Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase

Nature Nanotechnology volume 15, pages 761–767 (2020)Cite this article

Subjects

A Publisher Correction to this article was published on 08 September 2020

This article has been updated

Abstract

The phase transition most commonly observed is probably melting, a transition from ordered crystalline solids to disordered isotropic liquids. In three dimensions, melting is a single, first-order phase transition. In two-dimensional systems, however, theory predicts a general scenario of two continuous phase transitions separated by an intermediate, oriented liquid state, the so-called hexatic phase with short-range translational and quasi-long-range orientational orders. Such hexatic phases occur in colloidal systems, Wigner solids and liquid crystals, all composed of real-matter particles. In contrast, skyrmions are countable soliton configurations with non-trivial topology and these quasi-particles can form two-dimensional lattices. Here we show, by direct imaging with cryo-Lorentz transmission electron microscopy, that magnetic field variations can tune the phase of the skyrmion ensembles in Cu2OSeO3 from a two-dimensional solid through the long-speculated skyrmion hexatic phase to a liquid. The local spin order persists throughout the process. Remarkably, our quantitative analysis demonstrates that the aforementioned topological-defect-induced crystal melting scenario well describes the observed phase transitions.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Real-space images and topology analysis of the SkL melting process.
Fig. 2: Translational order of the SkL under different magnetic fields.
Fig. 3: Orientational order of the SkL under different magnetic fields.
Fig. 4: Evolutions of characteristic parameters with magnetic field throughout the melting process.
Fig. 5: Phase diagram of nano-slab Cu2OSeO3.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request due to the huge volume (over a terabyte) of the raw data collected in this study.

Code availability

The computer codes that support the findings of this study are available from the corresponding authors upon reasonable request.

Change history

  • 08 September 2020

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.

References

  1. Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966).

    Article  CAS  Google Scholar 

  2. Mermin, N. D. Crystalline order in two dimensions. Phys. Rev. 176, 250–254 (1968).

    Article  Google Scholar 

  3. Kosterlitz, J. M. & Thouless, D. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973).

    Article  CAS  Google Scholar 

  4. Kosterlitz, J. M. Kosterlitz–Thouless physics: a review of key issues. Rep. Prog. Phys. 79, 026001 (2016).

    Article  Google Scholar 

  5. Berezinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems. Sov. Phys. JETP 32, 493–500 (1971).

    Google Scholar 

  6. Berezinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610 (1972).

    Google Scholar 

  7. Ryzhov, V. N., Tareyeva, E. E., Fomin, Y. D. & Tsiok, E. N. Berezinskii–Kosterlitz–Thouless transition and two-dimensional melting. Phys.-Usp 60, 857 (2017).

    Article  CAS  Google Scholar 

  8. Halperin, B. I. & Nelson, D. R. Theory of two-dimensional melting. Phys. Rev. Lett. 41, 121–124 (1978); erratum 41, 519 (1978).

    Article  CAS  Google Scholar 

  9. Nelson, D. R. & Halperin, B. I. Dislocation-mediated melting in two dimensions. Phys. Rev. B 19, 2457–2484 (1979).

    Article  CAS  Google Scholar 

  10. Young, A. P. Melting and the vector Coulomb gas in two dimensions. Phys. Rev. B 19, 1855–1866 (1979).

    Article  CAS  Google Scholar 

  11. Knighton, T. et al. Evidence of two-stage melting of Wigner solids. Phys. Rev. B 97, 085135 (2018).

    Article  CAS  Google Scholar 

  12. Brock, J. D. et al. Orientational and positional order in a tilted hexatic liquid-crystal phase. Phys. Rev. Lett. 57, 98–101 (1986).

    Article  CAS  Google Scholar 

  13. Cheng, M., Ho, J. T., Hui, S. W. & Pindak, R. Observation of two-dimensional hexatic behavior in free-standing liquid-crystal thin films. Phys. Rev. Lett. 61, 550–553 (1988).

