported phases, including higher-energy structures that act as common ancestors of the widely discussed cubic and orthorhombic reference phases.

Figures (3)

• Figure 1: Crystal structure of relevant low-energy phases of hafnia. The inactive oxygens are coloured in yellow. In several phases, including mI (a) and oIII (b), every other oxygen column remains in 4-fold coordinated sites (inactive), while neighbouring columns undergo significant displacements and become 3-fold coordinated, resulting either in arrangements with no net dipole (in orange, a) or with a finite electric dipole (in red, b).
• Figure 2: Energy and volume of hafnia phases with pressure. Evolution of energy (a) and volume (b) of the most relevant phases of hafnia as a function of pressure. Many low energy polymorphs (including the monoclinic ground state mI and the ferroelectric phase oIII) transform smoothly into the oVII phase. The tetragonal tI phase transforms smoothly into the cubic cI phase. The antipolar oVIII phase destabilizes at relatively low pressures into the tI phase. Panel c shows the hafnia family tree based on the transformations with pressure. Each box represents one phase of hafnia. On the upper right corner the numbers indicate the number of formula units in the primitive cell (before the colon) and the coordination number of Hf of that phase (after the colon). The number in the upper left corner shows the energy with respect to the ground state in meV/f.u. at zero pressure. The phases are ordered vertically by increasing relative energy. Phases that show instabilities (at $P=0$ GPa) are marked with solid lines with arrows pointing to the phases to which the condensation of these instabilities give rise. Dashed lines indicate parent/daughter relationships that we found via crystallographic analysis where the leading distortion is not a soft phonon mode of the parent structure. The character of the leading distortion(s) is marked on each arrow. The encircled distortions mark soft-mode condensations that result in symmetry recovery.
• Figure 3: Phonons and crystal structure of relevant phases of hafnia which are saddle points of the energy. For oXIV we show the conventional unit cell; its primitive cell contains only 4 formula units.