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Toroidic phase transitions in a direct-kagome artificial spin ice

Nature Nanotechnology (2024)Cite this article

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Abstract

Ferrotoroidicity—the fourth form of primary ferroic order—breaks both space and time-inversion symmetry. So far, direct observation of ferrotoroidicity in natural materials remains elusive, which impedes the exploration of ferrotoroidic phase transitions. Here we overcome the limitations of natural materials using an artificial nanomagnet system that can be characterized at the constituent level and at different effective temperatures. We design a nanomagnet array as to realize a direct-kagome spin ice. This artificial spin ice exhibits robust toroidal moments and a quasi-degenerate ground state with two distinct low-temperature toroidal phases: ferrotoroidicity and paratoroidicity. Using magnetic force microscopy and Monte Carlo simulation, we demonstrate a phase transition between ferrotoroidicity and paratoroidicity, along with a cross-over to a non-toroidal paramagnetic phase. Our quasi-degenerate artificial spin ice in a direct-kagome structure provides a model system for the investigation of magnetic states and phase transitions that are inaccessible in natural materials.

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Fig. 1: Quasi-degenerate direct-kagome ASI with well-defined toroidal moments.
Fig. 2: Three thermal phases and phase transitions.
Fig. 3: Lattice-constant-dependent experiments.
Fig. 4: Accessing effective thermal states by tuning the lattice constant.
Fig. 5: Visualizing toroidic phase transitions from magnetic structure factors.

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Data availability

All of the data supporting this study are available via the FigShare public repository at https://doi.org/10.6084/m9.figshare.24270862 (ref. 60).

Code availability

The code of Monte Carlo simulations used in this study is available via the Zenodo public repository at https://doi.org/10.5281/zenodo.10825074 (ref. 61).

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant no. 62288101 to H.W., 62274086 to Y.-L.W., 12204434 to Y.D. and 62271245 to X.T.), the National Key R&D Program of China (grant no. 2021YFA0718802 to H.W. and Y.-L.W., and 2023YFF0718400 to Y.D.), a Postdoctoral Fellowship Program of CPSF, and Jiangsu Outstanding Postdoctoral Program to W.-C.Y. and Y.-Y.L. The work of C.N. was performed under the auspices of the US DOE through Los Alamos National Laboratory, operated by Triad National Security, LLC (contract no. 229892333218NCA000001) and financed by a grant from the DOE-LDRD office.

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  1. These authors contributed equally: Wen-Cheng Yue, Zixiong Yuan.

Authors and Affiliations

  1. School of Electronic Science and Engineering, Nanjing University, Nanjing, China

    Wen-Cheng Yue, Zixiong Yuan, Peiyuan Huang, Yizhe Sun, Tan Gao, Yang-Yang Lyu, Xuecou Tu, Sining Dong, Liang He, Xun Cao, Lin Kang, Huabing Wang, Peiheng Wu & Yong-Lei Wang

  2. Purple Mountain Laboratories, Nanjing, China

    Wen-Cheng Yue, Zixiong Yuan, Peiyuan Huang, Tan Gao, Yang-Yang Lyu, Xuecou Tu, Huabing Wang, Peiheng Wu & Yong-Lei Wang

  3. Research Institute of Superconductor Electronics, Nanjing University, Nanjing, China

    Wen-Cheng Yue, Zixiong Yuan, Peiyuan Huang, Tan Gao, Yang-Yang Lyu, Xuecou Tu, Sining Dong, Lin Kang, Huabing Wang, Peiheng Wu & Yong-Lei Wang

  4. National Key Laboratory of Spintronics, Nanjing University, Suzhou, China

    Yizhe Sun, Sining Dong, Liang He & Yong-Lei Wang

  5. College of Metrology Measurement and Instrument, China Jiliang University, Hangzhou, China

    Ying Dong

  6. Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

    Cristiano Nisoli

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Contributions

W.-C.Y. and Y.-L.W. conceived the project. W.-C.Y., Z.Y., X.T., L.H. and L.K. fabricated the samples. W.-C.Y., Z.Y. and Y.S. performed the thermal annealing. W.-C.Y. and Z.Y. performed MFM imaging. Z.Y. and C.N. performed the Monte Carlo simulations and the theoretical analysis. W.-C.Y. and P.H. performed the micromagnetic simulations. W.-C.Y., Z.Y. and Y.-Y.L. performed the statistical analysis. Z.Y. calculated the MSF. W.-C.Y., Z.Y., Y.D., S.D., C.N. and Y.-L.W. analysed and interpreted the data. W.-C.Y., Z.Y., S.D., H.W., C.N. and Y.-L.W. wrote and edited the manuscript. X.C., H.W., P.W., C.N. and Y.-L.W. supervised the project.

