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.

  • Review Article
  • Published:

Advances in artificial spin ice

Nature Reviews Physics volume 2, pages 13–28 (2020)Cite this article

Subjects

An Author Correction to this article was published on 12 January 2024

An Author Correction to this article was published on 20 January 2020

This article has been updated

Abstract

Artificial spin ices consist of nanomagnets arranged on the sites of various periodic and aperiodic lattices. They have enabled the experimental investigation of a variety of fascinating phenomena such as frustration, emergent magnetic monopoles and phase transitions that have previously been the domain of bulk spin crystals and theory, as we discuss in this Review. Artificial spin ices also show promise as reprogrammable magnonic crystals and, with this in mind, we give an overview of the measurements of fast dynamics in these magnetic metamaterials. We survey the variety of geometries that have been implemented, in terms of both the form of the nanomagnets and the lattices on which they are placed, including quasicrystalline systems and artificial spin systems in 3D. Different magnetic materials can also be incorporated to modify anisotropies and blocking temperatures, for example. With this large variety of systems, the way is open to discover new phenomena, and we complete this Review with possible directions for the future.

Key points

  • Artificial spin ices are metamaterials made up of coupled nanomagnets arranged on different lattices that exhibit a number of interesting phenomena, such as emergent magnetic monopoles, collective dynamics and phase transitions.

  • The motion of emergent magnetic monopoles in an artificial spin system can be controlled with external stimuli such as magnetic and electric fields, strain, temperature gradients and electric currents, which is of potential interest for future devices.

  • The ability to create thermally active artificial spin ices with fluctuating moments at room temperature makes it possible to explore the rich phase diagrams with phases that are determined by the geometry, temperature and disorder.

  • Signatures of the magnetic configurations are given by the specific spin-wave resonances in artificial spin ice, which offer a platform for programmable spin-wave devices, in particular magnonic crystals.

  • The established artificial spin ices consist of elongated nanoscale magnets arranged on the square and the kagome lattices, but these have now diversified. New geometries include different lattices, not only periodic but also aperiodic, different magnet shapes and anisotropies, and 3D structures.

  • Future work involves developments in fabrication and characterization methods, the study of artificial spin systems with new geometries and combinations of materials, and the development of applications including computation, data storage, encryption and reconfigurable microwave circuits.

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

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

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

Fig. 1: Square and kagome artificial spin ice.
Fig. 2: Propagation and chirality of monopole-like magnetic charges in artificial square and kagome ice.
Fig. 3: Magnetic order and phase transitions.
Fig. 4: Fast dynamics in square and kagome artificial spin ices.
Fig. 5: Experimental realizations of selected artificial spin-ice geometries.
Fig. 6: The key directions for future research in artificial spin ice are interlinked.

Similar content being viewed by others

Change history

References

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

    ADS  Google Scholar 

  2. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79, 2554–2557 (1997).

    ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

  5. 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).

    Google Scholar 

  6. 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).

    Google Scholar 

  7. Gilbert, I. et al. Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice. Nat. Phys. 10, 670–675 (2014).

    Google Scholar 

  8. Morrison, M. J., Nelson, T. R. & Nisoli, C. Unhappy vertices in artificial spin ice: new degeneracies from vertex frustration. New J. Phys. 15, 045009 (2013).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  10. Kapaklis, V. et al. Melting artificial spin ice. New J. Phys. 14, 035009 (2012).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  13. Krawczyk, M. & Grundler, D. Review and prospects of magnonic crystals and devices with reprogrammable band structure. J. Phys. Condens. Matter 26, 123202 (2014).

    Google Scholar 

  14. Lao, Y. et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys. 14, 723–727 (2018).

    Google Scholar 

  15. Gilbert, I. et al. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat. Phys. 12, 162–165 (2016).

    Google Scholar 

  16. Farhan, A. et al. Thermodynamics of emergent magnetic charge screening in artificial spin ice. Nat. Commun. 7, 12635 (2016).

    ADS  Google Scholar 

  17. Farmer, B. et al. Direct imaging of coexisting ordered and frustrated sublattices in artificial ferromagnetic quasicrystals. Phys. Rev. B 93, 134428 (2016).

    ADS  Google Scholar 

  18. Shi, D. et al. Frustration and thermalization in an artificial magnetic quasicrystal. Nat. Phys. 14, 309–314 (2018).

    Google Scholar 

  19. Leo, N. et al. Collective magnetism in an artificial 2D XY spin system. Nat. Commun. 9, 2850 (2018).

    ADS  Google Scholar 

  20. Schildknecht, D., Heyderman, L. J. & Derlet, P. M. Phase diagram of dipolar-coupled XY moments on disordered square lattices. Phys. Rev. B 98, 064420 (2018).

    ADS  Google Scholar 

  21. Östman, E. et al. Interaction modifiers in artificial spin ices. Nat. Phys. 14, 375–379 (2018).

    Google Scholar 

  22. Streubel, R. et al. Spatial and temporal correlations of XY macro spins. Nano Lett. 18, 7428–7434 (2018).

    ADS  Google Scholar 

  23. Velten, S. et al. Vortex circulation patterns in planar microdisk arrays. Appl. Phys. Lett. 110, 262406 (2017).

    ADS  Google Scholar 

  24. Louis, D. et al. A tunable magnetic metamaterial based on the dipolar four-state Potts model. Nat. Mater. 17, 1076–1080 (2018).

