Chiral edge mode in the coupled dynamics of magnetic solitons in a honeycomb lattice
CChiral edge mode in the coupled dynamics of magnetic solitons in a honeycomb lattice
Se Kwon Kim and Yaroslav Tserkovnyak
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Dated: November 13, 2018)Motivated by a recent experimental demonstration of a chiral edge mode in an array of spinninggyroscopes, we theoretically study the coupled gyration modes of topological magnetic solitons,vortices and magnetic bubbles, arranged as a honeycomb lattice. The soliton lattice under suitableconditions is shown to support a chiral edge mode like its mechanical analogue, the existence ofwhich can be understood by mapping the system to the Haldane model for an electronic system.The direction of the chiral edge mode is associated with the topological charge of the constituentsolitons, which can be manipulated by an external field or by an electric-current pulse. The directioncan also be controlled by distorting the honeycomb lattice. Our results indicate that the lattices ofmagnetic solitons can serve as reprogrammable topological metamaterials.
Introduction. —The term metamaterials refer to a classof man-made composite materials which can offer func-tionalities beyond those found in nature via collectivedynamics of constituent elements [1]. Inspired by therobust edge states in the topological electronic phasessuch as quantum Hall states [2], topological metameteri-als with analogous edge states have been proposed andrealized in optical [3], acoustic [1], magnetic [4], and me-chanical systems [5]. In particular, it has recently beenshown theoretically [6] and experimentally [7] that a hon-eycomb lattice of spinning gyroscopes can support a chi-ral edge mode that is protected from small perturbationssuch as lattice distortions and thus can be identified asa topological mechanical metamaterial. As discussed inRef. [7], an open challenge for its practical applicationsis to find a feasible way to keep gyroscopes spinning.Quantum-mechanically, nature has already endowed usa permanent gyroscope: spin of a particle. This intrin-sic angular momentum manifests itself macroscopicallythrough the gyrotropic force in the dynamics of mag-netic solitons with topologically nontrivial textures suchas magnetic bubbles (also known as skyrmions) and vor-tices [8]. These solitons and their dynamics have at-tracted much attention of physicists due to their funda-mental properties [9] and technological promise [10, 11].In particular, the collective gyration modes of arrays ofvortex disks have been studied theoretically [12] and ex-perimentally [13–16] as reprogrammable metamaterialswhose functionalities can be controlled by changing vor-tices’ polarities and chiralities [17].When viewing topological magnetic solitons as gyro-scopes, it is natural to expect that a honeycomb latticeof the solitons can support a chiral edge mode as its me-chanical analogues [6, 7]. In this Letter, we verify theexpectation both by numerically solving the equations ofmotion for the dynamics of coupled solitons and by map-ping the system to the Haldane model for an electron ingraphene, which is known to exhibit the quantum Halleffect [18]. We also show that the direction of the edgemode can be controlled either by changing the topolog-ical charge of the solitons or by distorting the geometry ( a ) ( b )( c ) ( d ) Q = 12 Q = Q = 1 Q = FIG. 1. Schematic illustrations of (a) a vortex with the topo-logical charge Q = 1 /
2, (b) a vortex with Q = − /
2, (c) amagnetic bubble with Q = 1, and (d) a magnetic bubble with Q = − of the honeycomb lattice. We conclude the Letter withan experimental outlook, including a possibility of thethermal chirality control using ferrimagnets [19]. Model. —We consider a two-dimensional array of mag-netic solitons such as vortices and magnetic bubbles,which are characterized by their topological charges, Q ≡ π (cid:90) dxdy n · ( ∂ x n × ∂ y n ) , (1)which measures how many times the unit vector n alongthe direction of the local magnetization wraps the unitsphere. The elementary topological charges of vorticesand magnetic bubbles are Q = ± / Q = ±
1, re-spectively, the sign of which is determined by the internalstructure. See Fig. 1 for schematic illustrations of them.The slow motion of the solitons can be described by theirpositions, R j ≡ ( X j , Y j ), which are assumed to be sub-jected to the restoring force toward equilibrium positions, R j ≡ ( X j , Y j ). The low-energy dynamics of the coupledsolitons can be described by Thiele’s equation [8] withinthe approximation of the rigid soliton texture: G ˆ z × ˙ U j − αD ˙ U j + F j = 0 , (2)where U j ≡ R j − R j is the displacement of the soli-ton from the equilibrium position, G ≡ − πstQ is the a r X i v : . [ c ond - m a t . o t h e r] J u l (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π ⦿ xyz (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π ⦿ k x ! / ! . .