    Article  CAS  Google Scholar 

  14. Murray, C. A. & Van Winkle, D. H. Experimental observation of two-stage melting in a classical two-dimensional screened Coulomb system. Phys. Rev. Lett. 58, 1200–1203 (1987).

    Article  CAS  Google Scholar 

  15. Kusner, R. E., Mann, J. A., Kerins, J. & Dahm, A. J. Two-stage melting of a two-dimensional collodial lattice with dipole interactions. Phys. Rev. Lett. 73, 3113–3116 (1994).

    Article  CAS  Google Scholar 

  16. von Grünberg, H.-H., Keim, P. & Maret, G. in Soft Matter (eds Gompper, G. & Schick, M.) 41–86 (Wiley, 2007).

  17. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).

    Article  Google Scholar 

  18. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).

    Article  CAS  Google Scholar 

  19. Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y. Observation of skyrmions in a multiferroic material. Science 336, 198–201 (2012).

    Article  CAS  Google Scholar 

  20. Kézsmárki, I. et al. Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor GaV4S8. Nat. Mater. 14, 1116–1122 (2015).

    Article  Google Scholar 

  21. Skyrme, T. H. R. A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962).

    Article  CAS  Google Scholar 

  22. Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).

    Article  Google Scholar 

  23. Wild, J. et al. Entropy-limited topological protection of skyrmions. Sci. Adv. 3, e1701704 (2017).

    Article  Google Scholar 

  24. Han, J. H., Zang, J., Yang, Z., Park, J.-H. & Nagaosa, N. Skyrmion lattice in a two-dimensional chiral magnet. Phys. Rev. B 82, 094429 (2010).

    Article  Google Scholar 

  25. Timm, C., Girvin, S. M. & Fertig, H. A. Skyrmion lattice melting in the quantum Hall system. Phys. Rev. B 58, 10634–10647 (1998).

    Article  CAS  Google Scholar 

  26. Nishikawa, Y., Hukushima, K. & Krauth, W. Solid–liquid transition of skyrmions in a two-dimensional chiral magnet. Phys. Rev. B 99, 064435 (2019).

    Article  CAS  Google Scholar 

  27. Adams, T. et al. Long-wavelength helimagnetic order and skyrmion lattice phase in Cu2OSeO3. Phys. Rev. Lett. 108, 237204 (2012).

    Article  CAS  Google Scholar 

  28. Seki, S. et al. Formation and rotation of skyrmion crystal in the chiral-lattice insulator Cu2OSeO3. Phys. Rev. B 85, 220406 (2012).

    Article  Google Scholar 

  29. White, J. S. et al. Electric-field-induced skyrmion distortion and giant lattice rotation in the magnetoelectric insulator Cu2OSeO3. Phys. Rev. Lett. 113, 107203 (2014).

    Article  CAS  Google Scholar 

  30. Huang, P. et al. In situ electric field skyrmion creation in magnetoelectric Cu2OSeO3. Nano Lett. 18, 5167–5171 (2018).

    Article  CAS  Google Scholar 

  31. Rajeswari, J. et al. Filming the formation and fluctuation of skyrmion domains by cryo-Lorentz transmission electron microscopy. Proc. Natl Acad. Sci. USA 112, 14212–14217 (2015).

    Article  CAS  Google Scholar 

  32. Nelson, D. R., Rubinstein, M. & Spaepen, F. Order in two-dimensional binary random arrays. Philos. Mag. A 46, 105–126 (1982).

    Article  CAS  Google Scholar 

  33. Bernard, E. P. & Krauth, W. Two-step melting in two dimensions: first-order liquid-hexatic transition. Phys. Rev. Lett. 107, 155704 (2011).

    Article  Google Scholar 

  34. Zahn, K., Lenke, R. & Maret, G. Two-stage melting of paramagnetic colloidal crystals in two dimensions. Phys. Rev. Lett. 82, 2721–2724 (1999).