    ADS  Google Scholar 

  25. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).

    ADS  Google Scholar 

  26. Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).

    ADS  Google Scholar 

  27. Kadowaki, H. et al. Observation of magnetic monopoles in spin ice. J. Phys. Soc. Jpn. 78, 103706 (2009).

    ADS  Google Scholar 

  28. Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7. Science 326, 411–414 (2009).

    ADS  Google Scholar 

  29. Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  31. León, A. Heavy and light monopoles in magnetic reversion in artificial spin ice. Curr. Appl. Phys. 13, 2014–2018 (2013).

    ADS  Google Scholar 

  32. Gilbert, I. et al. Direct visualization of memory effects in artificial spin ice. Phys. Rev. B 92, 104417 (2015).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  34. Farhan, A., Derlet, P. M., Anghinolfi, L., Kleibert, A. & Heyderman, L. J. Magnetic charge and moment dynamics in artificial kagome spin ice. Phys. Rev. B 96, 064409 (2017).

    ADS  Google Scholar 

  35. Silva, R. C. et al. Nambu monopoles interacting with lattice defects in a two-dimensional artificial square spin ice. Phys. Rev. B 87, 014414 (2013).

    ADS  Google Scholar 

  36. Thonig, D. & Henk, J. Pinning of thermal excitations at defects in artificial dipolar arrays: A theoretical investigation. J. Magn. Magn. Mater. 386, 117–124 (2015).

    ADS  Google Scholar 

  37. Drisko, J., Marsh, T. & Cumings, J. Topological frustration of artificial spin ice. Nat. Commun. 8, 14009 (2017).

    ADS  Google Scholar 

  38. Vedmedenko, E. Y. Dynamics of bound monopoles in artificial spin ice: How to store energy in Dirac strings. Phys. Rev. Lett. 116, 077202 (2016).

    ADS  Google Scholar 

  39. Budrikis, Z. et al. Domain dynamics and fluctuations in artificial square ice at finite temperatures. New J. Phys. 14, 035014 (2012).

    ADS  Google Scholar 

  40. Mól, L. A. S., Moura-Melo, W. A. & Pereira, A. R. Conditions for free magnetic monopoles in nanoscale square arrays of dipolar spin ice. Phys. Rev. B 82, 054434 (2010).

    ADS  Google Scholar 

  41. Chern, G.-W., Reichhardt, C. & Nisoli, C. Realizing three-dimensional artificial spin ice by stacking planar nano-arrays. Appl. Phys. Lett. 104, 013101 (2014).

    ADS  Google Scholar 

  42. Möller, G. & Moessner, R. Artificial square ice and related dipolar nanoarrays. Phys. Rev. Lett. 96, 237202 (2006).

    ADS  Google Scholar 

  43. Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature 540, 410–413 (2016).

    ADS  Google Scholar 

  44. Farhan, A. et al. Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice. Sci. Adv. 5, eaav6380 (2019).

    ADS  Google Scholar 

  45. Henley, C. L. The “Coulomb phase” in frustrated systems. Annu. Rev. Condens. Matter Phys. 1, 179–210 (2010).

    ADS  Google Scholar 

  46. Ribeiro, I. R. B. et al. Realization of rectangular artificial spin ice and direct observation of high energy topology. Sci. Rep. 7, 13982 (2017).

    ADS  Google Scholar 

  47. Loreto, R. P. et al. Experimental and theoretical evidences for the ice regime in planar artificial spin ices. J. Phys. Condens. Matter 31, 025301 (2019).

    ADS  Google Scholar 

  48. Perrin, Y., Canals, B. & Rougemaille, N. Quasidegenerate ice manifold in a purely two-dimensional square array of nanomagnets. Phys. Rev. B 99, 224434 (2019).

    ADS  Google Scholar 

  49. Schanilec, V., Perrin, Y., Denmat, S. L., Canals, B. & Rougemaille, N. Artificial vertex systems by design. Preprint at arXiv https://arxiv.org/abs/1902.00452 (2019).

  50. Takatsu, H. et al. Two-dimensional monopole dynamics in the dipolar spin ice Dy2Ti2O7. J. Phys. Soc. Jpn. 82, 073707 (2013).

    ADS  Google Scholar 

  51. Otsuka, H., Takatsu, H., Goto, K. & Kadowaki, H. Scaling ansatz for the ac magnetic response in two-dimensional spin ice. Phys. Rev. B 90, 144428 (2014).

    ADS  Google Scholar 

  52. Gliga, S., Kákay, A., Hertel, R. & Heinonen, O. G. Spectral analysis of topological defects in an artificial spin-ice lattice. Phys. Rev. Lett. 110, 117205 (2013).

    ADS  Google Scholar 

  53. Bhat, V. S., Heimbach, F., Stasinopoulos, I. & Grundler, D. Magnetization dynamics of topological defects and the spin solid in a kagome artificial spin ice. Phys. Rev. B 93, 140401 (2016).