30 2 ⇡/ a ⇡/ a ⇡/a h y i LL ⦿ xyz ! / ! . . h y i LL ⊗ ( a ) ( b )( c ) ( d ) k x ⇡/ a ⇡/ a ⇡/a FIG. 2. (a) A schematic illustration of physically separatedferromagnetic disks (drawn as circles) in a honeycomb latticewith zigzag edges; vortices in disks have the polarity p = 1 andthus the topological charge Q = 1 /
2. The gray lines betweencircles represent the magnetostatic interactions between vor-tices. The arrows along the edges represent the directions ofthe chiral mode. (b) The one-dimensional dispersion for thecoupled gyration modes of the system shown in (a), which isobtained by solving Eq. (4) numerically. The symbol a repre-sents the distance between the second-nearest neighbors. Thecolor represents the average vertical position (cid:104) y (cid:105) weighted bythe amplitude squared corresponding to each mode. (c), (d)Analogous figures for vortices with the polarity p = −
1, cor-responding to the topological charge Q = − / gyrotropic coefficient, s ≡ M s /γ is the spin density ofthe magnet, M s is the saturation magnetization, γ isthe gyromagnetic ratio, t is the thickness of the mag-net, αD ≡ αcst is the viscous coefficient, c is a dimen-sionless geometric factor determined by the exact profileof the solitons, α is the Gilbert damping constant [20],and F j ≡ − ∂U/∂ U j with U the potential energy as afunction of the displacements. Here, the first term is thegyrotropic force proportional the spin density, which iscrucial for the analogy between the lattice of magneticsolitons and the lattice of mechanical gyroscopes; thesecond term is the viscous force; the third term is theconservative force.To the quadratic order in the displacements, the en-ergy of the system is modeled by U = (cid:80) j K U j / (cid:80) j (cid:54) = k U jk /
2, where the first term is the pinning poten-tial parametrized by the spring constant
K > U jk = I (cid:107) ( d jk ) U (cid:107) j U (cid:107) k − I ⊥ ( d jk ) U ⊥ j U ⊥ k . (3)Here, d jk ≡ | R j − R k | is the distance between two soli-tons in the absence of the interaction; u (cid:107) j ≡ ˆ e jk · U j is the projection of the displacement U j onto the line connecting two solitons, described by the unit vectorˆ e jk ≡ ( R k − R j ) /d jk ; U ⊥ j ≡ (ˆ z × ˆ e jk ) · U j is the pro-jection of U j onto the line perpendicular to ˆ e jk ; I (cid:107) ( d jk )and I ⊥ ( d jk ) parametrize the corresponding interactions,which are attractive (repulsive) if the value is positive(negative). We will simplify the subsequent discussion byassuming that the interactions are much weaker than thepinning potential, | I (cid:107) | , | I ⊥ | (cid:28) K . The interaction of thisform can capture the magnetostatic interactions betweentwo vortices in separated disks [12] and the exchange-mediated interactions between two magnetic bubbles, aswill be explained further below. Thiele’s equation (2)with the inter-vortex interactions in Eq. (3) has beenemployed successfully to describe the observed dynamicsof an array of vortex disks [13, 14].We are interested in the dispersion of the normal modesof the coupled gyration dynamics of solitons. Since themain effects of the viscous force with the small Gilbertdamping α (cid:28) U j ≡ ( u j , v j ) [Eq. (2)] can be written as the followingcoupled equations for u j and v j :0 = sgn( Q ) (cid:18) ˙ v j − ˙ u j (cid:19) − ω (cid:18) u j v j (cid:19) − (cid:88) k ∈(cid:104) j (cid:105) (cid:18) ζ + ξ cos 2 θ jk ξ sin 2 θ jk ξ sin 2 θ jk ζ − ξ cos 2 θ jk (cid:19) (cid:18) u k v k (cid:19) , (4)where ω ≡ K/ | G | is the gyration frequency of an isolatedsoliton, (cid:104) j (cid:105) represents the set of the nearest neighbors ofthe soliton j , θ jk is the angle of the direction ˆ e jk fromthe x axis, ζ ≡ ( I (cid:107) − I ⊥ ) / | G | , and ξ ≡ ( I (cid:107) + I ⊥ ) / | G | . Theassumption that the interaction is much weaker than thepinning potential translates into ζ, ξ (cid:28) ω . Vortex honeycomb lattice. —A ferromagnetic disk of asuitable size can harbor a magnetic vortex in its groundstate [21]. A vortex is characterized by the polarity p = ±
1, which is the direction of the magnetization atits core, and the chirality, c = ±
1, which describes theclockwise ( c = −