    Article  CAS  Google Scholar 

  35. Guillamón, I. et al. Enhancement of long-range correlations in a 2D vortex lattice by an incommensurate 1D disorder potential. Nat. Phys. 10, 851–856 (2014).

    Article  Google Scholar 

  36. Zahn, K. & Maret, G. Dynamic criteria for melting in two dimensions. Phys. Rev. Lett. 85, 3656–3659 (2000).

    Article  CAS  Google Scholar 

  37. Beekman, A. J. et al. Dual gauge field theory of quantum liquid crystals in two dimensions. Phys. Rep. 683, 1–110 (2017).

    Article  CAS  Google Scholar 

  38. Janson, O. et al. The quantum nature of skyrmions and half-skyrmions in Cu2OSeO3. Nat. Commun. 5, 5376 (2014).

    Article  CAS  Google Scholar 

  39. Bergman, D., Alicea, J., Gull, E., Trebst, S. & Balents, L. Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets. Nat. Phys. 3, 487–491 (2007).

    Article  CAS  Google Scholar 

  40. del Campo, A. & Zurek, W. H. Universality of phase transition dynamics: topological defects from symmetry breaking. Int. J. Mod. Phys. A 29, 1430018 (2014).

    Article  Google Scholar 

  41. Deutschländer, S., Dillmann, P., Maret, G. & Keim, P. Kibble–Zurek mechanism in colloidal monolayers. Proc. Natl Acad. Sci. USA 112, 6925–6930 (2015).

    Article  Google Scholar 

Download references

Acknowledgements

We thank T. Giamarchi for very insightful discussions and J. White for estimating the longitudinal skyrmion correlation length in Cu2OSeO3 from small-angle neutron scattering data. We are grateful to D. Laub and B. Bártová for help with sample fabrication. This work was supported by the Swiss National Science Foundation (SNSF) through project 166298, the Sinergia network 171003 for Nanoskyrmionics and the National Center for Competence in Research 157956 on Molecular Ultrafast Science and Technology (NCCR MUST), as well as ERC project HERO. P.H. also acknowledges financial support from the Young Talent Support Plan of Xi’an Jiaotong University and the National Natural Science Foundation of China (project 11904277). L.H. and A.R. acknowledge financial support by the DFG within CRC1238 (C02) through project 277146847.

Author information

Author notes

  1. These authors contributed equally: Ping Huang, Thomas Schönenberger.

Authors and Affiliations

  1. State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an, China

    Ping Huang

  2. Laboratory for Quantum Magnetism (LQM), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Ping Huang, Thomas Schönenberger & Henrik M. Rønnow

  3. Laboratory for Ultrafast Microscopy and Electron Scattering (LUMES), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Ping Huang & Fabrizio Carbone

  4. Centre Interdisciplinaire de Microscopie Électronique (CIME), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Marco Cantoni

  5. Institut für Theoretische Physik, Universität zu Köln, Köln, Germany

    Lukas Heinen & Achim Rosch

  6. Crystal Growth Facility, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Arnaud Magrez

Authors
  1. Ping Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Schönenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marco Cantoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lukas Heinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Arnaud Magrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Achim Rosch
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Fabrizio Carbone
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Henrik M. Rønnow
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

P.H., F.C. and H.M.R. conceived the research. P.H. and M.C. designed the study. A.M. synthesized the crystalline samples. P.H., T.S. and M.C. fabricated the LTEM samples and performed the LTEM experiments. P.H., T.S., H.M.R. and F.C. analysed the data. L.H. and A.R. performed the micromagnetic simulations. All authors contributed to interpretation of the data. P.H., H.M.R., F.C. and T.S. wrote the paper with contributions from all authors.

Corresponding authors

Correspondence to Ping Huang or Henrik M. Rønnow.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Real space LTEM image of an area containing more than 30,000 skyrmions.

