The modification of geometry and interactions in two-dimensional magnetic nanosystems has enabled a range of studies addressing the magnetic order [1][2][3][4][5][6] , collective low-energy dynamics 7,8 and emergent magnetic properties 5,9,10 in, for example, artificial spin-ice structures. The common denominator of all these investigations is the use of Ising-like mesospins as building blocks, in the form of elongated magnetic islands. Here, we introduce a new approach: single interaction modifiers, using slave mesospins in the form of discs, within which the mesospin is free to rotate in the disc plane 11. We show that by placing these on the vertices of square artificial spin-ice arrays and varying their diameter, it is possible to tailor the strength and the ratio of the interaction energies. We demonstrate the existence of degenerate ice-rule-obeying states in square artificial spin-ice structures, enabling the exploration of thermal dynamics in a spin-liquid manifold. Furthermore, we even observe the emergence of flux lattices on larger length scales, when the energy landscape of the vertices is reversed. The work highlights the potential of a design strategy for two-dimensional magnetic nano-architectures, through which mixed dimensionality of mesospins can be used to promote thermally emergent mesoscale magnetic states.Lithographic techniques can be used to fabricate magnetic nanoarrays, in which the interaction between the elements can be chosen by, for example, the distance between the islands. This approach has been used in a number of previous works, addressing both the order and dynamics of magnetic nanostructures [1][2][3][4][5][6][7][8]12 . In the specific case of square artificial spin ice (SASI), this approach has even enabled tailoring of the thermal dynamics and relaxation 8,[13][14][15] , as well as experimental realizations 9 of the degenerate square-ice model 16 . The distance and thereby the coupling strength for nearest and nextnearest neighbours are different in SASI (d 1 ≠ d 2 ; see Fig. 1), resulting in the loss of degeneracy. As a consequence, the ice-rule-obeying vertices, with two islands pointing in-two islands pointing out, are split into two groups (T I and T II ) with different energies (E I < E II ). One way to remedy this shortcoming is to shift parts of the lattice in the third dimension 9,17,18 . An alternative way to modify the energy landscape is to introduce an interaction modifier, as illustrated in Fig. 1b. In these modified SASI (mSASI) arrays, all islands have the same distance, or gap G, to the interaction modifier. While a height offset might seem the obvious choice for manipulating the coupling strengths between the islands, the use of interaction modifiers at the vertices of artificial spin-ice structures is not only lithographically much easier to achieve, but also opens up completely new avenues for tailoring their energy landscapes. Instead of having a system consisting of only one type of island, we use two subsystems with widely different shape anisotropies a...

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The properties of natural and artificial assemblies of interacting elements, ranging from Quarks to Galaxies, are at the heart of Physics. The collective response and dynamics of such assemblies are dictated by the intrinsic dynamical properties of the building blocks, the nature of their interactions and topological constraints. Here we report on the relaxation dynamics of the magnetization of artificial assemblies of mesoscopic spins. In our model nano-magnetic system - square artificial spin ice – we are able to control the geometrical arrangement and interaction strength between the magnetically interacting building blocks by means of nano-lithography. Using time resolved magnetometry we show that the relaxation process can be described using the Kohlrausch law and that the extracted temperature dependent relaxation times of the assemblies follow the Vogel-Fulcher law. The results provide insight into the relaxation dynamics of mesoscopic nano-magnetic model systems, with adjustable energy and time scales, and demonstrates that these can serve as an ideal playground for the studies of collective dynamics and relaxations.

We present a direct experimental investigation of the thermal ordering in an artificial analogue of an asymmetric two dimensional Ising system composed of a rectangular array of nano-fabricated magnetostatically interacting islands. During fabrication and below a critical thickness of the magnetic material the islands are thermally fluctuating and thus the system is able to explore its phase space. Above the critical thickness the islands freeze-in resulting in an arrested thermalized state for the array. Determining the magnetic state of the array we demonstrate a genuine artificial twodimensional Ising system which can be analyzed in the context of nearest neighbour interactions.

A hot-deformed Nd-Fe-B sample has been chosen for the investigation of interaction domains by means of magnetic force microscopy. During the imaging process, a magnetic field of up to 6 T was applied in situ along the easy axis of magnetization. The thermally demagnetized state presents a regular pattern of interaction domains with an average width of about 1 μm but with a much larger length scale. Starting from the thermally demagnetized state, magnetization along the initial magnetization curve occurs via sequential switching of neighboring grain columns at the peripheries of the interaction domains. Demagnetization of a saturated sample takes place through the nucleation and expansion of a patchy domain pattern with a much larger extension and a substructure in the lateral range of the underlying grain size. Reversal processes under an applied magnetic field also take place at the borders of the domains.