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

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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.

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Figure 2. Various geometries for triangular artificial spin ice. Several early papers considered collinear (in-plane magnetized) islands arranged on a planar triangular lattice (a). Mól et al. used the same type of islands, but placed them along the edges of a triangular lattice (b). Finally, several papers have used circular islands made from materials with out-of- plane magnetization (e.g., Pt/Co multilayers). Arranging these islands on a triangular lattice yields a frustrated triangular- lattice antiferromagnetic system (c).

Like quasicrystal materials, these two-dimensional artificial quasicrystals have long-range order but lack translational symmetry, and like artificial spin ice, the vertices in artificial quasicrystals generally obey an ice rule dictating the number of moments pointing into or out of a vertex, and they have a large number of low-energy configurations produced by frustration19. The connected quasicrystal lattices differ from artificial spin ice comprised of disconnected islands, however, in that short-range exchange interactions also contribute to the energy hierarchy of vertex configurations, and magnetization reversal may occur via domain wall propagation.

3. VERTEX FRUSTRATION In 2013, Morrison and coworkers introduced the concept of vertex frustration in artificial spin ice21, examples of which is shown in Figure 3. In a non-vertex-frustrated lattice, the nanomagnets in each vertex (clusters of several converging islands) can arrange their moments such that the magnetostatic energy of the vertex is minimized. But in a vertex- frustrated array, interactions with the neighboring portions of the lattice prevent some of the vertices from achieving the lowest-energy state. Vertex frustration is similar to ordinary geometrical frustration because, in both cases, constraints imposed by the lattice geometry frustrate the system and lead to a degeneracy of low energy states. Vertex frustration differs from ordinary geometrical frustration because the basic units being frustrated are the vertices rather than the individual island moments. Using vertex frustration, one can design a large number of complex lattice structures that can be specifically tuned to exhibit a physical phenomenon of interest. The original theory paper of Morrison et al. described the structure of half a dozen vertex-frustrated lattices and predicted their ground states21, only a few have yet been experimentally studied.

The vertex-frustrated lattice which has received the most scrutiny is the shakti lattice21-23. Like many vertex-frustrated lattices, the shakti lattice is a modification of the square lattice, in this case with one quarter of the islands removed. The lattice, shown in Figure 3a, contains both three- and four-island vertices, and the lattice geometry prevents all of the vertices from reaching their ground state configurations. By considering the relative energy cost of the various possible defects, one can show that the best possible energy minimization involves placing half of the three-island vertices in defect configurations. This means that each square plaquette (one plaquette is indicated in light blue in Figure 3a) of the lattice will contain two ground-state vertices and two excited vertices, and there will be six ways of arranging the excited vertices, all with the same energy. This degeneracy in vertex arrangements mimics the original frustration of natural spin ice, something that two-dimensional artificial square spin ice is unable to do because of geometry constraints and the anisotropy of the dipolar interaction. Gilbert et al. were able to directly observe this “ice rule for vertices” using magnetic force microscopy to image a thermally annealed sample of shakti artificial spin ice22. The three-island vertices

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centered ellipse on the would overlap another such circlewith centered of on any The islands comprising the arrays are ellipses respectively. The arrays were generated by with placing islands major and minor axes of orientations 470 drawn from nm a and random 170 uniform nm distribution. An island location was rejected if a 590 nm some some preliminary measurements of artificial spin glass, a nanomagnet array similar the model random frustration spin of glasses. symmetry symmetry or, in the case of the artificial quasic symmetry. Frustrated magnetism is not, however, limited to geometrically frustrated instance, regular are lattices. alloys Spin in glasses, which for magnetic ions are distributed ra interactions between these isolated moments vary in strength and sign and can frustrate one another. Here All All of the types of artificial spin ice described up to this point consist of regular lattices that possess translational defined defined ground state at a temperature at which the vertex energy configurations. Correlations between moments in the dimensional Ising disordered model, so in vertex this case, vertex frustration leads to lattice The tetris lattice This lattice contains alternating bands of vertex thermalization, the non similar to those monopole sliding symmetry thewithin charge also possess a magnetic charge and were observed to organize themselves into antiferromagnetic

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Figure 4. Artificial spin glass. A scanning electron micrograph of the randomly located and oriented islands is shown in (a) for a sample with 7500 islands arranged in a 60 µm × 60 µm area. Panel (b) depicts the corresponding SEMPA image, which reveals island moments pointing in random directions with no type of order present.

