Topological Magnon Modes in Patterned Ferrimagnetic Insulator Thin

Letter Cite This: Nano Lett. 2018, 18, 3032−3037

pubs.acs.org/NanoLett

Topological Magnon Modes in Patterned Ferrimagnetic Insulator Thin Films Yun-Mei Li,†,‡ Jiang Xiao,§,∥ and Kai Chang*,†,‡,⊥ †

SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China College of Materials Science and Optoelectronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China § Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China ∥ Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China ⊥ Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China ‡

S Supporting Information *

ABSTRACT: Manipulation of magnons opens an attractive direction in the future energy-efficient information processing devices. Such quasi-particles can transfer and process information free from the troublesome Ohmic loss in conventional electronic devices. Here, we propose to realize topologically protected magnon modes using the interface between the patterned ferrimagnetic insulator thin films of different configurations without the Dzyaloshinskii-Moriya interaction. The interface thus behaves like a perfect waveguide to conduct the magnon modes lying in the band gap. These modes are immune to backscattering even in sharply bent tracks, robust against the disorders, and maintain a high degree of coherence during propagation. We design a magnonic Mach−Zehnder interferometer, which realizes a continuous change of magnon signal with varying external magnetic field or driving frequency. Our results pave a new way for realizing topologically protected magnon waveguide and finally achieving a scalable low-dissipation spintronic devices and even the magnonic integrated circuit. KEYWORDS: Magnon, topological phase, interfacial state, Mach−Zehnder interferometer

C

spintronics, aiming at reducing energy dissipation without compromising processing speed.9−12 In comparison with traditional magnetic metals, magnetic insulators, for example, yttrium−iron garnet (YIG), offer us a new platform to transfer and manipulate information by magnons without using conduction electrons, avoiding the Joule heating completely.11,13,14 The frequencies of the magnons or spin waves range from gigahertz (GHz) to terahertz (THz), and the propagation distance is up to millimeters.15−18 The dispersion and group velocity of magnons can also be engineered, which can form magnon waveguide19−22 and give rise to the chiral magnon modes when considering the Dzyaloshinskii-Moriya interaction22−28 or dipole interactions.29,30 The magnon dispersions can also be controlled straightforwardly by using the magnonic crystals, in which the magnetic properties are modulated periodically in real space.11,12,31,32 In this Letter, we propose a topologically protected magnonic waveguide using the interface between two artificial magnonic crystals with different topological charge. The

urrently there are significant interests to explore topological phases in various physical systems. These topological phases are characterized by integer topological invariants,1−4 which can not be changed continuously. Therefore, the physical properties associated with the topological invariant are robust against to perturbations. The integer topological invariants such as the Chern number indicate the nontrivial band structures of bulk crystals and the existence of topologically protected edge states, which has been proposed for applications in, for example, spintronics and quantum computation. Very recently, most efforts in this field have been devoted to extend topological concepts to photon,5 polariton,6 and even acoustic artificial systems.7,8 In comparison with the electron system, the photonic and acoustic systems can behave more cleanly because of the lacking of interactions between photons and acoustic waves. For device purposes, it could be more easily to construct photonic and acoustic artificial crystals with flexible and tunable edge states because the characteristic size of artificial lattices in these systems is much larger than that in electron system. As collective excitations of magnetic order in magnetic materials, magnons, that is, spin waves, are considered as a promising candidate as information carrier in the field of © 2018 American Chemical Society

Received: February 4, 2018 Revised: April 18, 2018 Published: April 20, 2018 3032

DOI: 10.1021/acs.nanolett.8b00492 Nano Lett. 2018, 18, 3032−3037

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Nano Letters

a perpendicular anisotropy field,33,34 which can be absorbed into Hext.) The magnetization M(r,t) can be divided into the out-of-plane (static) and the in-plane (dynamical) components, M(r,t) ≃ Msez + m(r,t), where |m(r,t)| ≪ Ms. By defining a complex field m(r,t) = mx(r,t) − imy(r,t), the equation of motion for m(r,t) can be obtained from the standard Landau− Lifshitz−Gilbert equation as (see Supporting Information for the details)