1) or counterclockwise ( c = 1) in-planecurling of the magnetization around the core. The po-larity of a vortex is related to its topological charge by Q = p/
2. The polarization and the chirality can be inde-pendently controlled by an external field or an electric-current pulse [17]. Throughout the Letter, we use vor-tices with the positive chirality c = 1, which are shownin Figs. 1(a) and (b).Let us consider vortices in physically separated disksthat are arranged as a honeycomb lattice. See Fig. 2for illustrations. The displacements of vortices fromthe centers of disks generate the magnetostatic chargesby altering the magnetization profile from the groundstate [22]. The magnetostatic energy associated withthese charges engender the inter-vortex interaction givenby Eq. (3) [12]. The magnitudes of the interactions de-cay as I (cid:107) ( d ) , I ⊥ ( d ) ∼ d − , which justifies our nearest-neighbor model in Eq. (4). Let us take the experimentalvalues for the parameters from Ref. [14] to obtain thenormal-mode dispersion from Eq. (4). For two permalloydisks of the radius R = 500nm, the thickness t = 50nm,and the center-to-center distance d = 1075nm, the pa-rameters are given by K ∼ × − J/m , ω ∼ I (cid:107) ∼ × − J/m , and I ⊥ ∼ × − J/m . These mea-sured values agree with the theoretical estimations [12].The corresponding parameters in our model [Eq. (4)] aregiven by ζ ∼ . ω and ξ ∼ . ω .We solve Eq. (4) for a honeycomb ribbon with peri-odic boundary conditions along the x direction and withzigzag terminations at the top y = L and the bottom y = 0. The color represents the average vertical positionof the mode, (cid:104) y (cid:105) ≡ (cid:80) j Y j | U j | / (cid:80) j | U j | . The one-dimensional dispersions for the normal modes are shownin Fig. 2(b) and (d) for vortices with the polarity p = 1and p = −
1, respectively. The results show that each sys-tem supports the chiral edge mode lying within the bulkgap, which rotates the boundary in the same directionas individual solitons precess. Two polarities, p = 1 and p = −
1, are related by the magnetization flip and thusby the time reversal. The chiralities of the edge modesare opposite accordingly.
Mapping to the Haldane model. —The existence of thechiral edge mode can be understood analytically by map-ping Eq. (4) to the Haldane model of the quantum Halleffect [18]. The analogous mapping is given in Ref. [7]for mechanical gyroscopes, which we adopt here for mag-netic solitons. For simplicity, we will consider the caseof the negative topological charge, Q = − /
2. We be-gin by casting Eq. (4) in terms of the complex variable, ψ j ≡ u j + iv j : i ˙ ψ j = ω ψ j + (cid:88) k ∈(cid:104) j (cid:105) ( ζψ k + ξe iθ jk ψ ∗ k ) . (5)The equation can be interpreted as the Schr¨odinger equa-tion for the wavefunction of an electron in a tight-bindingmodel. Since the interactions are much weaker than thepinning potential, ζ, ξ (cid:28) ω , we can use perturbationtheory to find the equation in terms of ψ j alone by elim-inating its complex conjugate. To that end, let us ex-pand the complex variable as ψ j ( t ) = χ j ( t ) exp( − iω t ) + φ j ( t ) exp( iω t ) where χ j ( t ) and φ j ( t ) change over timeslowly on the time scale set by the frequency ω . Then,by matching the coefficients of exp( iω t ) and exp( − iω t )in Eq. (4), we obtain i ˙ χ j = (cid:80) k ∈(cid:104) j (cid:105) ( ζχ k + ξe iθ jk φ ∗ k )and i ˙ φ j = 2 ω φ j + (cid:80) k ∈(cid:104) j (cid:105) ( ζφ k + ξe iθ jk χ ∗ k ). Since theinteractions are weak, ζ, ξ (cid:28) ω , the soliton dynam- ics is mostly associated with the resonance frequency ω and thus | χ j | (cid:29) | φ j | . Using the approximation2 ω φ j ≈ − (cid:80) k ∈(cid:104) j (cid:105) ξe iθ jk χ ∗ k obtained from the latteryields the following equation: i ˙ ψ j = (cid:0) ω − ξ / ω (cid:1) ψ j + ζ (cid:88) k ∈(cid:104) j (cid:105) ψ k − ( ξ / ω ) (cid:88) l ∈(cid:104)(cid:104) j (cid:105)(cid:105) cos (cid:0) θ jl (cid:1) ψ l − i ( ξ / ω ) (cid:88) l ∈(cid:104)(cid:104) j (cid:105)(cid:105) sin (cid:0) θ jl (cid:1) ψ l , (6)where (cid:104)(cid:104) j (cid:105)(cid:105) is the set of the second-nearest neighbors of j , ¯ θ jl ≡ θ jk − θ kl is the relative angle from the bond k → l to the bond j → k with k between j and l . This equationis similar to the Haldane model for an electron in a hon-eycomb lattice [18, 23], which is a prototypical exampleexhibiting the quantum Hall effect. The difference is theterm in the second line, which is real and does not affectthe existence of the chiral edge mode [7].Let us explain how the chiral edge mode originates inthe above equation for electrons. The angle between theneighboring bonds is given by ¯ θ jl = ± π/