This implies the potential that much larger skyrmion systems can be achieved for the studies of collective behaviours in 2D compared to the conventional model systems.

Extended Data Fig. 2 Real space LTEM images and topology analysis of SkL at different magnetic fields.

a-d, LoG filtered LTEM images at H = 980 Oe, 1070 Oe, 1147 Oe, and 1160 Oe respectively. e-h, Voronoi diagrams corresponding to a-d respectively.

Extended Data Fig. 3 Pair distribution functions Gr(r) at different magnetic fields.

a,b, The solid phase. c–e, The hexatic phase. f,g, The liquid phase. The intensities are normalized.

Extended Data Fig. 4 Structure factor Sq(q) at different magnetic fields.

a,b, The solid phase. c–e, The hexatic phase. f,g, The liquid phase. The intensities are normalized.

Extended Data Fig. 5 Distribution of the orientational order parameter Ψ6(r).

a-c, Representative color maps derived from the distribution of Ψ6(r) at the solid (H = 667 Oe), the hexatic (H = 1108 Oe), and the liquid phase (H = 1203 Oe) respectively. The hue and the brightness of the color maps denote the phase angle and the amplitude of Ψ6(r) respectively, as illustrated by the inset in a. d-f, The corresponding Ψ6(r) distribution shown by arrows.

Extended Data Fig. 6 Distribution of topological defects in a skyrmion liquid.

a, Thin lines show the Delaunay triangulation, where each node indicates a skyrmion. Red and blue dots indicate respectively 5- and 7- sites. Unpaired 5-/7- sites are further labelled with a circle in the corresponding colors, whereas 5-7 pairs are connected with thick bonds. Defects with other coordinate numbers are indicated by green dots. b-d, Zoom-in of a 5-disclination, a dislocation and a 7-disclination respectively. Note the different numbers of lattice lines at the left and the right sides of the dislocation, as indicated by the light blue lines in c.

Extended Data Fig. 7 Skyrmion interaction potential in Cu2OSeO3 at different magnetic fields calculated by micromagnetic simulations.

The potential was calculated from simulations of a skyrmion lattice in a thin slab, see Supplementary Information for details. For larger magnetic fields the potential flattens considerably.

Extended Data Fig. 8 Summary of properties of skyrmion ensembles in different phases.

Real-space configurations, topology, Fourier transform, translational orders and orientational orders are summarized for the solid, the hexatic and the liquid phases respectively.

Extended Data Fig. 9 Melting hysteresis analysis at different magnetic field ramping rates.

a, Average orientational order parameter ∣Ψ6(r)∣ as a function of magnetic field at different field-ramping rates. The slow ramping rate at 3.3 Oe/s exhibits equilibrium behavior with negligible hysteresis, whereas higher ramping rates result in hysteresis. b and c show the real space distribution of domains at H = 1006 Oe when ramping the magnetic field up and down respectively at the rate of 113 Oe/s, as indicated by the dashed line in the bottom panel of a. The dramatically different distribution of domains fully illustrates the hysteresis in the melting process at a high field-ramping rate.

Extended Data Fig. 10 The SkL melting processes observed under different electron beam intensities.

η6 is the power-law decay exponent of the orientational correlation function G6(r) calculated from the real space SkL configurations throughout the melting process. The critical value of η6 → 1/4 is indicated in each panel by a line near the hexatic to liquid phase transition. Shaded areas are at the same magnetic fields as in Fig. 4 in the main text, indicating where the two transitions happen respectively. The results rule out the possibility of non-equilibrium behaviors induced by electron beam irradiation in our experiments.

Supplementary information

Supplementary Information

Supplementary Figs. 1 and 2, Discussion and refs. 1–5.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, P., Schönenberger, T., Cantoni, M. et al. Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase. Nat. Nanotechnol. 15, 761–767 (2020). https://doi.org/10.1038/s41565-020-0716-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41565-020-0716-3

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing