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Kibble–Zurek mechanism of Ising domains

Nature Physics volume 19, pages 1495–1501 (2023)Cite this article

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Abstract

The formation of topological defects after a symmetry-breaking phase transition is an overarching phenomenon that encodes the underlying dynamics. The Kibble–Zurek mechanism (KZM) describes these non-equilibrium dynamics of second-order phase transitions and predicts a power-law relationship between the cooling rates and the density of topological defects. It has been verified as a successful model in a wide variety of physical systems, including structure formation in the early Universe and condensed-matter materials. However, it is uncertain if the KZM mechanism is also valid for topologically trivial Ising domains, one of the most common and fundamental types of domain in condensed-matter systems. Here we show that the cooling rate dependence of Ising domain density follows the KZM power law in two different three-dimensional structural Ising domains: ferro-rotation domains in NiTiO3 and polar domains in BiTeI. However, although the KZM slope of NiTiO3 agrees with the prediction of the 3D Ising model, the KZM slope of BiTeI exceeds the theoretical limit, providing an example of steepening KZM slope with long-range dipolar interactions. Our results demonstrate the validity of KZM for Ising domains and reveal an enhancement of the power-law exponent for transitions of non-topological quantities with long-range interactions.

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Fig. 1: Crystal structure and ferro-rotational domains of NiTiO3.
Fig. 2: KZM of 3D Ising ferro-rotational domains in NiTiO3.
Fig. 3: Crystal structure and polar domains of BiTeI.
Fig. 4: KZM of 3D Ising polar domains in BiTeI.

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Data availability

Source data are available with this paper. Other data that support the findings of this study are available from the corresponding author on reasonable request.

Code availability

The codes for the domain analysis of this study are available from the corresponding author on reasonable request.

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Acknowledgements

We thank S. Lin and W. H. Zurek for their helpful comments on this work. The work at Rutgers University was supported by the US Department of Energy (DOE) under grant no. DOE:DE-FG02-07ER46382, and the work at Pohang University (C.W.) was supported by the National Research Foundation of Korea funded by the Ministry of Science and ICT (grant nos. 2022M3H4A1A04074153 and 2020M3H4A2084417). F.J.G.-R. acknowledges financial support from the European Commission FET-Open project AVaQus GA 899561. The X-ray powder diffraction measurements were supported by the US DOE, Office of Science, Office of Basic Energy Science (BES), Materials Sciences and Engineering Division, and the use of the Advanced Photon Source (APS) by the DOE BES Scientific User Facilities Division under contract no. DE-AC02-06CH11357.

Author information

Author notes

  1. Chandan De

    Present address: 2D Crystal Consortium, Materials Research Institute, The Pennsylvania State University, University Park, PA, USA

Authors and Affiliations

  1. Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, USA

    Kai Du, Xiaochen Fang, Fei-Ting Huang & Sang-Wook Cheong

  2. Laboratory for Pohang Emergent Materials, Department of Physics, Pohang University of Science and Technology, Pohang, Korea

    Choongjae Won, Chandan De & Sang-Wook Cheong

  3. Max Planck POSTECH Center for Complex Phase Materials, Department of Physics, Pohang University of Science and Technology, Pohang, Korea

    Choongjae Won

  4. Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea

    Chandan De

  5. X-ray Science Division, Argonne National Laboratory, Lemont, IL, USA

    Wenqian Xu

  6. Materials Science Division, Argonne National Laboratory, Lemont, IL, USA

    Hoydoo You

  7. Instituto de Física Fundamental IFF-CSIC, Madrid, Spain

    Fernando J. Gómez-Ruiz

  8. Donostia International Physics Center, San Sebastián, Spain

    Fernando J. Gómez-Ruiz & Adolfo del Campo

  9. Department of Physics and Materials Science, University of Luxembourg, Luxembourg, Luxembourg

    Adolfo del Campo

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Contributions

S.-W.C. initiated and guided the project. C.W. and C.D. grew the crystals. K.D. and X.F. prepared and measured the samples. K.D. performed the data analysis. F.-T.H. conducted the TEM measurements. W.X. and H.Y. conducted the high-temperature X-ray studies. F.J.G.-R. and A.D.C. performed the theoretical analysis. K.D., F.J.G.-R., A.D.C. and S.-W.C. wrote the paper.

