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Chiral Magnonic Edge States in Ferromagnetic Skyrmion Crystals Controlled by Magnetic Fields

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Chiral Magnonic Edge States in Ferromagnetic Skyrmion Crystals Controlled by Magnetic Fields PHYSICAL REVIEW RESEARCH 2, 013231 (2020) Chiral magnonic edge states in ferromagnetic skyrmion crystals controlled by magnetic fields Sebastián A. Díaz ,1 Tomoki Hirosawa ,2 Jelena Klinovaja,1 and Daniel Loss 1 1Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 2Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan (Received 5 November 2019; accepted 23 January 2020; published 28 February 2020) Achieving control over magnon spin currents in insulating magnets, where dissipation due to Joule heating is highly suppressed, is an active area of research that could lead to energy-efficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here we uncover that an external magnetic field can drive a topological phase transition in the spin-wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a low-energy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the point correspond to the breathing and counterclockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states. DOI: 10.1103/PhysRevResearch.2.013231 I. INTRODUCTION nascent field of topological magnonics [19–21] is ferromag- netic (FM) skyrmion crystals (SkXs) [22,23], whose magnetic A major obstacle to meeting the demand for ever smaller texture is shown in Fig. 1. Although their low-energy spin- logic and storage devices, based on expectations set by wave modes have been investigated before [24–27], only re- Moore’s law, is the pernicious effect of Joule heating [1]. cently were they predicted to host chiral magnonic edge states Two diverging approaches relying on spin degrees of freedom protected by the nontrivial topology of their bulk magnon instead of electric charges have come to the rescue. Embracing bands [28]. heating effects and trying to utilize them to produce electronic In contrast to electronic topological insulators, ground- spin currents is the goal of the field of spin caloritronics [2]. state magnetic phases such as FM SkXs, and consequently the Alternatively, avoiding displacing electric charges altogether spin-wave spectrum they support, depend crucially on applied can be achieved via spin currents carried by magnons in magnetic fields. Therefore, it is a nontrivial question whether insulating magnets [3–6]. While several magnonic analogs of a magnetic field can drive a topological phase transition in electronic phenomena that could help design the next gener- ation of energy efficient devices have already been identified the spin-wave spectrum without drastically modifying the [3,7–9], a growing niche for magnon spin current applications ground-state texture. Quite remarkably, we discover that there that stray from those inspired by conventional electronics has is a window of magnetic-field values where such a phase tran- recently emerged [10,11]. sition between two topologically distinct spin-wave spectra of Despite progress on launching and getting a handle on the the FM SkX can be observed. It involves two low-energy bulk directionality of magnon spin waves [12–15], reliable control magnon bands, which at the point correspond to the counter- of their propagation and robustness against sample imperfec- clockwise and breathing modes [26]. Experimental studies in tions are still lacking. Realizing such control could be attained bulk chiral magnets showed that the energy difference of these in topological magnon insulators [16–18], systems with a modes gets smaller with increasing magnetic field, but it does spatially periodic magnetic texture that support topologically not vanish upon reaching the boundary with the conical phase protected magnonic edge states. A promising platform for the [27]. On the other hand, our two-dimensional model predicts a critical magnetic-field value at which these two bands touch at the point, closing a magnon gap and signaling the topological phase transition. Increasing the magnetic field past this critical value reopens the gap, the bands exchange Chern Published by the American Physical Society under the terms of the numbers, and band inversion takes place in the vicinity of Creative Commons Attribution 4.0 International