artificial magnonic crystal is constructed on a ferrimagnetic thin film with triangular-shaped holes periodically located on a hexagonal lattice. The sign of the half-integer topological charge at each valley is determined by the rotation direction of the triangular holes (THs) with respect to the underlying lattice orientation. Such rotation breaks the mirror symmetry of the crystal, leading to a gap opening at the corners of the Brillouin zone. Thus, an interface between two magnetic crystals with opposite rotation directions of THs is expected to hold an nontrivial topological invariant. According to the “bulk-edge correspondence”, gapless chiral modes emerge in the band gap and propagate along the interface. These modes that are due to the topological protection are immune to the backscattering and ideal for low-dissipation information transport. Utilizing the topological magnonic edge modes, we design a magnonic Mach−Zehnder interferometer, realizing a continuous change of magnon signal by changing external magnetic field or driving frequency. By proper design, the interferometer can be a magnetometer to detect magnetic field as small as the geomagnetic field. Our proposal paves the way for designing topological protected magnonic devices to achieve energy efficient information processing. As depicted in Figure 1a, the physical system under consideration is a hexagonal magnonic crystal consisting of

i(1 + iβMs)

∂m(r, t ) z = γ(Hext + Hms )m − γMsλex2 ∇2 m − γMsh(r, t ) ∂t

(1)

∇·(λ2ex∇)m

where is the exchange field with exchange coupling strength parametrized by λ2ex. h(r,t) = hx − ihy, hx, hy are the xand y-component of the demagnetization field, respectively, and z Hms is the z-component of the magnetostatic field (see Supporting Information for the definitions). Here γ is the electron gyromagnetic ratio and β is the Gilbert damping constant. We model the boundaries of the THs with the exchange boundary condition n·∇m(r,t) = 0,35 where n is the normal direction of the boundary. Similar as solving the Schrödinger equation in periodic structure, we employ the ansatz m(r,t) = mk(r)ei(k·r−ωt), h(r,t) = hk(r)eik·r−iωt31 and neglect the damping term. The eq 1 becomes a secular equation ωmk (r) = γ[H0 − Msλex2 (2i k·∇ + ∇2 )]mk (r) − γMshk (r) (2)

where H0 = Hext + Hzms + Msλ2exk2 and the exchange boundary condition reads n·∇mk(r) + in·kmk(r) = 0. Because of the nontrivial boundary configurations, we calculate the band structures numerically using the finite element method (see Supporting Information for the details). The band structures of the hexagonal magnonic crystal with rotation angle α = 0° and 10° are shown in Figure 1b. The dispersion shows a pair of well-defined extrema (that is, the valleys) in each corner of the first Brillouin zone (FBZ), that is, in K and K′ valleys, which are related by the time-reversal symmetry. For α = 0, the magnon band structure exhibits a gapless feature at the corners of the FBZ. For α ≠ 0, the broken mirror symmetry lifts the degeneracies of the valley states, inducing a finite band gap. The gapless (α = 0) and the gapped (α ≠ 0) band structures are similar to the cases of graphene without and with staggered potentials on A and B sublattices.36−38 The in-plane magnetization of the valley states, the Kv and Kc states in the upper inset of Figure 1b, localized at the points with C3 symmetry, that is, the A and B points, as shown in Figure 1c. We denote the two orthogonal states as mA(r) and mB(r) with the corresponding eigenfrequencies ωA and ωB. The reversal of the rotation direction leads to the sign change of the effective band gap Δ = ωA − ωB = −sgn(α)ΔK, where ΔK = |ωA − ωB| is the absolute band gap. The state in the vicinity of K (K′) point can be expanded as mq(r) = cqAmA(r) + cqBmB(r), where q = k − K(′). By substituting the state mq(r) into eq 2, one can obtain the effective low-energy Hamiltonian near K and K′ points39 (see Supporting Information for the derivation)

Figure 1. (a) Schematic of hexagonal magnonic crystal consisting of rotated triangular-shaped holes with the rotation angle α. a1 and a2 are lattice vectors and the side length of the THs is L. A and B are the positions with C3 symmetry. (b) The magnon energy dispersions. The lattice constant |a1| = 34.6 nm and L = 20 nm. μ0Hext = 344 mT. The red dashed line shows the band structure when α = 0, whereas the blue solid line indicates the one when α = 10°. The film thickness is 4 nm. The up-middle inset is the zoom in around the K-point. The right inset shows the first Brillouin zone of the hexagonal magnonic crystal. (c) The in-plane magnetization distributions of Kv and Kc states. (d) Band gap dependence on the rotation angle.

triangular-shaped holes on the ferrimagnetic YIG thin film. The orientation of the THs has an angle α with the lattice vectors as defined in Figure 1a. An out-of-plane magnetic field Hext = Hextez (Hext > Ms) is applied to guarantee the equilibrium magnetization along with the z-direction. (The magneto crystalline anisotropy energy in YIG thin film is equivalent to

Heff = ω0q + vK (qxσx + τzqyσy) − sgn(α)

ΔK σz 2

(3)

where σ = (σx,σy,σz) are the Pauli matrix and τz = ± denotes the 1 valley index. ω0q = (ω Kc + ω K v )/2 + η0 q·K + 2 q2 , vK = η0|

(

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DOI: 10.1021/acs.nanolett.8b00492 Nano Lett. 2018, 18, 3032−3037

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Figure 2. Magnon chiral states in the presence of the interfaces. (a,b) Dispersions of chiral states (red curves) with horizontal direction interface (a) and vertical direction interface (b). The widths of the ribbon are 600 nm (a) and 693 nm (b). The rotation angle is α = ±10°. (c,d) Propagation of chiral state at frequency f = 14.3 GHz in the horizontal direction interface (c) and vertical direction interface (d). The rotation angle is α = ±10°. The blue strip represents the radio frequency antenna generating an alternating magnetic field that excites magnons. The top right inset is the zoom in figures of the interfacial states. In (d), down right inset shows the propagation direction. The size of the lattice in (c,d) is 1.2 μm × 1.2 μm.

Figure 3. Magnon propagation in various types of interfaces. The excitation frequency is f = 14.3 GHz. (a) The interface with two 90° corners. The symbol “−” and “+” in the down left inset denote the rotation directions of the THs. “−” means clockwise while “+” represents counterclockwise. The size of the square is 450 nm × 450 nm. The same for the subsequent figures. (b) Cross-junction interface. The rotation angle for (a,b) is α = ±20°. (c) Interfacial states with rotation α randomly distributed between 10° and 20°. The length of THs is 20 nm. (d) Interfacial states with the length of THs randomly distributed between 16 and 24 nm. The rotation angle is α = ±20°. The size of the lattice in (c,d) is 1.33 μm × 1.23 μm. 1

x = ∫ mA*(r)( −i∂x)mB(r)d3r . The pxAB|, η0 = 2γMsλ2ex, and pAB integral region is the unit cell. In order to describe the topological property of the magnonic crystal, we define the Berry curvature and Chern number in the second band as Ω(k) = ∇ k × ⟨m k |i∇ k |m k ⟩ and 1 C = 2π ∫ d 2 kΩ(k). The effective Hamiltonian in eq 3 has BZ been investigated in graphene system36 and the topological

charge in each valley is given by * = 2 τz sgn(α). The Chern number of the band is zero because of the time reversal symmetry. In the artificial magnonic crystal made of THs, the sign of the topological charge in each valley can be controlled by the rotation direction of the THs. Thus, we may easily create interfaces between two regions with different topologies simply 3034

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Figure 4. Magnon Mach−Zehnder interferometer. (a) Schematic of Mach−Zehnder interferometer. (b,c) Interference pattern with external magnetic field 344 mT (b) and 362 mT (c). The excitation frequency is 14.4 GHz. The rotation angle α = ±30°. The cell size is 1.5 μm × 1.26 μm.

have opposite topological charge, the intervalley scattering is forbidden for gapless magnons. On the basis of this feature, we design a magnonic beam splitter using a cross-junction interface as shown in Figure 3b. The thin film is partitioned into four quadrants, where the THs in the first and third quadrants are rotated clockwise (negative α) and those in the second and fourth quadrants are rotated counterclockwise (positive α). Such arrangement of THs forms a cross-road type interfaces. The value of topological charge associated with each path along the interfaces are marked with the solid (blue) and dashed (red) lines in the inset of Figure 3b. It is evident that the eastbound magnon on path 1 cannot make its way to path 3 but can be diverted into path 2 and path 4, that is, a beam splitter. Because of the locking of the valley degree of freedom to the propagation direction, the chiral modes are topologically protected and thus robust against the disorders. Here we consider two kinds of disorders, the uncertainty in rotation angles and sizes of the THs. In the first case, the rotation angles of THs randomly distribute between 10° and 20° but the rotation direction is kept unchanged. In the second case, the length of sides of the THs randomly distributes between 16 and 24 nm. It is well-known that in disordered systems, the waves are more sensitive to disorders at lower dimension.42,43 The chiral modes are effectively one-dimensional states, therefore one would expect these states to experience Anderson localization. Nevertheless, this behavior appears to be completely evaded by the topological protected states, just like the edges states in the disordered quantum Hall systems. Figure 3c,d shows that the waveguide effect of the interface remains with either type of disorder, no localized states, or standing waves formed along the interface. Above, we have proven the existence of the topological states and its robustness regardless of the shape and disorder. Now we investigate the possibility to construct the magnonic device. As previously proven, the crossroad of the interface is like a beam splitter. The beam is split into two beams at the cross. In coherent optics, the coherence between the two beams split from one beam is guaranteed when the optical path difference is smaller than the coherent length. Different from optics, the coherence between the split two magnon beams is ensured by the back scattering immunity feature of the topological states. Or equivalently speaking, the phase of magnons are kept undisturbed by the bend tracks and disorders. Therefore, we can also say that topological magnons in YIG have a very long coherent length as the damping is quite small. The interference of the split two beams in Figure 3b can be observed by proper path design. We construct a Mach−Zehnder interferometer (MZI), as shown in Figure 4a. The beam is split at the left crossroad and then meets at the right crossroad. The path

by drilling THs with opposite rotation angles on the two sides of the interface, as shown in the insets in Figure 2a,b. The topological charge is * = *1 − *2 = τz , where *1 and *2 are the topological charges at the two sides of the interface, respectively. Then each valley carries an integer topological charge. As a result, gapless chiral modes are expected at their interface. To confirm the existence of the gapless states, we numerically calculate the band structures for two types of interface configurations, that is, a horizontal and a vertical direction interface, as shown in Figure 2a,b. Both configurations contain two sets of gapless states with opposite group velocities. Therefore, such interfaces form ideal waveguides for the gapless states. For the case of horizontal interface as in Figure 2a, the two sets of gapless states are bounded to different valleys, and they move toward opposite directions along the interface, similar to the graphene system with stagger potentials.37,40 These chiral states are localized at the interface. Figure 2c,d shows the propagation of these chiral states in a patterned 4 nm thick film.41 In our simulations, we use a radio frequency antenna to excite magnons. For α = ±10°, the magnons are found to be localized and propagate along the interface as shown in Figure 2c,d. The effective width of the waveguide is determined by the gap. A larger gap (that is, a larger rotation angle) gives a narrower effective width. The effective width of the waveguide is about 200 nm for the angle α = ±10° while 150 nm for the angle α = ±20° and 100 nm for the angle α = ±30°. The values are larger than the value 5.7 nm reported in ref 19 and smaller than 340 nm reported in the recent experiment.20 The chiral states lies in the band gap, that is near K and K′ points. The states thus carry a crystal momentum K (or K′). In the vertical interface (see Figure 2d), only the chiral states near the red and blue points in Brillouin zone (see the down-left inset of Figure 2d) are allowed to propagate. In order to excite the magnons, the antenna should be placed titled respect to the horizontal axis with the angle 30°. In our simulations, the Gilbert damping effect is included in our micromagnetic simulation. The path length of the interface in Figure 2c,d is 1.2 μm, much smaller than the decay length in YIG thin film, approximately 860 μm,17 therefore the damping effect is quite small in this case. The existence of the gapless states are independent of the interface configurations. Meanwhile, the states moving in opposite directions hold different topological charges, which suppresses the backscattering during the propagation of magnons. Consequently, the magnon propagates nicely through a bent interface, as shown in Figure 3a. The rotation angle here is ±20°. The waveguide shows a narrower effective width because of a larger band gap. Because the two valleys 3035

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Nano Letters difference between two path is δl ≃ 20a, where a is lattice constant. At the right crossroad, there are also two allowed propagation paths, that is, path 3 and 4, which are marked by the blue dashed arrows. Interestingly, there is an intrinsic πphase difference between the two paths (see Supporting Information for the proof). By changing the excitation frequency or external bias magnetic field, we can change the wavevector and hence change the phase difference between the two paths. Let q be the wavevector away from K points. When q = 0 (magnetic field 344 mT), path 3 experiences constructive interference while path 4 experiences destructive interference (see Figure 4b). While q·δl = π (magnetic field 362 mT), the interference results are reversed, as clearly shown in Figure 4c. Experimentally, via a continuous adjustment of the external magnetic field Hext, one can achieve continuous change of q, resulting in a phenomenal continuous change of magnon signal at the detector. The topological modes are linear dispersed. q is proportional to the change of the magnetic field. By increasing the path difference δl, a smaller variation of the magnetic field will remarkably change the magnon signal at the detector. For example, by extending the length of the path 2 in Figure 4 a (path 2 can be any shape, a battlement-shaped interface can largely increase the path length), the path difference will be increased significantly by orders of a few hundred times larger. When δl = 4000a, the magnetic field sensitivity of the signal is as small as 1 Gs, comparable to the magnitude of the geomagnetic field. The above band structure analysis indicates that only the change of out-of-plane component of the magnetic field will affect the interference pattern. To detect the direction and magnitude of the unknown external magnetic field, we can rotate the MZI with the fixed magnetic field Hext = 344 mT kept always perpendicular to the YIG film. The interference pattern of MZI will be changed when the MZI is rotated respect to an unknown external magnetic field. From the variation of the interference pattern, one can determine the magnitude and direction of external magnetic fields. Because the interference patterns are sensitive to the magnetic fields, the magnonic MZI can thus be used as a magnetometer. The system we proposed should be quite practical to be realized experimentally using the nanofabrication techniques.31,44−46 The topological modes are tunable by changing the lattice constant and the size of THs or by tuning the external magnetic field. In our model, the spin wave mode we discussed is in the exchange-dominated region. Because the topological mode is inherently related to the symmetry and symmetry breaking, therefore we believe the topological interfacial magnon modes still can exist for the dipolardominated magnons, that is, the large lattice constant case (usually a few hundred nanometers to a few micrometers). In summary, we proposed a topologically protected magnonic waveguide constructed from the interface between two artificial magnonic crystals with opposite rotation directions of the THs. The rotation of the THs breaks the mirror symmetry of the crystal, leading to the gap opening and a half-integer topological charge at the corners of the FBZ. The interface between the two crystals holds an integer topological charge, giving rise to the topologically protected modes that lie in the band gap and propagate along the interface. These modes are immune to backscattering in bent tracks and disordered interfaces. We design a magnonic MZI, achieving a continuous change of magnon signal with varying external magnetic field or driving frequency. Although we consider YIG ferrimagnetic thin film here, our conclusions also hold for

ferromagnetic insulators. Our proposal paves a completely new way to construct magnon-based spintronic device and even magnonic integrated circuit.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.8b00492. The calculation of the band structures, the derivation of effective Hamiltonian, the details of micromagnetic simulation and the phase calculation in Mach−Zehnder interferometer (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone:+8682304438. ORCID

Yun-Mei Li: 0000-0002-8074-467X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y.-M.L. thanks Weichao Yu for his helpful discussions. Y.-M.L. and K.C. were supported by the MOST of China (Grants 2015CB921503, 2016YFE0110000, and 2017YFA0303400), and the NSFC (Grant 11434010). J.X. was supported by the National Natural Science Foundation of China under Grant 11722430 and National Key Research Program of China under Grant 2016YFA0300702.

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REFERENCES

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