2500, 5000, and 7500 islands were fabricated using a standard electron beam lithography process with a bilayer resist stack that is described elsewhere24. An artificial square spin ice array with 700 nm lattice constant was fabricated on the

same sample to verify thermalization of the island moments. A ≈6 nm Ni80Fe15Mo5 film was deposited on a doped silicon substrate at ≈0.05 nm/s via electron beam evaporation, with a ≈2 nm Pt capping layer to prevent oxidation, followed by liftoff. The configuration of the island moments was measured using scanning electron microscopy with polarization analysis (SEMPA). The Pt capping layer was removed with in situ Ar+ ion etching, and a few monolayers of Fe were evaporated to increase the magnetic contrast (this Fe film is thin enough to remain paramagnetic on the nonmagnetic regions of the sample while exchange coupling to the island magnetization and increasing the electron polarization measured by SEMPA). The square lattice, which was shown in Figure 1, is in its completely-ordered ground state, indicating that the island moments were able to thermally equilibrate during the thin film deposition28. The artificial spin glass array (Figure 4) shows islands magnetized in all directions. The random structure of the lattice precludes the type of spatially-periodic long-range magnetic order found in the square lattice (Figure 1b), though there is a small net magnetic moment for area of the sample shown in Figure 4 (Mx = 0.05 ± 0.02, My = -0.08 ± 0.02, where the magnetization is normalized such that Mx (My) = 1 if the sample magnetization is saturated in the x (y) direction and the uncertainty represents the standard error calculated from the standard deviation of the distribution of island moment directions). This artificial spin glass sample allows the direct visualization of the random frustration of magnetic moments present in spin glass. Future work on artificial spin glass samples may allow the quantification of frustration as a function of areal density of islands and the exploration of the role of random frustration in determining the system’s susceptibility to thermal fluctuations.

5. SUMMARY Artificial spin ice permits the detailed study of frustration on a microscopic level while providing the ability to tune the interactions between magnetic moments. The geometries described here, such as the various triangular lattices, quasicrystals, and vertex frustrated lattices, provide an interesting way to probe the physics of frustration that complements measurements on the conventional square and kagome artificial spin ices. Even these are not a complete catalog of the possibilities for artificial magnetic materials. Defects such as dislocations can be individually fabricated and measured in artificial spin ice. Drisko and coworkers demonstrated that dislocations in artificial square spin ice nucleated strings of excited vertices that extended either to another dislocation or the array edge29. Recently, a modified artificial spin ice lattice designed to allow eight different ordered configurations has been described30. The magnetic charges of the vertices in this system should allow information recording, and the states of the individual vertices can be

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modified with the combined effects of an applied magnetic field and a magnetized MFM tip. The possibility of three- dimensional artificial spin ice remains appealing from a theory perspective31-33, though the challenges of fabricating such a complicated lattice structure have limited its experimental realization34. The magnetic moments comprising an array need not even be Ising spins: artificial XY model arrays34 and an artificial Potts model (even utilizing the magnetocrystalline anisotropy of Fe films35) have already been demonstrated. While the direct visualizations of the ice rule and magnetic monopole-like excitations have already been achieved, new lattice geometries promise to provide even more insight into the physics of frustration.

6. ACKNOWLEDGEMENTS This project was supported by the National Institute of Standards and Technology, Center for Nanoscale Science and Technology under project numbers R13.0004.04 and N09.0017.07 (nanofabrication). I.G. acknowledges support from the National Research Council’s Research Associateship Program and thank John Unguris for assistance with SEMPA measurements.

REFERENCES

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