    ADS  Google Scholar 

  54. Bhat, V. S., Heimbach, F., Stasinopoulos, I. & Grundler, D. Angular-dependent magnetization dynamics of kagome artificial spin ice incorporating topological defects. Phys. Rev. B 96, 014426 (2017).

    ADS  Google Scholar 

  55. Chern, G.-W. & Mellado, P. Magnetic monopole polarons in artificial spin ices. EPL 114, 37004 (2016).

    ADS  Google Scholar 

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

    Google Scholar 

  57. León, A. Thermal phase transition in artificial spin ice systems induces the formation and migration of monopole-like magnetic excitations. Physica B Condens. Matter 500, 59–65 (2016).

    ADS  Google Scholar 

  58. Chavez, A. C., Barra, A. & Carman, G. P. Voltage control of magnetic monopoles in artificial spin ice. J. Phys. D Appl. Phys. 51, 234001 (2018).

    ADS  Google Scholar 

  59. Mellado, P., Petrova, O., Shen, Y. & Tchernyshyov, O. Dynamics of magnetic charges in artificial spin ice. Phys. Rev. Lett. 105, 187206 (2010).

    ADS  Google Scholar 

  60. Zeissler, K. et al. The non-random walk of chiral magnetic charge carriers in artificial spin ice. Sci. Rep. 3, 1252 (2013).

    Google Scholar 

  61. Pushp, A. et al. Domain wall trajectory determined by its fractional topological edge defects. Nat. Phys. 9, 505–511 (2013).

    Google Scholar 

  62. Omari, K. A. & Hayward, T. J. Chirality-based vortex domain-wall logic gates. Phys. Rev. Applied 2, 044001 (2014).

    ADS  Google Scholar 

  63. Garg, C. et al. Highly asymmetric chiral domain-wall velocities in Y-shaped junctions. Nano Lett. 18, 1826–1830 (2018).

    ADS  Google Scholar 

  64. Burn, D. M., Chadha, M., Walton, S. K. & Branford, W. R. Dynamic interaction between domain walls and nanowire vertices. Phys. Rev. B 90, 144414 (2014).

    ADS  Google Scholar 

  65. Walton, S. K. et al. Limitations in artificial spin ice path selectivity: the challenges beyond topological control. New J. Phys. 17, 013054 (2015).

    ADS  Google Scholar 

  66. Burn, D. M., Chadha, M. & Branford, W. R. Dynamic dependence to domain wall propagation through artificial spin ice. Phys. Rev. B 95, 104417 (2017).

    ADS  Google Scholar 

  67. Burn, D. M., Chadha, M. & Branford, W. R. Angular-dependent magnetization reversal processes in artificial spin ice. Phys. Rev. B 92, 214425 (2015).

    ADS  Google Scholar 

  68. Kwon, J. et al. Low field domain wall dynamics in artificial spin-ice basis structure. J. Appl. Phys. 118, 163907 (2015).

    ADS  Google Scholar 

  69. Sethi, P. et al. Direct observation of deterministic domain wall trajectory in magnetic network structures. Sci. Rep. 6, 19027 (2016).

    ADS  Google Scholar 

  70. Gartside, J. C. et al. Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing. Nat. Nanotechnol. 13, 53–58 (2018).

    ADS  Google Scholar 

  71. Rougemaille, N. et al. Chiral nature of magnetic monopoles in artificial spin ice. New J. Phys. 15, 035026 (2013).

    ADS  Google Scholar 

  72. Nisoli, C. et al. Effective temperature in an interacting vertex system: theory and experiment on artificial spin ice. Phys. Rev. Lett. 105, 047205 (2010).

    ADS  Google Scholar 

  73. Morgan, J. P. et al. Real and effective thermal equilibrium in artificial square spin ices. Phys. Rev. B 87, 024405 (2013).

    ADS  Google Scholar 

  74. Lammert, P. E. et al. Direct entropy determination and application to artificial spin ice. Nat. Phys. 6, 786–789 (2010).

    Google Scholar 

  75. Porro, J. M., Bedoya-Pinto, A., Berger, A. & Vavassori, P. Exploring thermally induced states in square artificial spin-ice arrays. New J. Phys. 15, 055012 (2013).

    ADS  Google Scholar 

  76. Zhang, S. et al. Crystallites of magnetic charges in artificial spin ice. Nature 500, 553–557 (2013).

    ADS  Google Scholar 

  77. Drisko, J., Daunheimer, S. & Cumings, J. FePd3 as a material for studying thermally active artificial spin ice systems. Phys. Rev. B 91, 224406 (2015).

    ADS  Google Scholar 

  78. Ke, X. et al. Energy minimization and ac demagnetization in a nanomagnet array. Phys. Rev. Lett. 101, 037205 (2008).

    ADS  Google Scholar 

  79. Budrikis, Z. et al. Disorder strength and field-driven ground state domain formation in artificial spin ice: experiment, simulation, and theory. Phys. Rev. Lett. 109, 037203 (2012).

    ADS  Google Scholar 

  80. Coey, J. M. D. Magnetism and Magnetic Materials (Cambridge Univ. Press, 2010).

  81. Osborn, J. A. Demagnetizing factors of the general ellipsoid. Phys. Rev. 67, 351–357 (1945).

    ADS  Google Scholar 

  82. Morley, S. A. et al. Vogel-Fulcher-Tammann freezing of a thermally fluctuating artificial spin ice probed by x-ray photon correlation spectroscopy. Phys. Rev. B 95, 104422 (2017).

