- Letter
- Open Access
Magnetic order and long-range interactions in mesoscopic Ising chains
Phys. Rev. B 111, L020408 – Published 16 January, 2025
DOI: https://doi.org/10.1103/PhysRevB.111.L020408
Abstract
We investigate the design of magnetic ordering in one-dimensional mesoscopic magnetic Ising chains by modulating long-range interactions. These interactions are affected by geometrical modifications to the chain, which adjust the energy hierarchy and the resulting magnetic ground states. Consequently, the magnetic ordering can be tuned between antiferromagnetic and antiferromagnetic dimer phases. These phases are experimentally observed in chains fabricated using both conventional electron-beam lithography and ion implantation techniques, demonstrating the feasibility of controlling magnetic properties at the mesoscale. The ability of attaining these magnetic structures by thermal annealing, underlines the potential of using such systems instead of simulated annealers in tackling combinatorial optimization tasks.
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Supplemental Material
References (32)
- R. Zhang and R. Willis, Thickness-dependent Curie temperatures of ultrathin magnetic films: Effect of the range of spin-spin interactions, Phys. Rev. Lett. 86, 2665 (2001).
- A. Taroni and B. Hjörvarsson, Influence of the range of interactions in thin magnetic structures, Eur. Phys. J. B 77, 367 (2010).
- L. J. Heyderman and R. L. Stamps, Artificial ferroic systems: Novel functionality from structure, interactions and dynamics, J. Phys.: Condens. Matter 25, 363201 (2013).
- N. Rougemaille and B. Canals, Cooperative magnetic phenomena in artificial spin systems: Spin liquids, Coulomb phase and fragmentation of magnetism–a colloquium, Eur. Phys. J. B 92, 62 (2019).
- C. Nisoli, V. Kapaklis, and P. Schiffer, Deliberate exotic magnetism via frustration and topology, Nat. Phys. 13, 200 (2017).
- B. E. Skovdal, N. Strandqvist, H. Stopfel, M. Pohlit, T. Warnatz, S. D. Slöetjes, V. Kapaklis, and B. Hjörvarsson, Temperature-induced collapse of spin dimensionality in magnetic metamaterials, Phys. Rev. B 104, 014434 (2021).
- R. P. Cowburn and M. E. Welland, Room temperature magnetic quantum cellular automata, Science 287, 1466 (2000).
- A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, Majority logic gate for magnetic quantum-dot cellular automata, Science 311, 205 (2006).
- E. Digernes, S. D. Slöetjes, A. Strømberg, A. D. Bang, F. K. Olsen, E. Arenholz, R. V. Chopdekar, J. K. Grepstad, and E. Folven, Direct imaging of long-range ferromagnetic and antiferromagnetic order in a dipolar metamaterial, Phys. Rev. Res. 2, 013222 (2020).
- U. B. Arnalds, J. Chico, H. Stopfel, V. Kapaklis, O. Bärenbold, M. A. Verschuuren, U. Wolff, V. Neu, A. Bergman, and B. Hjörvarsson, A new look on the two-dimensional Ising model: Thermal artificial spins, New J. Phys. 18, 023008 (2016).
- U. B. Arnalds, M. Ahlberg, M. S. Brewer, V. Kapaklis, E. T. Papaioannou, M. Karimipour, P. Korelis, A. Stein, S. Òlafsson, T. P. A. Hase, and B. Hjörvarsson, Thermal transitions in nano-patterned XY-magnets, Appl. Phys. Lett. 105, 042409 (2014).
- V. Kapaklis, U. B. Arnalds, A. Farhan, R. V. Chopdekar, A. Balan, A. Scholl, L. J. Heyderman, and B. Hjörvarsson, Thermal fluctuations in artificial spin ice, Nat. Nanotechnol. 9, 514 (2014).
- M. Pohlit, G. Muscas, I.-A. Chioar, H. Stopfel, A. Ciuciulkaite, E. Östman, S. D. Pappas, A. Stein, B. Hjörvarsson, P. E. Jönsson, and V. Kapaklis, Collective magnetic dynamics in artificial spin ice probed by ac susceptibility, Phys. Rev. B 101, 134404 (2020).