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Correspondence to Sining Dong, Huabing Wang, Cristiano Nisoli or Yong-Lei Wang.

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Nature Nanotechnology thanks Jack Gartside, Nicolas Rougemaille 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 SEM images of samples with various lattice constants.

Scale bar, 500 nm (a-h), 1 μm (i-l) and 2 μm (m).

Extended Data Fig. 2 The annealing process.

a The sample was heated from room temperature to an annealing temperature of 550 °C in 60 min, held for 15 min, then cooled from 550 °C to 100 °C with 0.3 °C min−1. b and c SEM (b) and MFM (c) images of a square ASI sample, which served as a reference and was annealed on the same substrate of the direct-kagome ASIs. The nanomagnet size is the same with that of the direct-kagome ASI. The lattice constant of the square ASI is 360 nm. The nearly perfect ground state proves the effectiveness of our annealing.

Extended Data Fig. 3 MFM images with various lattice constants.

a-m, The MFM imaging was conducted after a sample annealing process shown in Extended Data Fig. 1. Scale bar, 2 μm.

Extended Data Fig. 4 Vertex distributions extracted from Extended Data Fig. 3.

a-m, Vertices of Types I, II-α, III, II-β, and IV are shown in gray, light blue, green, gold and magenta, respectively. Scale bar, 2 μm.

Extended Data Fig. 5 Toroidal moment distributions corresponding to vertex distributions in Extended Data Fig. 4.

a-m, Red and blue denote positive and negative toroidal moments, respectively. Scale bar, 2 μm.

Extended Data Fig. 6 Spin MSF maps for all the samples.

a-m, Spin MSF maps for the samples with various lattice constants ranging from 300 nm to 1760 nm, respectively. The color scale refers to the intensity at a given point (qx, qy) of reciprocal space.

Extended Data Fig. 7 Toroidal moment MSF maps for all the samples.

a-m, Toroidal moment MSF maps corresponding to the spin MSF maps in Extended Data Fig. 6, respectively. The colour scale refers to the intensity at a given point (qx, qy) of reciprocal space.

Extended Data Fig. 8 Strong and weak Bragg peaks in spin MSF of the direct-kagome ASI.

a, The spin MSF map of ideal ferrotoroidic ordering. A weak Bragg peak and a strong Bragg peak are marked by vectors \({\overrightarrow{q}}_{1}\) and \({\overrightarrow{q}}_{2}\), respectively. The color scale refers to the intensity at a given point (qx, qy) of reciprocal space. b, the \({\overrightarrow{q}}_{1}\) peak originates from the spin scattering between neighboring nanomagnets with an angle of 120 degrees. c, the \({\overrightarrow{q}}_{2}\) peak originates from the spin scattering between neighboring nanomagnets with an angle of 60 degrees. d, line cuts of MSF maps across \({\overrightarrow{q}}_{1}\) and \({\overrightarrow{q}}_{2}\) peaks from (a). The different spin components, \({\overrightarrow{S}}^{\perp }\), perpendicular to \({\overrightarrow{q}}_{1}\) and \({\overrightarrow{q}}_{2}\) lead to distinct intensities in the MSF for \({\overrightarrow{q}}_{1}\) and \({\overrightarrow{q}}_{2}\) (refer to Supplemental Information for a detailed derivation).

Extended Data Fig. 9 MC simulations of specific heat and entropy under various coupling energies.

When the lowest excitation energy E1 approaching the second lowest excitation energy E2, the low-temperature phase transition peak is gradually merging into the high-temperature cross-over. Therefore, the quasi-degeneracy, requiring E1«E2, is critical for observing the low-temperature phase transition.

Supplementary information

Supplementary Information

Supplementary Figs. 1–3, Discussion and Tables 1–3.

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Yue, WC., Yuan, Z., Huang, P. et al. Toroidic phase transitions in a direct-kagome artificial spin ice. Nat. Nanotechnol. (2024). https://doi.org/10.1038/s41565-024-01666-6

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