    ADS  Google Scholar 

  83. Andersson, M. S. et al. Thermally induced magnetic relaxation in square artificial spin ice. Sci. Rep. 6, 37097 (2016).

    ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

  86. Morley, S. A. et al. Effect of FePd alloy composition on the dynamics of artificial spin ice. Sci. Rep. 8, 4750 (2018).

    ADS  Google Scholar 

  87. Montaigne, F. et al. Size distribution of magnetic charge domains in thermally activated but out-of-equilibrium artificial spin ice. Sci. Rep. 4, 5702 (2014).

    Google Scholar 

  88. 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).

    ADS  Google Scholar 

  89. 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).

    Google Scholar 

  90. Silva, R. C., Nascimento, F. S., Mól, L. A. S., Moura-Melo, W. A. & Pereira, A. R. Thermodynamics of elementary excitations in artificial magnetic square ice. New J. Phys. 14, 015008 (2012).

    ADS  Google Scholar 

  91. Levis, D., Cugliandolo, L. F., Foini, L. & Tarzia, M. Thermal phase transitions in artificial spin ice. Phys. Rev. Lett. 110, 207206 (2013).

    ADS  Google Scholar 

  92. Parakkat, V. M., Xie, K. & Krishnan, K. M. Tunable ground state in heterostructured artificial spin ice with exchange bias. Phys. Rev. B 99, 054429 (2019).

    ADS  Google Scholar 

  93. Chern, G.-W., Mellado, P. & Tchernyshyov, O. Two-stage ordering of spins in dipolar spin ice on the kagome lattice. Phys. Rev. Lett. 106, 207202 (2011).

    ADS  Google Scholar 

  94. Möller, G. & Moessner, R. Magnetic multipole analysis of kagome and artificial spin-ice dipolar arrays. Phys. Rev. B. 80, 140409 (2009).

    ADS  Google Scholar 

  95. Chern, G.-W. & Tchernyshyov, O. Magnetic charge and ordering in kagome spin ice. Phil. Trans. R. Soc. A 370, 5718–5737 (2012).

    ADS  MathSciNet  Google Scholar 

  96. Canals, B. et al. Fragmentation of magnetism in artificial kagome dipolar spin ice. Nat. Commun. 7, 11446 (2016).

    ADS  Google Scholar 

  97. Brooks-Bartlett, M. E., Banks, S. T., Jaubert, L. D. C., Harman-Clarke, A. & Holdsworth, P. C. W. Magnetic-moment fragmentation and monopole crystallization. Phys. Rev. X 4, 011007 (2014).

    Google Scholar 

  98. Sendetskyi, O. et al. Magnetic diffuse scattering in artificial kagome spin ice. Phys. Rev. B 93, 224413 (2016).

    ADS  Google Scholar 

  99. Farhan, A. et al. Thermally induced magnetic relaxation in building blocks of artificial kagome spin ice. Phys. Rev. B 89, 214405 (2014).

    ADS  Google Scholar 

  100. Branford, W. R., Ladak, S., Read, D. E., Zeissler, K. & Cohen, L. F. Emerging chirality in artificial spin ice. Science 335, 1597–1600 (2012).

    ADS  Google Scholar 

  101. Summers, B., Chen, Y., Dahal, A. & Singh, D. K. New description of evolution of magnetic phases in artificial honeycomb lattice. Sci. Rep. 7, 16080 (2017).

    ADS  Google Scholar 

  102. Glavic, A. et al. Spin solid versus magnetic charge ordered state in artificial honeycomb lattice of connected elements. Adv. Sci. 5, 1700856 (2018).

  103. Summers, B. et al. Temperature-dependent magnetism in artificial honeycomb lattice of connected elements. Phys. Rev. B 97, 014401 (2018).

    ADS  Google Scholar 

  104. Chioar, I. A. et al. Kinetic pathways to the magnetic charge crystal in artificial dipolar spin ice. Phys. Rev. B 90, 220407 (2014).

    ADS  Google Scholar 

  105. Liashko, S. Y., Jónsson, H. & Uzdin, V. M. The effect of temperature and external field on transitions in elements of kagome spin ice. New J. Phys. 19, 113008 (2017).

    ADS  Google Scholar 

  106. Petersen, C. F. et al. Tuning magnetic ordering in a dipolar square-kite tessellation. Appl. Phys. Lett. 112, 092403 (2018).

    ADS  Google Scholar 

  107. Chioar, I. A. et al. Nonuniversality of artificial frustrated spin systems. Phys. Rev. B 90, 064411 (2014).

    ADS  Google Scholar 

  108. Hamp, J., Moessner, R. & Castelnovo, C. Supercooling and fragile glassiness in a dipolar kagome Ising magnet. Phys. Rev. B 98, 144439 (2018).

    ADS  Google Scholar 

  109. Sklenar, J. et al. Field-induced phase coexistence in an artificial spin ice. Nat. Phys. 15, 191–195 (2018).

    Google Scholar 

  110. Carlotti, G. et al. From micro- to nanomagnetic dots: evolution of the eigenmode spectrum on reducing the lateral size. J. Phys. D Appl. Phys. 47, 265001 (2014).

