Fig. 1: Ferroelectric domains expected in hafnia.

a shows the tetragonal (t) phase of hafnia (center) and the four orthorhombic ferroelectric (o-III) variants it leads to. b shows the orthorhombic centrosymmetric phase (o-ref) we propose as reference (center) and the two o-III domains it leads to. Hafnium atoms are shown in blue. The active oxygens, responsible for the development of the spontaneous polarization, are shown in red; the other oxygens are shown in orange. Black arrows indicate spontaneous polarization, which goes against the displacement of the active oxygens from the reference structure. The computed polarization with respect to the t-phase for the structure marked with an asterisk in (a) is +0.54 C m⁻²; by contrast, the polarization with respect to the o-ref phase of the structure marked with an asterisk in (b) is −0.68 C m⁻², while the one marked with a dagger presents +0.68 C m⁻². (As shown in Supplementary Note 1, +0.54 C m⁻² and −0.68 C m⁻² differ by two polarization quanta.) Lattice constants are indicated.