- G. M. Macauley, G. W. Paterson, Y. Li, R. Macêdo, S. McVitie, and R. L. Stamps, Tuning magnetic order with geometry: Thermalization and defects in two-dimensional artificial spin ices, Phys. Rev. B 101, 144403 (2020).
- V.-D. Nguyen, Y. Perrin, S. Le Denmat, B. Canals, and N. Rougemaille, Competing interactions in artificial spin chains, Phys. Rev. B 96, 014402 (2017).
- H. Lou, W.-C. Yue, Z. Yuan, P. Huang, Y. Chen, Y.-L. Wang, S. Zhang, and D. Shi, Competing interaction induced phase diagram and phase transition in artificial triangular spin ice, Phys. Rev. B 108, 184433 (2023).
- Y. Perrin, B. Canals, and N. Rougemaille, Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice, Nature (London) 540, 410 (2016).
- E. Östman, H. Stopfel, I.-A. Chioar, U. B. Arnalds, A. Stein, V. Kapaklis, and B. Hjörvarsson, Interaction modifiers in artificial spin ices, Nat. Phys. 14, 375 (2018).
- A. Farhan, M. Saccone, C. F. Petersen, S. Dhuey, R. V. Chopdekar, Y.-L. Huang, N. Kent, Z. Chen, M. J. Alava, T. Lippert, A. Scholl, and S. v. Dijken, Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice, Sci. Adv. 5, eaav6380 (2019).
- S. Zhang, J. Li, I. Gilbert, J. Bartell, M. J. Erickson, Y. Pan, P. E. Lammert, C. Nisoli, K. K. Kohli, R. Misra, V. H. Crespi, N. Samarth, C. Leighton, and P. Schiffer, Perpendicular magnetization and generic realization of the Ising model in artificial spin ice, Phys. Rev. Lett. 109, 087201 (2012).
- I. A. Chioar, N. Rougemaille, and B. Canals, Ground-state candidate for the classical dipolar kagome Ising antiferromagnet, Phys. Rev. B 93, 214410 (2016).
- C. Vantaraki, P. Ström, T. T. Tran, M. P. Grassi, G. Fevola, M. Foerster, J. T. Sadowski, D. Primetzhofer, and V. Kapaklis, Magnetic metamaterials by ion-implantation, Appl. Phys. Lett. 125, 202403 (2024).
- A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. 4, 107133 (2014).
- L. Aballe, M. Foerster, E. Pellegrin, J. Nicolas, and S. Ferrer, The ALBA spectroscopic LEEM-PEEM experimental station: Layout and performance, J. Synchrotron Radiat. 22, 745 (2015).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.111.L020408 for the calculation of energy per mesospin for infinite range interactions. We further present the FFT intensities for the implanted chains. We also provide details for the calculation of the experimental probabilities and the fitted Boltzmann distribution.
- W. Selke, The ANNNI model–theoretical analysis and experimental application, Phys. Rep. 170, 213 (1988).
- X. Zhang, Y. Lao, J. Sklenar, N. S. Bingham, J. T. Batley, J. D. Watts, C. Nisoli, C. Leighton, and P. Schiffer, Understanding thermal annealing of artificial spin ice, APL Mater. 7, 111112 (2019).
- C. Bybee, D. Kleyko, D. E. Nikonov, A. Khosrowshahi, B. A. Olshausen, and F. T. Sommer, Efficient optimization with higher-order Ising machines, Nat. Commun. 14, 6033 (2023).
- N. Mohseni, P. L. McMahon, and T. Byrnes, Ising machines as hardware solvers of combinatorial optimization problems, Nat. Rev. Phys. 4, 363 (2022).
- T. Wang, L. Wu, P. Nobel, and J. Roychowdhury, Solving combinatorial optimisation problems using oscillator based Ising machines, Nat. Comput. 20, 287 (2021).
- J. Si, S. Yang, Y. Cen, J. Chen, Y. Huang, Z. Yao, D.-J. Kim, K. Cai, J. Yoo, X. Fong, and H. Yang, Energy-efficient superparamagnetic Ising machine and its application to traveling salesman problems, Nat. Commun. 15, 3457 (2024).
- S. Bhanja, D. K. Karunaratne, R. Panchumarthy, S. Rajaram, and S. Sarkar, Non-Boolean computing with nanomagnets for computer vision applications, Nat. Nanotechnol. 11, 177 (2016).