Scaling Behavior and Beyond Equilibrium in the Hexagonal Manganites

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Scaling Behavior and Beyond Equilibrium in the Hexagonal Manganites

S. M. Griffin, M. Lilienblum, K. T. Delaney, Y. Kumagai, M. Fiebig, and N. A. Spaldin

Phys. Rev. X 2, 041022 – Published 27 December 2012

Abstract

We show that the improper ferroelectric phase transition in the multiferroic hexagonal manganites displays appropriate symmetry-breaking characteristics for testing the Kibble-Zurek mechanism originally proposed to describe early-universe phase transitions. We present an analysis of the Kibble-Zurek theory of topological defect formation applied to the hexagonal manganites, discuss the conditions determining the range of cooling rates in which Kibble-Zurek behavior is expected, and show that recent literature data are consistent with our predictions. Finally, we explore experimentally the crossover out of the Kibble-Zurek regime and find a surprising reversal of the scaling behavior.

Received 23 July 2012

DOI:https://doi.org/10.1103/PhysRevX.2.041022

This article is available under the terms of the Creative Commons Attribution 3.0 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

Authors & Affiliations

S. M. Griffin¹, M. Lilienblum¹, K. T. Delaney², Y. Kumagai¹, M. Fiebig¹, and N. A. Spaldin¹

¹Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
²Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA

Popular Summary

Topological defects occur in many fields of physics, from magnetic vortices in solid-state materials to cosmic strings that are proposed to have formed in the early Universe just after the Big Bang. What is the mechanism for their formation and distribution? Two theoretical physicists, Kibble and Zurek, proposed several decades ago a unifying mechanism in which topological defects emerge from certain (mathematically specific) types of transitions that take systems in question from their high-symmetry states to certain lower-symmetry ones. An important consequence follows from this so-called Kibble-Zurek mechanism: The density of the topological defects has a power-law dependence on the rate at which the system in question passes through the transition. Demonstrating this mechanism experimentally on cosmological scales is out of reach, and demonstrating it in condensed-matter systems has proved elusive. In this paper, we report the first unambiguous experimental demonstration of the Kibble-Zurek mechanism in the solid-state hexagonal manganite materials and propose this class of materials as a laboratory platform where the analogue of cosmic-string formation in the early Universe can be studied.

Hexagonal manganites have earned their own place of distinction among condensed-matter systems for their multiferroism—the simultaneous occurrence of ferroelectricity and magnetic ordering. Upon cooling, a ferroelectricity transition takes place between structures with zero and nonzero macroscopic electric polarization. In this work, we have shown that the lattice symmetries associated with the transition set the right symmetry conditions for the Kibble-Zurek mechanism to operate. In addition, the resulting topological defects—vortices of electric polarization—that form at the transition can be readily detected because they are associated with the formation of easily observable ferroelectric domains. By controlling and varying the rate of cooling across the transition temperature of the material and by monitoring with polarization force microscopy the density of polarization vortices and its dependence on the cooling rate, we have established that the density of vortex formation is consistent with the Kibble-Zurek scaling. Adding to the richness of physics in these materials, a crossover is also found of the defect-formation physics from the Kibble-Zurek scaling to a new regime at very fast cooling.

Our work represents a significant advance in the fundamental understanding of topological-defect generation in solid-state materials. We hope that it will also stimulate studies of the beyond–Kibble-Zurek scaling in a cosmological context.

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Vol. 2, Iss. 4 — October - December 2012

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Images

Figure 1
(a) High-symmetry $P 6_{3} / m m c$ structure of $R {MnO}_{3}$ before the onset of trimerization. (b) Action of the $K_{3}$ trimerization mode on the $R$ ions and ${MnO}_{5}$ trigonal bipyramids. The insets below the image of the main structure emphasize that outward trimerization results in a downward shift of the corresponding $R$ ion, whereas inward trimerization results in an upward shift. (c) The subsequent additional displacements of the $R$ ions (blue arrows) in the $Γ_{2}^{-}$ mode provide the ferroelectricity. Note that, once the orientation of the trimerization mode is set, the spontaneous polarization can emerge in only one direction. (d) Typical domain structure measured using piezoresponse force microscopy. The black and white regions correspond to opposite orientations of the ferroelectric polarization along the $z$ axis. Note that the domain structure is isotropic, in spite of the layered crystal structure.Reuse & Permissions
Figure 2
Mexican-hat potential-energy surface of the hexagonal manganites. At high energy (the peak of the hat), the energy is independent of the angle of trimerization, and the system has continuous $U (1)$ symmetry. At lower energy (in the brim of the hat), six of the trimerization angles become favorable (white circles), and the symmetry reduces to the sixfold discrete symmetry described by $Z_{6}$ .Reuse & Permissions
Figure 3
Domain formation and the Kibble-Zurek mechanism. Above $T_{C}$ , fluctuating regions of lateral extension $d$ occur with uniform orientation of the emerging order parameter (fuzzy patches). At high temperature ( $T_{>} > T_{C} + Δ T_{f}$ ), the size of the correlated regions is determined by the correlation length (purple curve). At temperature $T_{f} = T_{C} + Δ T_{f}$ , a freeze-out of the lateral extension $d$ begins, and below the freeze-out temperature, the lateral extension of the fluctuating regions can no longer match the diverging correlation length. The size of the fluctuating regions at temperatures $T_{<} < T_{C} + Δ T_{f}$ is set by the correlation length at the freeze-out temperature, $ξ_{f} = ξ (T_{C} + Δ T_{f})$ , and corresponds to the “communication length,” which is the distance that information propagates during the time in which the system cools from $T_{C} + Δ T_{f}$ to $T_{C} - Δ T_{f}$ (red vertical lines). Below $T_{C} - Δ T_{f}$ , stable domains of lateral extension $ξ (T_{C} + Δ T_{f})$ form (indicated by solid black domain boundaries).Reuse & Permissions
Figure 4
(a) Areal vortex-core density as a function of cooling rate for slow cooling (red triangles, Ref. 45) and fast cooling (blue circles, this work). Note the turnover in the cooling-rate dependence of the vortex-core density occurring at around $10 K / \min $ . The red solid line is the result of our ab initio application of the Kibble-Zurek mechanism with parameters from first-principles calculations. (b)–(d) Distribution of ferroelectric domains in $z$ -oriented ${YMnO}_{3}$ samples after annealing cycles at different cooling rates. The images, which are obtained using PFM on an area $30 \times 30 μ m^{2}$ , reveal a striking “anti–Kibble-Zurek” behavior, with higher cooling rates leading to larger domains.Reuse & Permissions
Figure 5
Sketch of the annealing procedure. After preannealing, the samples are heated to 1420 K with a hold time of 24 h and subsequent cooling at rates between 0.1 and $1360 K / \min $ . After the annealing cycle, the samples are thinned by $5 - 10 μ m$ by polishing in order to access the domain structure formed in the true three-dimensional bulk of the sample.Reuse & Permissions

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