3, where the up-per (lower) sign is for the case when we have to turn right(left) to go from j to l . When the last term vanishes, twoelectronic bands associated with two sublattices toucheach other at two points in the momentum space, form-ing two Dirac cones. When the last term is finite, anelectron picks up a phase when hopping to its second-nearest neighbors and the sign of the accumulated phasedepends on whether the electron makes a left or rightturn to arrive at its neighbors. Via this path-dependentphase, the imaginary second-nearest hopping opens topo-logical gaps at the Dirac cones and engenders the chiraledge mode. The gap size is given by ∆ = 9 ξ / ω [23]. Magnetic bubble honeycomb lattice. —Now let us turnto a honeycomb soliton lattice, in which only the nearest-neighbor solitons connected by bonds are engineeredto interact. We consider distortions of the constituenthexagons, which preserve the lattice connectivity and thebond lengths corresponding to the original nearest neigh-bors. From the above discussions, the origin of the chiraledge mode is the staggered phase gathered by an elec-tron hopping between the second-nearest neighbors [18].As can be seen in Eq. (6), the sign of the phase can becontrolled by changing the angle ¯ θ jl between the neigh-boring bonds. Let us take examples shown in Fig. 3.In Fig. 3(a), sin 2¯ θ jl is positive (negative) if we make aright (left) turn to go from j to l ; In Fig. 3(c), sin 2¯ θ jl vanishes; In Fig. 3(e), the sign of sin 2¯ θ jl is opposite tothe case in Fig. 3(a) for all pairs of j and k . Since thestaggered phase changes its sign between (a) and (c) withvanishing in (b), we expect the change of the chirality ofthe topological edge mode from (a) to (c) via the gapclosing in (b) [7]. We verify this expected dependence ofthe chirality on the shape of the constituent hexagons by (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π (cid:1) π (cid:1) π π ⦿ k x ! / ! . .
30 2 ⇡/ a ⇡/ a ⇡/a h y i L h y i L ( a ) ( b )( c ) ( d ) k x ⇡/ a ⇡/ a ⇡/a h y i L k x ⇡/ a ⇡/ a ⇡/a ( e ) ( f ) ⊗ ⊗⊗⊗ ! / ! . . ! / ! . . FIG. 3. (a) A schematic illustration of magnetic bubbles withthe topological charge Q = 1 (drawn as circles) in a hon-eycomb lattice with zigzag edges. The black lines betweencircles represent the exchange-coupled interactions betweenmagnetic bubbles connected by the magnetic strips. (b) Theone-dimensional dispersion for the coupled gyration modes forthe system shown in (a). (c), (d) Analogous figures when theangles between nearest bonds are multiples of π/
2, for whichthe dispersion for the bulk is gapless and thus the topologicaledge mode is not supported. (e), (f) Analogous figures forthe system exhibiting the chiral edge mode in the oppositedirection to (a) and (b). numerically solving Eq. (4) below by taking the approachused in Ref. [7], which has studied the analogous problemfor gyroscope lattices.Let us consider a honeycomb lattice of magnetic bub-bles with the topological charge Q = 1, which can appearas a ground state of a magnetic disk with perpendicularanisotropy [24]. The constituent disks are connected byferromagnetic strips so that neighboring magnetic bub-bles can interact with each other via the exchange en-ergy. The coupled gyration modes of magnetic bub-bles in one-dimensional magnetic strip have been stud-ied by micromagnetic simulations in Ref. [25], accord-ing to which the dominant contributions to the inter-action comes from the exchange energy. We model theexchange-driven (repulsive) interaction as a function ofthe distance, f ( R j , R k ) = f ( | R j − R k | ) by followingRefs. [26]. To the second order in the displacements, theinteraction can be written in the form of Eq. (3) with I (cid:107) = − f (cid:48)(cid:48) and