license. Further the gap-closing point. Furthermore, topologically protected distribution of this work must maintain attribution to the author(s) chiral magnonic edge states within this gap are only supported and the published article’s title, journal citation, and DOI. in the phase below the critical field. Therefore, an external 2643-1564/2020/2(1)/013231(9) 013231-1 Published by the American Physical Society DÍAZ, HIROSAWA, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 013231 (2020) partitioned as r = R + r j. The Hamiltonian of free spin waves is then obtained, by taking a lattice Fourier transform, as the O(S) piece in the expansion 1 H = S † H (k) + E . (2) SW 2 ki ij k j 0 k;i, j − · + Here = (a , a† )T , a = √1 e ik (R r j )a , E = ki ki −ki k j N R R j 0 − 1 2 NS i i, with N the total number of magnetic unit cells, (k) − (k) H = ij ij , ij(k) −∗ − ∗ − (3) ij( k) ij( k) where (k) = 1 [J (k)e+ · e+ + D (k) · e+ × e+], (k) ij 2 ij i j ij i j ij = δ − 1 + · − + · + × − = ij i 2 [Jij(k)ei e j Dij(k) ei e j ], i j[Jij FIG. 1. Ferromagnetic skyrmion crystal texture. Triangular crys- = 3 · 3 + = · 3 × 3 − 3 · = μ (k 0)ei e j Dij(k 0) ei e j ] bJei zˆ, with b g B tal of ferromagnetic Néel skyrmions obtained from the classical −ik·(R+r −r ) B/JS, J (k) = J + , e i j , and mutatis mutandis ground-state texture of the spin-lattice Hamiltonian model for D/J = ij R R ri r j { 1, 2, 3} 1.0andb = 0.8. The out-of-plane component of the moments is for Dij(k). The orthonormal frame e j e j e j is defined at 3 color coded. each magnetic unit cell site j, where e j is parallel to the ± = 1 ± 2 classical ground-state spin texture and e j e j ie j . magnetic field could be used to control the propagation of A paraunitary Bogoliubov transformation Tk diagonalizes robust magnon spin currents with a defined directionality the above spin-wave Hamiltonian yielding along the edge of FM SkXs in thin films. † 1 H = S E λ α α λ + + E , (4) SW k kλ k 2 0 II. SPIN WAVES IN A FERROMAGNETIC k,λ SKYRMION CRYSTAL α† where kλ creates magnons with crystal momentum k at The FM SkX is obtained from the classical ground-state the λth band whose energy is given by Ekλ. Each band can texture of spin-S operators on a triangular lattice (with lattice be classified by their Chern number, a topological index constant a) governed by the Hamiltonian defined in reciprocal space, given by an integral over the first Cλ = i dk μν {Pλ k ∂ Pλ k ∂ Pλ k }/ π 1 Brillouin zone, Tr ( )[ kμ ( )] kν ( ) 2 , † H =− (Jr,r Sr · Sr + Dr,r · Sr × Sr ) = σ σ with Pλ(k) Tk λ zTk z a projection operator and 2 r,r ( λ) = δλ δ [16]. ij i ij We numerically compute the bulk magnon band structure + gμ B S · zˆ, (1) B r and the Chern number of the lowest-energy bands as a func- r tion of external magnetic field. Using magnon coherent states = including nearest-neighbor exchange Jr,r J and interfacial (details in Appendix B), we also obtain the time evolution Dzyaloshinskii-Moriya (DM) interactions Dr,r = D zˆ × (r − of the real-space deformation of the FM SkX for those same r )/|r − r | as well as a coupling to an external magnetic bands. field Bzˆ, where g is the gyromagnetic ratio and μB is the Bohr magneton. Focusing on zero temperature, we use Monte III. NEARLY FLAT AND DISPERSIVE BANDS Carlo simulated annealing [29] followed by further time evo- lution employing a restricted Landau-Lifshitz-Gilbert (LLG) Now we discuss the magnon bands. Figures 2(a)–2(d) show equation [30] to obtain the classical ground-state spin texture the lowest-energy bulk magnon bands for increasing values of as a function of the magnetic field. The magnetic moment the (rescaled) external magnetic field b. The majority of the texture of the FM SkX is related to the classical ground-state bands are dispersive, but a few of them appear to be nearly spin texture by Mr =−g.s.|Sr|g.s., whose direction mˆ r = flat. As the magnetic field increases, the nearly flat bands are Mr/|Mr| is shown in Fig. 1. shifted to higher energies, while the dispersive bands remain Spin waves are described by rotating the spin quantization almost unaffected. Moreover, overlapping bands belonging axis at each lattice site along the direction of the FM SkX to these two different classes hybridize less the lower their spin texture so that the rotated spin operators model collective energy is.
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