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

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.