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Correspondence to Sang-Wook Cheong.

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Nature Physics thanks Quintin Meier and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Ferro-rotational domains after polishing in NiTiO3.

(a) CDIC image of NiTiO3 surface after polishing. (b) Corresponding atomic force microscopy topography image of the blue dotted region in (a). (c) Corresponding line profile of the red dashed line in (b).

Extended Data Fig. 2 Confirming ferro-rotational domains in NiTiO3 by TEM.

(a) Selected area electron diffraction (SAED) pattern of quenched NiTiO3 single crystal along [\(\bar{1}2\)0]. Side-view dark-field TEM images by selecting spots (b) (214) and (c) (211), showing CCW and CW ferro-rotational domains. The zone [\(\bar{1}\)20] is indexed in respect to CCW ferro-rotational domain and become zone [\(\bar{2}\)10] in respect to CW domain. Note that the Bragg spot (214) is converted into (12\(\bar{4}\)) in the reciprocal space by the twofold rotation as two ferro-rotation domains do in real space. This produces a corresponding contrast difference between the two ferro-rotational domains that allows us to uniquely confirm the existence of ferro-rotational domains.

Extended Data Fig. 3 CDIC images and corresponding black/white images of ferro-rotation domains for KZM plot in NiTiO3 crystals.

Their image sizes and cooling rates are labeled individually. Images of crystals with the same cooling rate are grouped by black squares, and their average domain density is used for the KZM plot in Fig. 2(a). The cooling rate of the quenched crystal was estimated to be 15000°C/h at its ferro-rotational transition temperature. Due to its small domain size, the domain pattern of the quenched sample was imaged by AFM topography instead of the typical CDIC optical microscope method.

Extended Data Fig. 4 High-temperature X-ray powder diffraction of BiTeI.

(a) Lattice constants and the cell volume as a function of temperature upon heating up to 550°C. Dashed black curves are first derivatives of corresponding lattice parameters, which show clear anomalies at 460°C. (b) Selected X-ray diffraction spectra of BiTeI near the transition temperature upon heating, demonstrating continuous and smooth shifts of X-ray peaks. (c) (100) X-ray diffraction peak of BiTeI at selected temperatures upon heating with no signs of phase coexistence near the transition temperature. The continuous evolution of lattice parameters without abrupt jumps and signs of phase coexistence near the transition unambiguously confirms the second-order nature of this phase transition.

Extended Data Fig. 5 PFM/TEM images and corresponding black/white images of polar domains for KZM plot in BiTeI.

Their image sizes and cooling rates are labeled individually. Images with the same scale are grouped by black squares to facilitate an easier comparison.

Extended Data Fig. 6 Ruling out artifacts from the coarsening effect and chemical defects.

One 0.5°C/h-cooled (green data point) and one 20°C/h-cooled (blue data point) BiTeI crystal with an additional coarsening at 400°C for 300 hours are plotted with regularly-collected KZM data. No significant changes in domain density are found after excessive coarsening. An additional post-annealing (40°C/h, pink data point) can convert a previously slowly-cooled (0.5°C/h) crystal to a state with a high domain density, which illustrates polar domains are not pinned by chemical defects.

Extended Data Fig. 7 Relaxation time as a function of reduced temperature with possible small dynamical critical exponent z in BiTeI.

For 3D Ising model, the spatial critical exponent v ≈ 0.63. The surprisingly small dynamical critical exponent z from our experiments indicates that the relaxation time (τ) near the polar transition in BiTeI has a much broader and slower response to the reduced temperature Tred =(T-Tc)/Tc, where Tc is the polar transition temperature.

Extended Data Fig. 8 Temperature-dependent resistivity and the expected resistivity of BiTeI at the transition.

The measured resistivity of BiTeI can be well fitted by the exponential grow function with R2 = 0.99989. Based the fitting model, the elevated resistivity of BiTeI near the transition temperature at Tc ≈470°C can potentially reach an order of 0.01 Ω•cm, which may enable non-vanishing effects of long-range dipolar interactions.

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Du, K., Fang, X., Won, C. et al. Kibble–Zurek mechanism of Ising domains. Nat. Phys. 19, 1495–1501 (2023). https://doi.org/10.1038/s41567-023-02112-5

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