Local structure and its implications for the relaxor ferroelectric ${\mathrm{Cd}}_{2}{\mathrm{Nb}}_{2}{\mathrm{O}}_{7}$

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Local structure and its implications for the relaxor ferroelectric ${Cd}_{2} {Nb}_{2} O_{7}$

Daniel Hickox-Young, Geneva Laurita, Quintin N. Meier, Daniel Olds, Nicola A. Spaldin, Michael R. Norman, and James M. Rondinelli

Phys. Rev. Research 4, 033187 – Published 6 September 2022

Abstract

The relaxor ferroelectric transition in ${Cd}_{2} {Nb}_{2} O_{7}$ is thought to be described by the unusual condensation of two $Γ がんま$ -centered phonon modes, ${Γ がんま}_{4}^{-}$ and ${Γ がんま}_{5}^{-}$ . However, their respective roles have proven to be ambiguous, with disagreement between ab initio studies, which favor ${Γ がんま}_{4}^{-}$ as the primary mode, and global crystal refinements, which point to ${Γ がんま}_{5}^{-}$ instead. Here, we resolve this issue by demonstrating from x-ray pair distribution function measurements that locally, ${Γ がんま}_{4}^{-}$ dominates, but globally, ${Γ がんま}_{5}^{-}$ dominates. This behavior is consistent with the near degeneracy of the energy surfaces associated with these two distortion modes found in our own ab initio simulations. Our first-principles calculations also show that these energy surfaces are almost isotropic, providing an explanation for the numerous structural transitions found in ${Cd}_{2} {Nb}_{2} O_{7}$ , as well as its relaxor behavior. Our results point to several candidate descriptions of the local structure, some of which demonstrate two-in/two-out behavior for Nb displacements within a given Nb tetrahedron. Although this suggests the possibility of a charge analog of spin ice in ${Cd}_{2} {Nb}_{2} O_{7}$ , our results are more consistent with a Heisenberg-like description for dipolar fluctuations rather than an Ising one. We hope this encourages future experimental investigations of the Nb and Cd dipolar fluctuations, along with their associated mode dynamics.

Received 1 July 2022
Accepted 18 August 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.033187

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Ferroelectricity

Pyrochlores Relaxor ferroelectrics

Density functional calculations Density functional theory X-ray pair-distribution function analysis

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Daniel Hickox-Young ¹, Geneva Laurita ², Quintin N. Meier ³, Daniel Olds⁴, Nicola A. Spaldin ⁵, Michael R. Norman ^6,*, and James M. Rondinelli ^1,7,†

¹Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
²Department of Chemistry and Biochemistry, Bates College, Lewiston, Maine 04240, USA
³Université Grenoble Alpes, CEA, LITEN, 17 rue des Martyrs, 38054 Grenoble, France
⁴National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
⁵Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
⁶Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
⁷Northwestern-Argonne Institute of Science and Engineering (NAISE), Northwestern University, Evanston, Illinois 60208, USA

^*norman@anl.gov
^†jrondinelli@northwestern.edu

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Vol. 4, Iss. 3 — September - November 2022

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Figure 1
(a) ${NbO}_{6}$ and ${Cd}_{2} O$ sublattice positions in the ideal $F d \bar{3} m$ structure and subsequent (b) ${Γ がんま}_{4}^{-}$ and (c) ${Γ がんま}_{5}^{-}$ mode distortions that lead to the low-temperature $I m a 2 - α あるふぁ$ polar structure. Adapted from Laurita et al. [7].
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Figure 2
Calculated energies for ${Cd}_{2} {Nb}_{2} O_{7}$ in various space groups relative to the cubic phase. The bars colored in blue have a dominant ${Γ がんま}_{4}^{-}$ component, the ones in red a dominant ${Γ がんま}_{5}^{-}$ component, and the variant in green is driven by an $X_{4}$ mode. The four structures on the right without bars are associated with modes that are stable, and therefore do not lower the energy. Updated from Laurita et al. [7].
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Figure 3
[(a), (b)] Contour maps of the energy surfaces for the three-dimensional (a) ${Γ がんま}_{4}^{-}$ and (b) ${Γ がんま}_{5}^{-}$ distortion modes. (c) Energy reduction relative to the cubic phase as a function of distortion amplitude along the (a,0,0) (solid curves) and (a,a,0) (dot-dashed curves) order parameter directions for ${Γ がんま}_{4}^{-}$ and ${Γ がんま}_{5}^{-}$ . The inset shows the two-dimensional energy surfaces defined by the order parameter (a,b,0) for ${Γ がんま}_{4}^{-}$ and ${Γ がんま}_{5}^{-}$ . (d) Energy reduction relative to the cubic phase with constant distortion amplitude, r, varying the order parameter direction for ${Γ がんま}_{4}^{-}$ and ${Γ がんま}_{5}^{-}$ through one half of a great circle, that is, (a,a,b) with $2 a^{2} + b^{2}$ = $r^{2}$ , following the path shown by the black curve on the sphere in the inset.
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Figure 4
(a) Goodness of fit ( $R_{w}$ ) as a function of temperature for a series of candidate space groups with a fit $r_{max}$ of 5.7 Å, 12.0 Å, and 30 Å. The fit against the cubic $F d \bar{3} m$ structure is shown as black circles, space groups with primarily ${Γ がんま}_{4}^{-}$ distortion modes are shown as blue symbols, space groups with primarily ${Γ がんま}_{5}^{-}$ distortion modes are shown as red symbols, and the space group with a primary ${Γ がんま}_{5}^{+}$ distortion mode ( $I m m a$ ) is shown as yellow symbols. (b) Local fits (1.7–5.7 Å) of the XPDF data against the $I 4_{1} m d$ and $I \bar{4} 2 d$ space groups at 240 K, 180 K, and 120 K. The $M$ -O correlations are indicated by the red short-dash lines and the $M - M^{'}$ correlations by the blue long-dash lines.
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Figure 5
Fits of the XPDF data collected at 300 K against proposed structures for ${Cd}_{2} {Nb}_{2} O_{7}$ : (a) cubic $F d \bar{3} m$ , (b) orthorhombic $F d d 2$ , (c) orthorhombic $I m a 2 - α あるふぁ$ , (d) orthorhombic $I m a 2 - β べーた$ , and (e) monoclinic $C c$ . Fit ranges are indicated by the range shown in each panel, $M$ -O correlations of interest are shown as short red dashes, and $M - M / M - M^{'}$ correlations of interest are shown as long blue dashes.
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Figure 6
[(a), (b)] Displacement pattern of the relaxed $F d d 2$ structure relative to the high-symmetry cubic structure ( $F d \bar{3} m$ ) for the (a) Nb sublattice and (b) Cd-O sublattice. [(c), (d)] Displacement pattern of the relaxed ${Γ がんま}_{4}^{-}$ -dominated $I m a 2 - β べーた$ structure relative to the high-symmetry cubic structure ( $F d \bar{3} m$ ) for the (c) Nb sublattice and (d) Cd-O sublattice.
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Physical Review Research

Local structure and its implications for the relaxor ferroelectric Cd2Nb2O7

Daniel Hickox-Young, Geneva Laurita, Quintin N. Meier, Daniel Olds, Nicola A. Spaldin, Michael R. Norman, and James M. Rondinelli

Phys. Rev. Research 4, 033187 – Published 6 September 2022

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Local structure and its implications for the relaxor ferroelectric ${Cd}_{2} {Nb}_{2} O_{7}$