Magnetic order and energy-scale hierarchy in artificial spin-ice structures

Henry Stopfel, Erik Östman, Ioan-Augustin Chioar, David Greving, Unnar B. Arnalds, Thomas P. A. Hase, Aaron Stein, Björgvin Hjörvarsson, and Vassilios Kapaklis

Phys. Rev. B 98, 014435 – Published 30 July 2018

Abstract

In order to explain and predict the properties of many physical systems, it is essential to understand the interplay of different energy scales. Here we present investigations of the magnetic order in thermalized artificial spin-ice structures, with different activation energies of the interacting Ising-like elements. We image the thermally equilibrated magnetic states of the nanostructures using synchrotron-based magnetic microscopy. By comparing results obtained from structures with one or two different activation energies, we demonstrate a clear impact on the resulting magnetic order. The differences are obtained by the analysis of the magnetic spin structure factors, in which the role of the activation energies is manifested by distinct short-range order. These results highlight the potential of artificial spin-ice structures to serve as model systems for designing various energy-scale hierarchies and investigating their impact on the collective dynamics and magnetic order.

Received 5 June 2018
Revised 29 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.014435

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Henry Stopfel^1,*, Erik Östman¹, Ioan-Augustin Chioar¹, David Greving², Unnar B. Arnalds³, Thomas P. A. Hase², Aaron Stein⁴, Björgvin Hjörvarsson¹, and Vassilios Kapaklis¹

¹Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
²Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
³Science Institute, University of Iceland, 107 Reykjavik, Iceland
⁴Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA

^*Corresponding author: henry.stopfel@gmail.com

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Vol. 98, Iss. 1 — 1 July 2018

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Images

Figure 1
A schematic representation of the Shakti (SH) and modified-Shakti (mSH) lattices along with magnetic states representative of each temperature regime, generally defined by the Curie ( $T_{C}$ ) and the blocking temperatures ( $T_{B 1 / B 2}$ ). Elements in their paramagnetic state are gray, superparamagnetic elements are light blue, and the frozen elements are represented with a magnetic north (blue) and south (red) pole, reflecting their final magnetic orientation. The gray-scale images to the left are representative PEEM-XMCD snapshots obtained at 65 K, the lowest temperature reached during the cooling.
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Figure 2
Long-mesospin correlations in the Shakti lattice. The network-averaged magnetic correlations for long mesospins spatially oriented vertically/horizontally are shown in the left/right graph. In blue/red the correlations of long mesospins which are oriented in a string and in light blue/orange the correlations of the long mesospins diagonally oriented are represented, as indicated in the inset. All correlations show that already the first neighbor is randomly oriented and therefore the magnetic orientations of the long mesospins are not correlated. Error bars represent one standard deviation.
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Figure 3
Vertex type definition and statistics for SH and mSH lattices. SEM image of the SH (a) and mSH (b) lattice. The different vertex types are indicated by their respective colors. The real-space base vectors for the Shakti lattice are defined along the $x$ and $y$ directions and each has a length of four times the lattice spacing ( $d = 600 nm$ ). (c) Definition of the vertex states for all different vertex types. While both lattices share the vertex type with a coordination number of four (green), the mSH lattice contains vertices with coordination number three (red) and two (orange). In the SH lattice the threefold-coordinated vertices (blue) include a long mesospin. The fourfold- (d) and threefold-coordinated (e) vertex statistics for SH (dark blue) and mSH (dark red) lattices show no significant differences. Error bars represent one standard deviation.
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Figure 4
Spin structure factor maps computed for the fourfold-coordinated vertex sublattices of the mSH (left) and the SH (right) lattices. The real-space lattice vectors as well as the subset of fourfold-coordinated vertices (red islands) are indicated in the insets.
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Figure 5
Spatially limited spin structure factor as indicator for short-range order. (a) Illustration of the mesospin subset used in the SSF calculations. The real-space lattice vectors ( $a_{1}, a_{2}$ ) which are used for the reciprocal lattice units (r.l.u.) are also shown. The real-space input window size for the averaged SSF $I_{1}$ [red; SSF (b) and (d)] and $I_{9}$ [blue; SSF (c) and (e)] are highlighted. (b)–(e) The averaged SSF maps for SH [(b) and (c)] and mSH [(d) and (e)] lattices, with reduced real-space input size according to Eq. (2). (f) Diagonal line profiles for the SSF of different input sizes for the SH (dark blue) and mSH (dark red) lattices. The modulation of the intensity in the SSF maps at $(q_{x}, q_{y}) = (2, 2)$ [r.l.u.] is clearly distinguishable for the case of the SH lattice and an indication of a different order for these length scales.
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Figure 6
Spin arrangements across the long mesospins/twofold-coordinated vertices. (a) Schematic representation of all mesospins contained within two neighboring threefold-coordinated vertices (gray numbered islands). We classify the arrangements of all these mesospins into three classes, denoted as $A_{1}$ , $A_{2}$ , and $A_{3}$ . The abundance values of $A_{1}$ and $A_{2}$ are averages of the four possible spin arrangements in these classes. One representative spin arrangement for each of the three classes is depicted in (a). The magnetization directions of islands without arrows are not in consideration for the specific spin arrangement and can be magnetized either way. The coupled fourfold-coordinated vertices are indicated by the light-green islands. The number of coupled fourfold-coordinated vertices across the type-I $_{2}$ vertex or a long mesospin is two for $A_{1}$ , three for $A_{2}$ , and all four for $A_{3}$ . (b) The abundance of these three classes is plotted for the SH (dark blue) and the mSH (dark red) lattices; the abundance for each spin arrangement in a random mesospin state is represented with a gray inside bar. The abundance reveals the distinct differences in the arrangement of the mesospins around the type-I $_{2}$ vertex and strongly indicates a defined tiling for the fourfold-coordinated vertices around the long mesospins and the buildup of a short-range order. Error bars represent one standard deviation.
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