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Relation between microscopic interactions and macroscopic properties in ferroics

Nature Nanotechnology volume 15, pages 896–900 (2020)Cite this article

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Abstract

The driving force in materials to spontaneously form states with magnetic or electric order is of fundamental importance for basic research and device technology. The macroscopic properties and functionalities of these ferroics depend on the size, distribution and morphology of domains; that is, of regions across which such uniform order is maintained1. Typically, extrinsic factors such as strain profiles, grain size or annealing procedures control the size and shape of the domains2,3,4,5, whereas intrinsic parameters are often difficult to extract due to the complexity of a processed material. Here, we achieve this separation by building artificial crystals of planar nanomagnets that are coupled by well-defined, tuneable and competing magnetic interactions6,7,8,9. Aside from analysing the domain configurations, we uncover fundamental intrinsic correlations between the microscopic interactions establishing magnetically compensated order and the macroscopic manifestations of these interactions in basic physical properties. Experiment and simulations reveal how competing interactions can be exploited to control ferroic hallmark properties such as the size and morphology of domains, topological properties of domain walls or their thermal mobility.

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Fig. 1: Model of a ferroic crystal with zero net magnetization and two competing microscopic interactions.
Fig. 2: Phase diagram revealing the correlation between short-range and long-range order.
Fig. 3: Correlation between short-range order and domain formation.
Fig. 4: Macroscopic physical properties determined by the microscopic interactions.

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Data availability

The data that support the figures and other findings of this study can be found here (https://doi.org/10.3929/ethz-b-000429489).

Code availability

The programme codes that support the figures and other findings of this study can be found here (https://doi.org/10.3929/ethz-b-000429490). Additional data and information are available from the corresponding authors upon reasonable request.

References

  1. Wadhawan, V. K. Introduction to Ferroic Materials (CRC Press, 2000).

  2. Hubert, A. & Schäfer, R. Magnetic Domains: The Analysis of Magnetic Microstructures (Springer, 2009).

  3. Seul, M. & Andelman, D. Domain shapes and patterns: the phenomenology of modulated phases. Science 267, 476–483 (1995).

    Article  CAS  Google Scholar 

  4. Farztdinov, M. M. Structure of antiferromagnets. Sov. Phys. Uspekhi 7, 855–876 (1965).

    Article  Google Scholar 

  5. Fiebig, M., Fröhlich, D., Leute, S. & Pisarev, R. V. Second harmonic spectroscopy and control of domain size in antiferromagnetic YMnO3. J. Appl. Phys. 83, 6560–6562 (1998).

    Article  CAS  Google Scholar 

  6. Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013).

    Article  CAS  Google Scholar 

  7. Heyderman, L. J. & Stamps, R. L. Artificial ferroic systems: novel functionality from structure, interactions and dynamics. J. Phys.: Condens. Matter 25, 363201 (2013).

    CAS  Google Scholar 

  8. Rougemaille, N. & Canals, B. Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism – a colloquium. Eur. Phys. J. B. 92, 62 (2019).

    Article  Google Scholar 

  9. Skjærvø, S. H., Marrows, C. H., Stamps, R. L. & Heyderman, L. J. Advances in artificial spin ice. Nat. Rev. Phys. 2, 13–28 (2020).

    Article  Google Scholar 

  10. Kittel, C. Physical theory of ferromagnetic domains. Rev. Mod. Phys. 21, 541–583 (1949).

    Article  Google Scholar 

  11. Sander, D. et al. The 2017 magnetism roadmap. J. Phys. D. Appl. Phys. 50, 363001 (2017).

    Article  Google Scholar 

  12. Jungwirth, T. et al. The multiple directions of antiferromagnetic spintronics. Nat. Phys. 14, 200–203 (2018).

    Article  CAS  Google Scholar 

  13. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nat. Nanotechnol. 11, 231–241 (2016).

    Article  CAS  Google Scholar 

  14. Wadley, P. et al. Electrical switching of an antiferromagnet. Science 351, 587–590 (2016).

    Article  CAS  Google Scholar 

  15. Gomonay, H. V. & Loktev, V. M. Shape-induced phenomena in finite-size antiferromagnets. Phys. Rev. B. 75, 174439 (2007).

    Article  Google Scholar 

  16. Lehmann, J., Donnelly, C., Derlet, P. M., Heyderman, L. J. & Fiebig, M. Poling of an artificial magneto-toroidal crystal. Nat. Nanotechnol. 14, 141–144 (2019).

    Article  CAS  Google Scholar 

  17. Luttinger, J. M. & Tisza, L. Theory of dipole interaction in crystals. Phys. Rev. 70, 954–964 (1946).

    Article  CAS  Google Scholar 

  18. Kraemer, C. et al. Dipolar antiferromagnetism and quantum criticality in LiErF4. Science 336, 1416–1419 (2012).

    Article  CAS  Google Scholar 

  19. Alkadour, B., Mercer, J. I., Whitehead, J. P., Southern, B. W. & van Lierop, J. Dipolar ferromagnetism in three-dimensional superlattices of nanoparticles. Phys. Rev. B. 95, 214407 (2017).