    ADS  Google Scholar 

  111. Li, Y. et al. Thickness dependence of spin wave excitations in an artificial square spin ice-like geometry. J. Appl. Phys. 121, 103903 (2017).

    ADS  Google Scholar 

  112. Sklenar, J., Bhat, V. S., DeLong, L. E. & Ketterson, J. B. Broadband ferromagnetic resonance studies on an artificial square spin-ice island array. J. Appl. Phys. 113, 17B530 (2013).

    Google Scholar 

  113. Ma, F., Wu, Y. & Zong, B. Micromagnetic investigation of microwave permeability of magnetic artificial spin ice. Mater. Sci. Appl. 5, 991–995 (2014).

    Google Scholar 

  114. Montoncello, F. et al. Mutual influence between macrospin reversal order and spin-wave dynamics in isolated artificial spin-ice vertices. Phys. Rev. B 97, 014421 (2018).

    ADS  Google Scholar 

  115. Mironov, V. L., Skorohodov, E. V. & Blackman, J. A. Magnetic states and ferromagnetic resonance in geometrically frustrated arrays of multilayer ferromagnetic nanoparticles ordered on triangular lattices. J. Appl. Phys. 115, 184301 (2014).

    ADS  Google Scholar 

  116. Gliga, S., Kákay, A., Heyderman, L. J., Hertel, R. & Heinonen, O. G. Broken vertex symmetry and finite zero-point entropy in the artificial square ice ground state. Phys. Rev. B 92, 060413 (2015).

    ADS  Google Scholar 

  117. Behncke, C. et al. Tunable geometrical frustration in magnonic vortex crystals. Sci. Rep. 8, 186 (2018).

    ADS  Google Scholar 

  118. Jungfleisch, M. B. et al. High-frequency dynamics modulated by collective magnetization reversal in artificial spin ice. Phys. Rev. Applied 8, 064026 (2017).

    ADS  Google Scholar 

  119. Bhat, V. S. & Grundler, D. Angle-dependent magnetization dynamics with mirror-symmetric excitations in artificial quasicrystalline nanomagnet lattices. Phys. Rev. B 98, 174408 (2018).

    Google Scholar 

  120. Kruglyak, V. V. et al. Formation of the band spectrum of spin waves in 1D magnonic crystals with different types of interfacial boundary conditions. J. Phys. D Appl. Phys. 50, 094003 (2017).

    ADS  Google Scholar 

  121. Szulc, K. et al. Remagnetization in arrays of ferromagnetic nanostripes with periodic and quasiperiodic order. Phys. Rev. B 99, 064412 (2019).

    ADS  Google Scholar 

  122. Lisiecki, F. et al. Reprogrammability and scalability of magnonic Fibonacci quasicrystals. Phys. Rev. Applied 11, 054003 (2019).

    ADS  Google Scholar 

  123. Iacocca, E., Gliga, S., Stamps, R. L. & Heinonen, O. Reconfigurable wave band structure of an artificial square ice. Phys. Rev. B 93, 134420 (2016).

    ADS  Google Scholar 

  124. Iacocca, E. & Heinonen, O. Topologically nontrivial magnon bands in artificial square spin ices with Dzyaloshinskii-Moriya interaction. Phys. Rev. Applied 8, 034015 (2017).

    ADS  Google Scholar 

  125. Zhou, X., Chua, G. L., Singh, N. & Adeyeye Adekunle, O. Large area artificial spin ice and anti-spin ice Ni80Fe20 structures: static and dynamic behavior. Adv. Funct. Mater. 26, 1437–1444 (2016).

    Google Scholar 

  126. Schneider, T. et al. Programmability of Co-antidot lattices of optimized geometry. Sci. Rep. 7, 41157 (2017).

    ADS  Google Scholar 

  127. Mamica, S., Krawczyk, M. & Grundler, D. Nonuniform spin-wave softening in two-dimensional magnonic crystals as a tool for opening omnidirectional magnonic band gaps. Phys. Rev. Applied 11, 054011 (2019).

    ADS  Google Scholar 

  128. Mamica, S., Zhou, X., Adeyeye, A., Krawczyk, M. & Gubbiotti, G. Spin-wave dynamics in artificial anti-spin-ice systems: Experimental and theoretical investigations. Phys. Rev. B 98, 054405 (2018).

    ADS  Google Scholar 

  129. Grundler, D. Nanomagnonics. J. Phys. D Appl. Phys. 49, 391002 (2016).

    Google Scholar 

  130. Li, J. et al. Comparing artificial frustrated magnets by tuning the symmetry of nanoscale permalloy arrays. Phys. Rev. B 81, 092406 (2010).

    ADS  Google Scholar 

  131. Farhan, A. et al. Nanoscale control of competing interactions and geometrical frustration in a dipolar trident lattice. Nat. Commun. 8, 995 (2017).

    ADS  Google Scholar 

  132. Macêdo, R., Macauley, G. M., Nascimento, F. S. & Stamps, R. L. Apparent ferromagnetism in the pinwheel artificial spin ice. Phys. Rev. B 98, 014437 (2018).

