Thermally induced magnetic relaxation in building blocks of artificial kagome spin ice

Alan Farhan, Armin Kleibert, Peter M. Derlet, Luca Anghinolfi, Ana Balan, Rajesh V. Chopdekar, Marcus Wyss, Sebastian Gliga, Frithjof Nolting, and Laura J. Heyderman

Phys. Rev. B 89, 214405 – Published 9 June 2014

Abstract

We have performed a study of thermally driven magnetic relaxation in building blocks of artificial kagome spin ice. For room-temperature measurements, we observe that low-energy states are accessed with high efficiency, particularly in structures with strong dipolar coupling and with low thicknesses. With carefully tuned heating experiments, we demonstrate how thermally active artificial spin ice systems relax magnetically from higher-energy states and eventually fall into low-energy states. The methods applied in our work offer the possibility to observe the thermodynamics of artificial spin ice systems in real space and time, and provide a way to directly investigate the nature of complex stochastic processes.

Received 28 November 2013
Revised 14 May 2014

DOI:https://doi.org/10.1103/PhysRevB.89.214405

Authors & Affiliations

Alan Farhan^1,2, Armin Kleibert^3,*, Peter M. Derlet⁴, Luca Anghinolfi^1,2,5, Ana Balan³, Rajesh V. Chopdekar^1,3, Marcus Wyss^1,6, Sebastian Gliga^1,2, Frithjof Nolting³, and Laura J. Heyderman^1,2,†

¹Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, Switzerland
²Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
³Swiss Light Source, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
⁴Condensed Matter Theory Group, NUM, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
⁵Laboratory for Neutron Scattering, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
⁶Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

^*armin.kleibert@psi.ch
^†laura.heyderman@psi.ch

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Vol. 89, Iss. 21 — 1 June 2014

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Images

Figure 1
Three-ring building blocks of artificial kagome spin ice: (a) SEM image of a three-ring structure with nanomagnets of length $L$ = 470 nm, width $W$ = 170 nm, and thickness $d$ = 5 nm patterned within a lattice parameter $a$ = 500 nm. (b) Corresponding magnetic contrast image from which the orientation of the nanomagnet moments can be determined.
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Figure 2
(a) The six degenerate ice-rule moment configurations at a vertex of artificial kagome spin ice. (b)–(f) Schematics of building block structures of artificial kagome spin ice and some of their low-energy states shown together with their degeneracy, $D$ , and energy, $Δ E$ , relative to the ground-state energy for a lattice parameter $a$ = 500 nm. While all shown states belong to the first (lowest-energy) band, all structures possess a ground state, as indicated, and with an increasing system size there is an increasing number of degenerate low-energy states belonging to this band.
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Figure 3
(a) Thickness of the permalloy film measured at different positions across a wedge sample. (b)–(d) Percentages of low-energy states observed at different thicknesses $d$ as a function of the lattice parameter $a$ at $T = 300$ K (filled circles). At lower film thicknesses in (c) and (d), we observe spontaneous configurational changes within the time scale of XMCD imaging. Recording image sequences, consisting of 10 images per sequence, we determine the average percentages of low-energy states achieved with the standard deviations given as error bars. Colored lines correspond to thermal occupancies of low-energy states for one-, two- and three-ring structures derived from Eq. (2). The filled circles are the PEEM data and both the experimental measurements as well as the simulations were performed for lattice parameters $a$ = 500, 560, 600, and 700 nm.
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Figure 4
(a)–(h) Thermal relaxation of a three-ring structure ( $a = 500$ nm and $T = 330$ K), going from an energetically higher state to one of 24 degenerate low-energy states within 168 seconds. Blue arrows in (a) to (h) represent reversed nanomagnet moments compared to the initial moment configuration in (a). The time $t$ and the energy relative to the ground state $Δ E$ are shown to visualize the evolution of the system within the energy landscape. (i) Percentages of low-energy states achieved (closed shapes) in arrays of kagome building blocks after thermal annealing involving heating up to 330 K and cooling down to 300 K following relaxation. While the number of rings (up to three) does not seem to have a large effect on the numbers of low-energy states achieved, an increasing lattice parameter (decreasing dipolar coupling) significantly affects the occupancy of the lowest energy states, which can be explained by a strong decrease in the energy gap between the first and the second energy bands with decreasing dipolar coupling. The percentages of low-energy states achieved following a demagnetizing protocol [21] are given for comparison (open squares, circles and triangles).
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Figure 5
Density of states calculated for a seven-ring kagome building block structure ( $a = 500$ nm) revealing a gap between the first and the second energy bands. The upper inset demonstrates the systematic decrease of the energy gap with increasing system size. The lower inset focuses on the first (lowest) energy band, with the black line being a plot of the Boltzmann factor at a temperature of 330 K. Multiplying the Boltzmann factor (black line) by the calculated density of states gives the predicted thermal occupancy at $T = 330$ K.
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Figure 6
(a)–(h) Thermal relaxation of a seven-ring artificial kagome spin ice structure ( $a = 500$ nm), from a high-energy state to an external flux closure state (low-energy state), where perimeter nanomagnets have all their moments aligned head to tail. This is followed by a number of moment reorientations mainly in interior nanomagnets, leading to transitions between several low-energy states of similar energies $Δ E$ relative to the ground state (i)–(l). The ground state was never observed. Blue arrows in (b)–(l) represent reversed moments of nanomagnets, compared to the initial configuration in (a).
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