Invited Paper Exploring frustrated magnetism with artificial spin ice Ian Gilbert*a and B. Robert Ilica aCenter for Nanoscale Science and Technology, National Institute of Standards and Technology, 100 Bureau Dr., MS6202, Gaithersburg, MD 20899 ABSTRACT Nanomagnet arrays known as artificial spin ice provide insight into the microscopic details of frustrated magnetism because, unlike natural frustrated magnets, the individual moments can be experimentally resolved and the lattice geometry can be easily tuned. Most studies of artificial spin ice focus on two lattice geometries, the square and the kagome lattices, due to their direct correspondence to natural spin ice materials such as Dy2Ti2O7. In this work, we review experiments on these more unusual lattice geometries and introduce a new type of nanomagnet array, artificial spin glass. Artificial spin glass is a two-dimensional array of nanomagnets with random locations and orientations and is designed to elucidate the more complex frustration found in spin glass materials. Keywords: Artificial spin ice, frustrated magnetism, spin ice, spin glass 1. INTRODUCTION When the interactions between the microscopic components of a condensed matter system (e.g., the atomic spins in a magnetic material) cannot all simultaneously be satisfied, the system is said to be frustrated. This competition between interactions produces a broad range of interesting phenomena1. Examples of geometrically frustrated magnetic materials 2 are the spin ices Ho2Ti2O7 and Dy2Ti2O7 . The pyrochlore crystal lattice of these materials includes a network of corner- sharing tetrahedra with Dy ions located at each tetrahedron corner, and crystal field constrains the rare earth ions’ moments to point directly into or out of the tetrahedra. The rare earth spins’ ferromagnetic coupling is frustrated in this geometry, because there is no way to place four spins on the four corners of a tetrahedron such that they all point head- to-tail. This geometrical frustration produces a six-fold degenerate “compromise” configuration in which two spins point into and two spins point out of each tetrahedron3. The frustration of spin ice causes two particularly interesting effects. First, the degeneracy associated with this two-in, two-out “ice rule” gives spin ice a residual entropy that persists down to the lowest experimentally-accessible temperatures4. Second, the elementary excitations of spin ice behave like magnetic monopoles5. An excitation occurs wherever the ice rule is broken, and if one considers the spins not as point dipoles but as separate north and south magnetic poles, the adjacent tetrahedra (three-in, one-out or vice versa) will have a net magnetic charge. These monopole excitations can move apart by reversing a chain of spins called (again in analogy with magnetic monopoles) a Dirac string. 6 Artificial spin ice was developed as a mesoscopic analog to spin ice systems such as Dy2Ti2O7 . Elongated islands a few hundred nanometers long and made from a ferromagnetic material such as permalloy (Ni81Fe19) are fabricated in frustrated lattices to model spin ice. The islands contain a single ferromagnetic domain that is constrained by the island’s shape anisotropy to point along the island’s long axis, which makes the island moment behave like a giant Ising spin. Artificial spin ice possesses two advantages over natural spin ice. First, because the samples are fabricated using electron beam lithography, the sample geometry can be easily tuned. The island shape and size can be modified to change the properties of the moments, the lattice constant can be tuned over a wide range, and the lattice geometry can be changed at will. Such tailoring of interactions is not possible with natural materials like spin ice. Second, the exact configuration of the individual island moments can be imaged using techniques such as magnetic force microscopy (MFM), something that is not possible for atomic spins in bulk crystals. The first experiments on artificial spin ice demonstrated that when nanomagnets were arranged on a frustrated square lattice (an example of which is shown in Figure 1), an ice rule analogous to that found in spin ice resulted6. The vertices (sites at which several islands converge) of the square lattice exhibited a strong preference for configurations obeying the two-moments-in, two-moments-out ice rule. Further experiments have shown monopole-like excitations similar to those found in spin ice7,8. The majority of artificial spin ice investigations have considered the square and kagome lattices. Since these results have been described recently in Spintronics IX, edited by Henri-Jean Drouhin, Jean-Eric Wegrowe, Manijeh Razeghi, Proc. of SPIE Vol. 9931, 99311P · © 2016 SPIE · CCC code: 0277-786X/16/$18 · doi: 10.1117/12.2237000 Proc. of SPIE Vol. 9931 99311P-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/30/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx , a111111111111 reelel i i i i i i i ie' I bjell ellei jell leVellf I1Ì111111111111i sl1l1liliilll 11111111111111e - - - -- - - e1ÌÌÌÌÌÌÌÌIIÌÌÌlÌi` Ì ÌÏ ÌÌ riel !a!