    ADS  Google Scholar 

  133. Li, Y. et al. Superferromagnetism and domain-wall topologies in artificial “pinwheel” spin ice. ACS Nano 13, 2213–2222 (2019).

    Google Scholar 

  134. Yu, L. et al. Monte Carlo simulation on a new artificial spin ice lattice consisting of hexagons and three-moment vertices. AIP Adv. 7, 085211 (2017).

    ADS  Google Scholar 

  135. Lehmann, J., Donnelly, C., Derlet, P. M., Heyderman, L. J. & Fiebig, M. Poling of an artificial magneto-toroidal crystal. Nat. Nanotechnol. 14, 141–144 (2018).

    ADS  Google Scholar 

  136. Wang, Y.-L. et al. Rewritable artificial magnetic charge ice. Science 352, 962–966 (2016).

    ADS  Google Scholar 

  137. Chern, G.-W., Morrison, M. J. & Nisoli, C. Degeneracy and criticality from emergent frustration in artificial spin ice. Phys. Rev. Lett. 111, 177201 (2013).

    ADS  Google Scholar 

  138. Stopfel, H. et al. Magnetic order and energy-scale hierarchy in artificial spin-ice structures. Phys. Rev. B 98, 014435 (2018).

    ADS  Google Scholar 

  139. Nisoli, C. in Frustrated Materials and Ferroic Glasses (eds Lookman, T. & Ren, X.) 57-99 (Springer International Publishing, 2018).

  140. Stamps, R. L. Artificial spin ice: The unhappy wanderer. Nat. Phys. 10, 623–624 (2014).

    Google Scholar 

  141. Mydosh, J. A. Spin Glasses: An Experimental Introduction. (Taylor & Francis, 1993).

  142. Saccone, M. et al. Towards artificial Ising spin glasses: thermal ordering in randomized arrays of Ising-type nanomagnets. Phys. Rev. B 99, 224403 (2019).

    ADS  Google Scholar 

  143. Islam, Z. et al. Reinvestigation of long-range magnetic ordering in icosahedral Tb-Mg-Zn. Phys. Rev. B 57, R11047–R11050 (1998).

    ADS  Google Scholar 

  144. Chernikov, M. A., Bernasconi, A., Beeli, C., Schilling, A. & Ott, H. R. Low-temperature magnetism in icosahedral Al70Mn9Pd21. Phys. Rev. B 48, 3058–3065 (1993).

    ADS  Google Scholar 

  145. Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266-271 (1974).

  146. Vedmedenko, E. Y., Oepen, H. P. & Kirschner, J. Decagonal quasiferromagnetic microstructure on the Penrose tiling. Phys. Rev. Lett. 90, 137203 (2003).

    ADS  Google Scholar 

  147. Vedmedenko, E. Y., Grimm, U. & Wiesendanger, R. Interplay between magnetic and spatial order in quasicrystals. Philosoph. Mag. 86, 733–739 (2006).

    ADS  Google Scholar 

  148. Bhat, V. S. et al. Controlled magnetic reversal in permalloy films patterned into artificial quasicrystals. Phys. Rev. Lett. 111, 077201 (2013).

    ADS  Google Scholar 

  149. Brajuskovic, V., Barrows, F., Phatak, C. & Petford-Long, A. K. Real-space observation of magnetic excitations and avalanche behavior in artificial quasicrystal lattices. Sci. Rep. 6, 34384 (2016).

    ADS  Google Scholar 

  150. Brajuskovic, V., Addi, A., Phatak, C. & Petford-Long, A. K. Observation of transient states during magnetization reversal in a quasicrystal artificial spin ice. Phys. Rev. B 98, 094424 (2018).

    ADS  Google Scholar 

  151. Barrows, F., Brajuskovic, V., Petford-Long, A. K. & Phatak, C. Emergent magnetic ordering and topological frustration in quasicrystal artificial spin ices. Phys. Rev. B 99, 094424 (2019).

    ADS  Google Scholar 

  152. Perron, J. et al. Extended reciprocal space observation of artificial spin ice with x-ray resonant magnetic scattering. Phys. Rev. B 88, 214424 (2013).

    ADS  Google Scholar 

  153. Lee, J. C. T. et al. Textured heterogeneity in square artificial spin ice. Phys. Rev. B 99, 024406 (2019).

    ADS  Google Scholar 

  154. Mistonov, A. A. et al. Magnetic structure of the inverse opal-like structures: small angle neutron diffraction and micromagnetic simulations. J. Magn. Magn. Mater. 477, 99–108 (2019).

    ADS  Google Scholar 

  155. Chen, X. M. et al. Spontaneous magnetic superdomain wall fluctuations in an artificial antiferromagnet. Preprint at arXiv https://arxiv.org/abs/1809.05656 (2018).

  156. Santos-Filho, J. B., Plascak, J. A., Sobrinho, M. C. & Araujo Batista, T. S. Phase diagram of the XY Vector Blume–Emery–Griffiths model on a Kagome lattice by Monte Carlo simulation. Physica A Stat. Mech. Appl. 503, 844–848 (2018).

