Non-stochastic switching and emergence of magnetic vortices in artificial quasicrystal spin ice
Introduction
Geometrically frustrated (GF) systems are of high interest from a fundamental physics perspective, since all competing magnetic interactions cannot be simultaneously optimized down to T = 0 K; this circumstance forces the system to adopt one of many nearly degenerate low-energy states that comprise a manifold of high residual entropy [1]. The study of GF systems was initiated by the discovery of residual entropy in water ice [2]. When the common Ih phase of water ice forms, oxygen atoms order, but hydrogen atoms do not. Each oxygen ion obeys the two “Bernal–Fowler ice rules”: (1) There should be one proton per O–O bond on average. (2) Two protons must be located close to the oxygen ion, and the remaining two protons must be situated further away [3]. This “ice rule” is termed the “two-in, two-out rule”. Pauling showed that the near-degeneracy among a large number of proton configurations that obey the ice rules is the cause of the observed residual entropy [4].
An analogous rule has been applied to natural “spin ice” (SI) such as rare earth pyrochlores in which local magnetic moments reside on the vertices of corner-sharing tetrahedra [5]. At two of the four vertices of a given tetrahedron, the magnetic moment of the rare earth ion points towards the center of the tetrahedron, and the other two point away from the center to minimize energy. Like water ice, the number of nearly degenerate configurations that obey the ice rule within a macroscopic sample is extremely large, which gives rise to a large residual entropy; hence spin ice materials are GF [6].
Advances in nanolithography allow one to fabricate mesoscopic patterned thin films for which theoretical concepts can be tested via direct manipulation of structural parameters; one such system is “artificial spin ice” (ASI) [7], which is a connected or disconnected 2D network of ferromagnetic (FM) segments arranged on a lattice in which magnetic interactions among the individual segments are frustrated. ASI are therefore 2D analogues of real, GF 3D materials (e.g., pyrochlores). Two types of ASI having either periodic square or Kagome lattices have been studied extensively, with a primary focus on observing a unique thermal ground state that is assumed to split off from the degenerate manifold [7], [8], [9], [10].
Recently, there has been growing interest towards understanding field-induced reversal of periodic ASI systems such as honeycomb or Kagome lattices [11], [12], [13], [14], [15]. Theoretical and experimental studies on honeycomb/Kagome ASI reveal “stochastic” switching and “avalanches” in which neighboring segments sequentially reverse along chains [12]. The critical field required to switch the magnetization of a particular segment depends upon the magnetization M, thickness t, width w, and angle θ between a segment’s long axis and the applied field direction. The formation of a vortex loop on an individual hexagon of the honeycomb lattice implies segments that are parallel to each other must have opposite magnetization directions, which minimizes stray magnetic field energy in an equilibrium remnant state. Indeed, closed vortex loops have been observed experimentally in Kagome ASI [15]. Therefore, the formation of vortex loops on a honeycomb lattice can initiate with the switching of any segment that is parallel to the applied field, making it a stochastic process.
The question now arises: Are there GF systems in which the reversal of individual segments and creation of vortex loops are non-stochastic? We anticipate such a system should have the following properties: (1) Pattern vertices should not be simply coordinated (e.g., symmetric twofold, threefold, fourfold, and sixfold). (2) A given vertex should not have a local environment identical to that of neighboring vertices.
Quasicrystals such as the Penrose P2 tiling (P2T) [16], [17], [18] possess the required characteristics (see Fig. 1): (a) P2T have vertices that are symmetric or asymmetric fivefold, asymmetric threefold, etc. (b) The P2T is aperiodic, which forbids vertices with identical environments. (c) A P2T has a GF topology [19].
We have therefore chosen to study permalloy thin films patterned into P2T antidot lattices that can also be described as multiply-connected, aperiodic nanowire networks (see Fig. 1). Herein, we report our experimental and simulation results for the magnetic reversal of a 3rd generation P2T tiling. This work extends our initial study of the static and dynamic magnetization of P2T [19].
Section snippets
Experimental details
We numerically generated P2T via the “deflation method” [20] using a modified graphics algorithm that was incorporated into our electron beam lithography (EBL) software. The long and short edges of Penrose “kites” were initialized at 3.45 and 2.13 μm, respectively, for Sample III134E. Five of these Penrose kites were then linked at a common vertex to form a “Penrose sun” configuration that we term the “0th generation of the deflation rule”. In successive iterations, each kite was replaced by two
Results, discussion and analysis
Fig. 2 shows DC magnetization data at T = 310 K and T = 5 K for 3rd generation sample III134E (w = 85 nm); the data are also compared with the results of a T = 0 K, OOMMF simulation (w = 100 nm) for a P2T of essentially the same geometry. The DC hysteresis curves exhibit increased coercivity for T = 5 K compared with the T = 310 K data; moreover, the T = 5 K DC magnetization curve for Sample III134E is in good agreement with the simulated data for a 3rd generation P2T.
It is important to note the qualitative
Conclusion
We have studied magnetic reversal in a 3rd generation P2T from an ASI perspective. Our experimental magnetization data (w = 85 nm) and simulation results (w = 100 nm) are in reasonable agreement, and indicate the presence of non-stochastic switching of Ising segments in the reversal regime. The temperature dependence of knee anomalies in the magnetization curves, the overlaps between three intervals of applied field in which segments having different orientations with respect to the applied field
Acknowledgments
Research at the University of Kentucky was supported by U.S. DoE Grant #DE-FG02-97ER45653 and U.S. NSF Grant #EPS-0814194, and the University of Kentucky Center for Computational Sciences.
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