`Ì 11111111111111ÌÌ Ì Ì Ì le IÌ Ì Ì Ìes 11111111111111 IÌ 111111111111111ÌÌÌÌÌiÌÌ1ÌÌÌ1 o 4D - - - - - - - - - - .I.ÌÌ Ì lÌIÌ Ì Ìi 1 Ì 1 .111111111111114, 411 410 4W deD 411 4D 411 4, 4D 4 1111 i111114/_i_1 ,11111111111111 AD AD 4D AD 411 4, 4D 4D 4D 4D 4, !!1 ! il11 i 1!i 11111111111111e Ill e e e 411 II iÌeÌ1Ì 1 1 1 1 1 1 1 1 1 1 11,r,,1 Ì Ì 1e1 Ì Ì ÌleÌ ! 11111111111 - e e e e e e .....Ì1aÌÌÌÌÌÌÌi Figure 1. Artificial square spin ice. Panel (a) shows a scanning electron micrograph of a lattice comprised of 470 nm × 170 nm elliptical islands arranged on a square array with lattice constant 700 nm. Panel (b) shows an image taken with scanning electron microscopy with polarization analysis (SEMPA), which reveals the direction of each island’s magnetization. The magnetization direction is color coded according to the color wheel inset in the lower right corner, and nonmagnetic areas are black. This particular sample is in its ground state, which is an ordered arrangement of island moments with two moments pointing into and two out of each vertex (the sites where four islands come together in the shape of a plus sign). several excellent review articles9,10, here we will focus on reviewing other, more novel lattice geometries, describing both completed experiments and further proposals. 2. TUNING GEOMETRICAL FRUSTRATION Many of the early studies of new lattice geometries utilized the triangular lattice in various forms. Several possibilities are shown in Figure 2. The first consists of collinear nanomagnets placed on the points of a triangular Bravais lattice, as shown in Figure 2a. Ising spins with equal nearest-neighbor antiferromagnet interactions on a lattice comprised of equilateral triangles is one of the first (and simplest) examples of geometrical frustration11. The anisotropy of the dipolar interactions of the in-plane-magnetized islands reduces the degree of frustration and permits several types of order to develop12, depending on the relative size of the two lattice constants (labeled x and d in Figure 2a). A detailed analysis of the correlations between island moments in the triangular lattice revealed that in some cases the sign of the correlation between two islands was opposite what one would expect based on the sign of the dipolar interaction13. This was attributed to indirect interactions between the two islands mediated by other, neighboring islands, analogous to the Ornstein-Zernike theory used to describe the structure of liquids. Another possible arrangement of islands is to place the long islands between the points of a triangular Bravais lattice, with the island long axes parallel to the lattice vectors (Figure 2b). This scenario was considered from a theoretical perspective by Mól and coworkers, who noted that such a system has several different types of magnetically-charged excitations (e.g., six-out, five-out, one-in, etc.). Furthermore, some of these magnetically-charged excitations are lower in energy than uncharged excitations, and the tension (energy per unit length) of strings of flipped moments connecting excitations can have a wide range of values. This artificial triangular spin ice has not yet been studied experimentally. The third possible triangular lattice arrangement (Figure 2c) is to place islands magnetized normal to the lattice on the points of a triangular lattice15,16 (or the related hexagonal and kagome lattices17). These arrays of perpendicularly-magnetized islands could be of significant interest in the context of frustrated magnetism because the dipolar interaction between two islands depends on distance only, and not on the angle between the islands’ long axes, as is the case for in-plane magnetized islands. Frustrated nanomagnet arrays are not restricted to periodic lattices. A number of works have examined artificial quasicrystals, in which (connected) permalloy bars are arranged along the edges of Penrose18,19 or Ammann tilings20. 2 Proc. of SPIE Vol. 9931 99311P-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/30/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx 0\1 \0\1 \1 \1 \1\ \0\1 \0\1 \1 \/ \1 / \/ \/ \/ \/ \/ %/\ /\/\/\/\/\/\/\\/ \1 \0\1 \1 \/ \1 \/\/\ /\/\/\/\/ 2 1\1 \1\1 \1\1 \1% \1 \/ \1 \/ \1 \/ \1 Figure 2. Various geometries for triangular artificial spin ice. Several early papers considered collinear (in-plane magnetized) islands arranged on a planar triangular