    ADS  Google Scholar 

  157. Streubel, R. et al. Magnetism in curved geometries. J. Phys. D Appl. Phys. 49, 363001 (2016).

    Google Scholar 

  158. Fernández-Pacheco, A. et al. Three-dimensional nanomagnetism. Nat. Commun. 8, 15756 (2017).

    ADS  Google Scholar 

  159. Lavrijsen, R. et al. Magnetic ratchet for three-dimensional spintronic memory and logic. Nature 493, 647–650 (2013).

    ADS  Google Scholar 

  160. Kadic, M., Bückmann, T., Stenger, N., Thiel, M. & Wegener, M. On the practicability of pentamode mechanical metamaterials. Appl. Phys. Lett. 100, 191901 (2012).

    ADS  Google Scholar 

  161. Williams, G. et al. Two-photon lithography for 3D magnetic nanostructure fabrication. Nano Res. 11, 845–854 (2018).

    Google Scholar 

  162. May, A., Hunt, M., Berg, A. V. D., Hejazi, A. & Ladak, S. Realisation of a frustrated 3D magnetic nanowire lattice. Commun. Phys. 2, 13 (2019).

    Google Scholar 

  163. Donnelly, C. et al. Element-specific X-ray phase tomography of 3D structures at the nanoscale. Phys. Rev. Lett. 114, 115501 (2015).

    ADS  Google Scholar 

  164. Frenzel, T., Kadic, M. & Wegener, M. Three-dimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).

    ADS  Google Scholar 

  165. Keller, L. et al. Direct-write of free-form building blocks for artificial magnetic 3D lattices. Sci. Rep. 8, 6160 (2018).

    ADS  Google Scholar 

  166. Fowlkes, J. D. et al. High-fidelity 3D-nanoprinting via focused electron beams: computer-aided design (3BID). ACS Appl. Nano Mater. 1, 1028–1041 (2018).

    Google Scholar 

  167. Mistonov, A. A. et al. Three-dimensional artificial spin ice in nanostructured Co on an inverse opal-like lattice. Phys. Rev. B 87, 220408 (2013).

    ADS  Google Scholar 

  168. Shishkin, I. S. et al. Nonlinear geometric scaling of coercivity in a three-dimensional nanoscale analog of spin ice. Phys. Rev. B 94, 064424 (2016).

    ADS  Google Scholar 

  169. Chopdekar, R. V. et al. Nanostructured complex oxides as a route towards thermal behavior in artificial spin ice systems. Phys. Rev. Mater. 1, 024401 (2017).

    Google Scholar 

  170. Cowburn, R. P. Property variation with shape in magnetic nanoelements. J. Phys. D Appl. Phys. 33, R1 (2000).

    ADS  Google Scholar 

  171. Wang, Y.-L. et al. Switchable geometric frustration in an artificial-spin-ice–superconductor heterosystem. Nat. Nanotechnol. 13, 560–565 (2018).

    ADS  Google Scholar 

  172. Rollano, V. et al. Topologically protected superconducting ratchet effect generated by spin-ice nanomagnets. Nanotechnology 30, 244003 (2019).

    ADS  Google Scholar 

  173. Torrejon, J. et al. Neuromorphic computing with nanoscale spintronic oscillators. Nature 547, 428–431 (2017).

    Google Scholar 

  174. Luo, Z. et al. Chirally coupled nanomagnets. Science 363, 1435–1439 (2019).

    ADS  Google Scholar 

  175. Le, B. L. et al. Understanding magnetotransport signatures in networks of connected permalloy nanowires. Phys. Rev. B 95, 060405 (2017).

    ADS  Google Scholar 

  176. Gypens, P., Leliaert, J. & Van Waeyenberge, B. Balanced magnetic logic gates in a kagome spin ice. Phys. Rev. Applied 9, 034004 (2018).

    ADS  Google Scholar 

  177. Caravelli, F. & Nisoli, C. Computation via interacting magnetic memory bites: integration of boolean gates. Preprint at arXiv https://arxiv.org/abs/1810.09190 (2018).

  178. Arava, H. et al. Engineering relaxation pathways in building blocks of artificial spin ice for computation. Phys. Rev. Applied 11, 054086 (2019).

    ADS  Google Scholar 

  179. Arava, H. et al. Computational logic with square rings of nanomagnets. Nanotechnology 29, 265205 (2018).

    ADS  Google Scholar 

  180. Pan, L., Laurita, N. J., Ross, K. A., Gaulin, B. D. & Armitage, N. P. A measure of monopole inertia in the quantum spin ice Yb2Ti2O7. Nat. Phys. 12, 361–366 (2016).

    Google Scholar 

  181. Pancaldi, M., Leo, N. & Vavassori, P. Selective and fast plasmon-assisted photo-heating of nanomagnets. Nanoscale 11, 7656–7666 (2019).

    Google Scholar 

  182. Hong, J., Lambson, B., Dhuey, S. & Bokor, J. Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits. Sci. Adv. 2, e1501492 (2016).

    ADS  Google Scholar 

  183. Khajetoorians, A. A. et al. Atom-by-atom engineering and magnetometry of tailored nanomagnets. Nat. Phys. 8, 497–503 (2012).

    Google Scholar 

  184. Libál, A., Reichhardt, C. & Reichhardt, C. J. O. Realizing colloidal artificial ice on arrays of optical traps. Phys. Rev. Lett. 97, 228302 (2006).

