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Magnetic ordering in out-of-plane artificial spin systems based on Archimedean lattices

A. Pac1,2,*, G. M. Macauley1,2,†,‡, J. R. Massey1,2, A. Kurenkov1,2, F. Mila3, P. M. Derlet1,4, and L. J. Heyderman1,2,§

  • 1Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
  • 2PSI Center for Neutron and Muon Sciences, 5232 Villigen PSI, Switzerland
  • 3Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
  • 4PSI Center for Scientific Computing, Theory and Data, 5232 Villigen PSI, Switzerland

  • *Contact author: aleksandra.pac@proton.me
  • †Contact author: gavin.macauley@psi.ch
  • ‡Present address: Department of Physics, Princeton University, Princeton, NJ 08540, USA.
  • §Contact author: laura.heyderman@psi.ch
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Phys. Rev. B 113, 144403 – Published 2 April, 2026

DOI: https://doi.org/10.1103/n2ns-j2hp

Abstract

Artificial spin systems, sometimes referred to as artificial spin ices, are arrays of coupled nanoscale magnets that order according to the lattice geometry, nanomagnet shape and magnetic anisotropy. Here, we characterize a family of artificial spin systems that are formed by placing arrays of out-of-plane nanomagnets on the vertices of the Archimedean lattices. On demagnetizing these lithographically fabricated nanomagnet arrays using a magnetic field protocol and subsequently imaging the magnetic configuration using magnetic force microscopy, we observe experimentally how these systems order. We compare our experimental results with those predicted by Monte Carlo simulations to assign an effective temperature to each lattice. We find that, for all of the lattices, the assigned effective temperature is above the transition temperature. This reflects the difficulty of obtaining system-spanning order in lattices with out-of-plane nanomagnets. We consider to what extent further-neighbor interactions affect the thermal properties and spin-spin correlations in each lattice, illustrating our results with four example lattices. We can divide the lattices into three main groups: bipartite lattices that admit a perfect antiferromagnetic ground state, frustrated lattices where ordering proceeds via a single step, and frustrated lattices with two-step ordering. Our work highlights the diversity of magnetic ordering that can be hosted in two-dimensional artificial spin systems with out-of-plane nanomagnets, and demonstrates the importance of including longer-range dipolar interactions to explain the magnetic ordering.

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Phys. Rev. B 113, 144403– Published 2 April, 2026
  • Received 9 March 2025
  • Revised 21 November 2025
  • Accepted 20 January 2026
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