    Article  Google Scholar 

  20. Sendetskyi, O. et al. Continuous magnetic phase transition in artificial square ice. Phys. Rev. B. 99, 214430 (2019).

    Article  CAS  Google Scholar 

  21. Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).

    Article  CAS  Google Scholar 

  22. Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363 (2010).

    Article  CAS  Google Scholar 

  23. Mengotti, E. et al. Real-space observation of emergent magnetic monopoles and associated dirac strings in artificial kagome spin ice. Nat. Phys. 7, 68–74 (2011).

    Article  CAS  Google Scholar 

  24. Farhan, A. et al. Exploring hyper-cubic energy landscapes in thermally active finite artificial spin-ice systems. Nat. Phys. 9, 375–382 (2013).

    Article  CAS  Google Scholar 

  25. Kapaklis, V. et al. Thermal fluctuations in artificial spin ice. Nat. Nanotechnol. 9, 514–519 (2014).

    Article  CAS  Google Scholar 

  26. Anghinolfi, L. et al. Thermodynamic phase transitions in a frustrated magnetic metamaterial. Nat. Commun. 6, 8278 (2015).

    Article  CAS  Google Scholar 

  27. Farhan, A. et al. Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013).

    Article  CAS  Google Scholar 

  28. Gliga, S. et al. Emergent dynamic chirality in a thermally driven artificial spin ratchet. Nat. Mater. 16, 1106–1111 (2017).

    Article  CAS  Google Scholar 

  29. Loreto, R. P. et al. Emergence and mobility of monopoles in a unidirectional arrangement of magnetic nanoislands. Nanotechnology 26, 295303 (2015).

    Article  CAS  Google Scholar 

  30. Arnalds, U. B. et al. A new look on the two-dimensional Ising model: thermal artificial spins. N. J. Phys. 18, 023008 (2016).

    Article  Google Scholar 

  31. Morgan, J. P., Stein, A., Langridge, S. & Marrows, C. H. Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nat. Phys. 7, 75–79 (2011).

    Article  CAS  Google Scholar 

  32. Nisoli, C. On thermalization of magnetic nano-arrays at fabrication. N. J. Phys. 14, 035017 (2012).

    Article  Google Scholar 

  33. Mól, L. A. et al. Magnetic monopole and string excitations in two-dimensional spin ice. J. Appl. Phys. 106, 063913 (2009).

    Article  Google Scholar 

  34. Bortz, A., Kalos, M. & Lebowitz, J. A new algorithm for Monte Carlo simulation of Ising spin systems. J. Computational Phys. 17, 10–18 (1975).

    Article  Google Scholar 

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Acknowledgements

We thank T. Lottermoser, S. Vélez Centoral, A. Cano and T. Weber for discussions. M.F. acknowledges funding by the Swiss National Science Foundation (project no. 200021-175926). J.L. and M.F. acknowledge funding by the ETH Research grant no. ETH-28 14-1 ‘Resonant optical magnetoelectric effect in magnetic nanostructures’. A.B. and M.F. acknowledge funding by the European Research Council (advanced grant no. 694955–INSEETO). N.L. and L.J.H. acknowledge funding by the Swiss National Science Foundation (project no. 200021-155917).

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Author notes

  1. These authors contributed equally: Jannis Lehmann, Amadé Bortis.

Authors and Affiliations

  1. Laboratory for Multifunctional Ferroic Materials, Department of Materials, ETH Zurich, Zurich, Switzerland

    Jannis Lehmann, Amadé Bortis & Manfred Fiebig

  2. Condensed Matter Theory Group, Paul Scherrer Institute, Villigen, Switzerland

    Peter M. Derlet

  3. Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, Zurich, Switzerland

    Peter M. Derlet, Claire Donnelly, Naëmi Leo & Laura J. Heyderman

  4. Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, Villigen, Switzerland

    Claire Donnelly, Naëmi Leo & Laura J. Heyderman

  5. Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Claire Donnelly

  6. Nanomagnetism Group, CIC nanoGUNE BRTA, Donostia—San Sebastián, Spain

    Naëmi Leo

Authors
  1. Jannis Lehmann
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  7. Manfred Fiebig
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Contributions

All authors contributed to the discussion and interpretation of the study. J.L., A.B. and M.F. wrote the manuscript with input from all coauthors. C.D. and N.L. fabricated the nanomagnetic structures. J.L. performed magnetic force microscopy experiments. A.B. and P.M.D. performed the Monte Carlo simulations. M.F. and L.J.H. supervised the study.

Corresponding authors

Correspondence to Jannis Lehmann, Amadé Bortis or Manfred Fiebig.

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The authors declare no competing interests.

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Peer review information Nature Nanotechnology thanks Will Branford, Olena Gomonay and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Lehmann, J., Bortis, A., Derlet, P.M. et al. Relation between microscopic interactions and macroscopic properties in ferroics. Nat. Nanotechnol. 15, 896–900 (2020). https://doi.org/10.1038/s41565-020-0763-9

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