Topologically Nontrivial Magnon Bands in Artificial Square Spin Ices with Dzyaloshinskii-Moriya Interaction

Ezio Iacocca and Olle Heinonen

Phys. Rev. Applied 8, 034015 – Published 20 September 2017

Abstract

Systems that exhibit topologically protected edge states are interesting both from a fundamental point of view as well as for potential applications, the latter because of the absence of backscattering and robustness to perturbations. It is desirable to be able to control and manipulate such edge states. Here, we show that artificial square ices can incorporate both features: an interfacial Dzyaloshinskii-Moriya interaction gives rise to topologically nontrivial magnon bands, and the equilibrium state of the spin ice is reconfigurable with different configurations having different magnon dispersions and topology. The topology is found to develop as odd-symmetry bulk and edge magnon bands approach each other so that constructive band inversion occurs in reciprocal space. Our results show that topologically protected bands are supported in square spin ices.

Received 21 December 2016

DOI:https://doi.org/10.1103/PhysRevApplied.8.034015

Physics Subject Headings (PhySH)

Frustrated magnetism

Ferromagnets

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ezio Iacocca^1,2,* and Olle Heinonen^3,4

¹Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA
²Department of Physics, Division for Theoretical Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden
³Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
⁴Northwestern-Argonne Institute for Science and Engineering, Evanston, Illinois 60208, USA

^*ezio.iacocca@colorado.edu

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Issue

Vol. 8, Iss. 3 — September 2017

Subject Areas

Magnetism

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Images

Figure 1
(a) Schematic of the square ASI unit cell. The composing nanoislands are identical with length $l$ , width $w$ , and thickness $t$ . The lattice constant $d$ is taken as the center-to-center distance between adjacent nanoislands. Interfacial DMI is imparted through a heavy-metal layer. (b) Schematic of a vortex (type-I) unit cell. (c) Schematic of a remanent (type-II) unit cell.
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Figure 2
Micromagnetically computed ground states for a series of Py nanoislands of different thicknesses with (a)–(c) $D = 0$ exhibiting $S$ states and (d)–(f) $D = 1 mJ / m^{2}$ exhibiting $C$ states. The gray scale and arrows represent the $\hat{z}$ and in-plane magnetization components, respectively.
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Figure 3
Even frequencies at the $Γ$ point as a function of $D$ for (a) Py nanoislands of thicknesses $t = 10$ , 15, and 20 nm and (b) ${Co}_{75} {Fe}_{25}$ nanoislands of thicknesses $t = 5$ , 10, and 15 nm. Good agreement is observed between the even bulk mode obtained semianalytically (solid blue curves) and micromagnetically (empty red circles). The even edge mode obtained semianalytically (dashed black curves) also agrees with those obtained micromagnetically (filled red circles). However, a qualitative deviation of the $D$ dependence for 20-nm-thick Py nanoislands [(a) right panel] is observed.
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Figure 4
Band diagram for odd modes in 10-nm-thick Py nanoislands. The FBZ path is depicted at the top of each column. Band diagrams for both $D = 0$ and $D = 0.75 mJ / m^{2}$ are shown. (a) Bulk odd modes exhibit a Dirac cone along the $Γ − X$ direction indicated by a red arrow. The symmetry between the paths ${FBZ}^{+}$ and ${FBZ}^{-}$ is broken when $D \neq 0$ . (b) Edge modes exhibit band touching in the $Γ − M$ direction, but for the edge mode bands, these do not accumulate Berry phase, and the band touching occurs both for the topologically trivial and nontrivial cases. The Chern number $c$ for each band is specified in each panel.
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Figure 5
Field dependence of the band diagram at $\vec{k} = π / d (0.25, 0.25)$ for (a) Py and (b) ${Co}_{75} {Fe}_{25}$ . The field is directed along the (1,0) direction in the top panels and along the (1,1) direction in the bottom panels. The red arrows indicate the appearance of a field-dependent Dirac cone at 18 mT for Py and 28 mT for ${Co}_{75} {Fe}_{25}$ .
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Figure 6
Angle dependence of the band diagram at $\vec{k} = π / d (0.25, 0.25)$ for (a) Py at $| H | = 18 mT$ and (b) ${Co}_{75} {Fe}_{25}$ at $| H | = 28 mT$ . Bands do not touch when $D = 0$ (top panels), while Dirac cones are observed in the odd-symmetry edge modes at $θ_{H} = π / 4$ and $3 π / 4$ (bottom panels).
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