    ADS  Google Scholar 

  185. Libál, A. et al. Ice rule fragility via topological charge transfer in artificial colloidal ice. Nat. Commun. 9, 4146 (2018).

    ADS  Google Scholar 

  186. Han, Y. et al. Geometric frustration in buckled colloidal monolayers. Nature 456, 898–903 (2008).

    ADS  Google Scholar 

  187. Le Cunuder, A., Frérot, I., Ortiz-Ambriz, A. & Tierno, P. Competing orders in colloidal kagome ice: Importance of the in-trap motion of the particles. Phys. Rev. B 99, 140405 (2019).

    Google Scholar 

  188. Nisoli, C. Unexpected phenomenology in particle-based ice absent in magnetic spin ice. Phys. Rev. Lett. 120, 167205 (2018).

    ADS  Google Scholar 

  189. Libál, A., Reichhardt, C. J. O. & Reichhardt, C. Creating artificial ice states using vortices in nanostructured superconductors. Phys. Rev. Lett. 102, 237004 (2009).

    ADS  Google Scholar 

  190. Xue, C. et al. Tunable artificial vortex ice in nanostructured superconductors with a frustrated kagome lattice of paired antidots. Phys. Rev. B 97, 134506 (2018).

    ADS  Google Scholar 

  191. Kang, S. H. et al. Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures. Phys. Rev. Lett. 112, 098701 (2014).

    ADS  Google Scholar 

  192. Conde-Rubio, A. et al. Geometric frustration in a hexagonal lattice of plasmonic nanoelements. Opt. Express 26, 20211–20224 (2018).

    ADS  Google Scholar 

  193. Gunnarsson, K. et al. Programmable motion and separation of single magnetic particles on patterned magnetic surfaces. Adv. Mater. 17, 1730–1734 (2005).

    Google Scholar 

  194. Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotechnol. 8, 152–156 (2013).

    ADS  Google Scholar 

  195. Ma, F., Reichhardt, C., Gan, W., Reichhardt, C. J. O. & Lew, W. S. Emergent geometric frustration of artificial magnetic skyrmion crystals. Phys. Rev. B 94, 144405 (2016).

    ADS  Google Scholar 

  196. Braun, H.-B. Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons. Adv. Phys. 61, 1–116 (2012).

    ADS  Google Scholar 

  197. Ryzhkin, I. A. Magnetic relaxation in rare-earth oxide pyrochlores. J. Exp. Theor. Phys. 101, 481–486 (2005).

    ADS  Google Scholar 

  198. Park, J. et al. Magnetic response of brickwork artificial spin ice. Phys. Rev. B 96, 024436 (2017).

    ADS  MathSciNet  Google Scholar 

  199. Pohle, R., Benton, O. & Jaubert, L. D. C. Reentrance of disorder in the anisotropic shuriken Ising model. Phys. Rev. B 94, 014429 (2016).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  201. Liu, Y. et al. Confined chemical fluid deposition of ferromagnetic metalattices. Nano Lett. 18, 546–552 (2018).

    ADS  Google Scholar 

  202. Chioar, I. A., Rougemaille, N. & Canals, B. Ground-state candidate for the classical dipolar kagome Ising antiferromagnet. Phys. Rev. B 93, 214410 (2016).

    ADS  Google Scholar 

  203. Zhang, S. et al. Perpendicular magnetization and generic realization of the Ising model in artificial spin ice. Phys. Rev. Lett. 109, 087201 (2012).

    ADS  Google Scholar 

  204. Lendinez, S. & Jungfleisch, M. B. Magnetization dynamics in artificial spin ice. J. Phys. Condens. Matter 32, 013001 (2020).

    ADS  Google Scholar 

  205. Ortiz-Ambriz, A. Nisoli, C., Reichhardt, C., Reichhardt, C. J. O. & Tierno, P. Ice rule and emergent frustration in particle ice and beyond. Preprint at arXiv https://arxiv.org/abs/1909.13534 (2019).

Download references

Acknowledgements

The authors are very grateful for the important insights from P. M. Derlet, H.-B. Braun, D. Grundler, V. Scagnoli, K. Hofhuis and Q. N. Meier, which were of great value in the preparation of this manuscript. L.J.H. and S.H.S. acknowledge financial support provided by the Swiss National Science Foundation, grant no. 200020_172774.

Reviewer information

Nature Reviews Physics thanks C. Nisoli, B. Hjörvarsson and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Author information

Authors and Affiliations

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

    Sandra H. Skjærvø & Laura J. Heyderman

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

    Sandra H. Skjærvø & Laura J. Heyderman

  3. School of Physics and Astronomy, University of Leeds, Leeds, UK

    Christopher H. Marrows

  4. Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada

    Robert L. Stamps

Authors
  1. Sandra H. Skjærvø
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher H. Marrows
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert L. Stamps
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Laura J. Heyderman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors have read, discussed and contributed to the writing of the manuscript.

Corresponding authors

Correspondence to Sandra H. Skjærvø or Laura J. Heyderman.

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.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Skjærvø, S.H., Marrows, C.H., Stamps, R.L. et al. Advances in artificial spin ice. Nat Rev Phys 2, 13–28 (2020). https://doi.org/10.1038/s42254-019-0118-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-019-0118-3

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