Realizing three-dimensional artificial spin ice by stacking planar nano-arrays

Abstract

Artificial spin ice is a frustrated magnetic two-dimensional nano-material, recently employed to study variety of tailor-designed unusual collective behaviours. Recently proposed extensions to three dimensions are based on self-assembly techniques and allow little control over geometry and disorder. We present a viable design for the realization of a three-dimensional artificial spin ice with the same level of precision and control allowed by lithographic nano-fabrication of the popular two-dimensional case. Our geometry is based on layering already available two-dimensional artificial spin ice and leads to an arrangement of ice-rule-frustrated units which is topologically equivalent to that of the tetrahedra in a pyrochlore lattice. Consequently, we show, it exhibits a genuine ice phase and its excitations are, as in natural spin ice materials, magnetic monopoles interacting via Coulomb law. PACS numbers:Spin ice materials, such as rare-earth pyrochlores and artificial spin ice, are magnetic systems in which frustrated interactions lead to complex (partial) orderings and unusual collective behaviours [1][2][3]. Magnetic ions in pyrochlore spin ice form a network of corner-sharing tetrahedra whose classical magnetic macro-spins minimize the local interaction energy by obeying the twoin-two-out ice rule proposed by Pauling for the proton orderings in water ice [4]: hence the name, spin ice. It has recently been demonstrated that elementary excitations over the disordered ice manifold of spin ice materials are emergent magnetic monopoles that fractionalize from local dipole excitations [5,6].The artificial counterparts of these natural materials, artificial spin ices [3,7], are nanostructured twodimensional (2D) arrays of single-domain ferromagnetic bars that behave like giant Ising spins. Collective behaviour of the nanomagnets can be controlled through appropriate choices of material, geometry and array topology. Because of their nano-scale interaction energies (∼ 10 3 − 10 5 K depending on the size of the nanomagnet and mutual spacing) they reveal, at accessible temperatures, emergent features which in natural materials are seen only at very low temperature. Following the pioneering work of Wang et al. on the two-dimensional square-ice array [7,8], artificial spin ices have been proposed and studied in diverse types of physical systems [9][10][11][12] and geometries such as honeycomb (kagome ice) [13][14][15][16][17][18][19][20], brickwork [21], triangular [22][23][24], and pentagonal lattices [25]. A systematic approach for designing 2D arrays with emergent ice-type frustration has also been proposed [26,27], and recently realized experimentally [29].Most of the experimental efforts in such artificial frustrated magnets has understandably focused on twodimensional systems: even with mature nanolithography, it remains a great challenge to integrate a full three-dimensional (3D) structure with oblique angles between nanobars such as the pyrochlore lattice. Recently an interesting realization of a 3D artificial s...

Hopfions in a lattice dimer model

“…realizing ASI on a cubic lattice with simultaneous magnetization flips around each square plaquette, may result in realizing ASI, which dynamics is the same as considered in this paper. Finally, we mention, that in the recent years, there were numerous efforts to create 3D artificial spin ice [29][30][31], and we emphasize that creating hopfions requires a lattice with just two layers.…”

Spin ice Thin Film: Surface Ordering, Emergent Square ice, and Strain Effects

“…However, they fail miserably on this score [70] as, with planar geometry, the band width of ice rules states is as large as the energy scale for producing topological defects. The band width can be reduced in model systems by varying the relative height between horizontal and vertical chains of nano-islands [70,71]. This is naturally what happens here with the pyrochlore geometry [41,45] which, in addition, by having magnetic moment orientations along the [111] axes of the cubic cell, possesses the high symmetry of the vertex configurations that ASI arrays lack.…”

Focus on artificial frustrated systems

“…The field continues to expand and move forward at a remarkable pace, exploring many new directions. These include the cross-over with magnonics in terms of the analysis of the spectra of high frequency dynamics [42], studies of disorder-induced criticality [43], proposals for three-dimensional systems [44], and the development by Budrikis et al of network theories to represent the changes of state in artificial spin ice systems [45]. From a more technological perspective, viewing these systems as information storage [33] or processing technologies could also lead to nanomagnetic logic architectures [46] based on frustrated arrays.…”