Ordering of Oxygen Vacancies and Related Ferroelectric Properties in ${\mathrm{HfO}}_{2\ensuremath{-}\ensuremath{\delta}}$

Ordering of Oxygen Vacancies and Related Ferroelectric Properties in ${HfO}_{2 - δ}$

Konstantin Z. Rushchanskii, Stefan Blügel, and Marjana Ležaić

Phys. Rev. Lett. 127, 087602 – Published 20 August 2021

Abstract

Using density functional theory combined with an evolutionary algorithm, we investigate ferroelectricity in substoichiometric ${HfO}_{2 - δ}$ with fixed composition $δ = 0.25$ . We find that oxygen vacancies tend to cluster in the form of two-dimensional extended defects, revealing several patterns of local relative arrangements within an energy range of 100 meV per Hf atom. Two lowest-energy patterns result in polar monoclinic structures with different transformation properties. The lowest one elastically transforms to the ferroelectric orthorhombic structure via a shear deformation, overcoming an energy barrier, which is more than twice lower than in the stoichiometric hafnia. The second-lowest structure transforms at smaller volumes to a nonpolar tetragonal one. We discuss the experimentally observed wake-up effect, fatigue, and imprint in ${HfO}_{2}$ -based ferroelectrics in terms of different local ordering of oxygen-vacancy extended defects, which favor specific crystallographic phases.

Received 26 October 2020
Revised 12 May 2021
Accepted 11 July 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.087602

Physics Subject Headings (PhySH)

Ferroelasticity Ferroelectricity Martensitic phase transition

Oxides Transition metal oxides

Density functional calculations Genetic algorithm

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Konstantin Z. Rushchanskii ^*, Stefan Blügel , and Marjana Ležaić

Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany

^*Corresponding author. k.rushchanskii@fz-juelich.de

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Vol. 127, Iss. 8 — 20 August 2021

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Figure 1
Equation of state for the selected groups of crystalline structures obtained with the evolutionary algorithm: The black line corresponds to the vacancy ordering leading to the ground-state monoclinic phase $M 1$ [see Fig. 2], which continuously transforms to the polar orthorhombic phase $O$ [responsible for ferroelectricity in pristine ${HfO}_{2}$ —see Fig. 2]. Red lines correspond to the vacancy ordering pattern favoring transformation of the low-energy monoclinic phase $M 2$ [see Fig. 2] into the nonpolar tetragonal metastable structure $T$ [see Fig. 2]. Green curves indicate intermediate phases with ordering of oxygen vacancies in tunnels in tetragonal $T T$ and monoclinic $T M$ hafnia [see Figs. 2 and 2]. The blue line displays the robust $I \bar{4} 2 m − {Hf}_{4} O_{7}$ pseudo-fluorite phase $P F$ , stemming from ${Hf}_{2} O_{3}$ , which might be responsible for the resistive switching in ${HfO}_{2 - δ}$ (see Refs. [34, 48]). Black dots indicate structures sampled by the evolutionary algorithm.
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Figure 2
Crystalline structures of the low-energy oxygen vacancy orderings in ${HfO}_{1.75}$ : (a) monoclinic ground state $M 1$ and (b) corresponding orthorhombic local minimum $O$ of martensitic transformation along the black curve in Fig. 1; (c) second-lowest-energy monoclinic phase $M 2$ and (d) corresponding martensitic tetragonal phase $T$ along the red line in Fig. 1; (e) tunnel structure in tetragonal phase $T T$ and (f) corresponding tunnel structure in monoclinic phase $T M$ along the green curves in Fig. 1. Shaded areas indicate the location of oxygen vacancies. The thin solid line shows the unit cell. Hf and O atoms are shown as the big green and small red spheres, respectively. Dashed lines in (e) and (f) highlight the difference in local structural distortion for tetragonal and monoclinic structures.
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Figure 3
Calculated energy barriers and change of spontaneous polarization along the transition path between the lowest-energy monoclinic phase $M 1$ and the metastable orthorhombic phase $O$ (black line). The $O^{+}$ -to- $M 1^{+}$ path describes the transformation between the orthorhombic and the monoclinic phase for one direction of polarization, $O^{-}$ to $M 1^{-}$ , the same for the opposite direction of polarization. The path $M 1^{-}$ to $M 1^{+}$ describes polarization switching within the polar $M 1$ phase. Ferroelectric switching within the orthorhombic phase (without the $O - M 1$ transformation) follows the $O^{-} - O^{0} - O^{+}$ path shown in red. The blue dashed line describes the transition from the monoclinic to the orthorhombic phase in stoichiometric hafnia. Numbers above the arrows are differences of polarization $P$ in the corresponding parts of the transformation path (in $μ C / {cm}^{2}$ ).
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Physical Review Letters

Ordering of Oxygen Vacancies and Related Ferroelectric Properties in HfO2−δ

Konstantin Z. Rushchanskii, Stefan Blügel, and Marjana Ležaić

Phys. Rev. Lett. 127, 087602 – Published 20 August 2021

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Ordering of Oxygen Vacancies and Related Ferroelectric Properties in ${HfO}_{2 - δ}$