I ⊥ = f (cid:48) /d jk . Since the parameters forthe magnetic bubble interactions are not known unlikethe well-studied vortex interactions, we adopt the pa-rameters for vortices: ζ = − . ω and ξ = − . ω , inwhich the minus sign represent the repulsive interactions.Equation (4) is solved for three hexagonal lattices com-posed of distorted hexagons. Fig. 3(b) shows the one-dimensional dispersion for the coupled magnetic bub-ble gyration when the angles are θ jk = π/ , π/ , π/ j at Y-shaped junctions. Thiscase is similar to the vortex honeycomb lattice composedof regular hexagons, and thus exhibits the chiral edgemode rotating the boundary counterclockwise, same asthe precession of individual magnetic bubbles. Fig. 3(d)shows the normal-mode dispersion when the angles are θ jk = 0 , π, π/ j . In thiscase, the last term in Eq. (6) vanishes and thus the bulkband is gapless; the topological edge mode does not ex-ist. Fig. 3(f) shows the dispersion when the angles are θ jk = − π/ , π/ , π/ Discussion. —We have shown that a honeycomb latticeof magnetic vortices and bubbles can exhibit a chiral edgemode via their coupled gyrations, the direction of whichcan be controlled by flipping the topological charge or bydistorting the lattice geometry. The dispersions of thecoupled vortex gyration have been investigated experi-mentally in several different arrangements including one-dimensional arrays of 5 disks [15] and two-dimensionalarrays of 50 ×
50 disks [16] by scanning transmission x-raymicroscopy, which leads us to believe that experimentalrealization of our proposal for a vortex honeycomb lat-tice is within the current experimental reach. The exper-imental exploration of the chiral edge mode in a magneticbubble lattice seems to be more challenging, as reflectedin the relative lack of an experimental study on the dy-namics of engineered magnetic bubble lattices.We would like to mention that there is a class of ferri-magnets which allows us to thermally control the chiral-ity of the edges modes. These are rare-earth transition-metal alloys such as GdFeCo and CoTb, possessing thespecial temperature referred to as the angular momen-tum compensation point, across which the gyromagneticratio changes its sign while keeping the magnetizationfinite [19]. By changing the sign of the gyromagneticratio, we can flip the sign of the gyrotropic force inEq. (2) [27]. Therefore, when disks harboring vortices aremade of such ferrimagnets, we should be able to changethe chirality of the edge mode by varying the temperatureacross the compensation point, providing an example oftemperature-driven topological phase transitions.This work was supported by the Army Research Officeunder Contract No. W911NF-14-1-0016. [1] G. Ma and P. Sheng, Sci. Adv. (2016), and referencestherein.[2] R. E. Prange and S. M. Girvin, eds., The QuantumHall Effect (Springer, New York, 1990), and referencestherein.[3] F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. ,013904 (2008); L. Lu, J. D. Joannopoulos, and M. Sol-jacic, Nat. Photonics , 821 (2014).[4] R. Shindou, J.-i. Ohe, R. Matsumoto, S. Murakami, andE. Saitoh, Phys. Rev. B , 174402 (2013); R. Shindou,R. Matsumoto, S. Murakami, and J.-i. Ohe, Phys. Rev.B , 174427 (2013).[5] E. Prodan and C. Prodan, Phys. Rev. Lett. , 248101(2009); L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys.Rev. Lett. , 225901 (2010); K. Sun, A. Souslov,X. Mao, and T. C. Lubensky, Proc. Natl. Acad. Sci. , 12369 (2012); C. L. Kane and T. C. Lubensky,Nat. Phys. , 39 (2014); B. G.-g. Chen, N. Upadhyaya,and V. Vitelli, Proc. Natl. Acad. Sci. , 13004 (2014);J. Paulose, B. G.-g. Chen, and V. Vitelli, Nat. Phys. ,153 (2015); R. S¨usstrunk and S. D. Huber, Science ,47 (2015).[6] P. Wang, L. Lu, and K. Bertoldi, Phys. Rev. Lett. ,104302 (2015).[7] L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M.Turner, and W. T. M. Irvine, Proc. Natl. Acad. Sci. , 14495 (2015).[8] A. A. Thiele, Phys. Rev. Lett. , 230 (1973).[9] A. Kosevich, B. Ivanov, and A. Kovalev, Phys. Rep. ,117 (1990), and references therein.[10] S. D. Bader, Rev. Mod. Phys. , 1 (2006), and referencestherein.[11] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. , 899(2013), and references therein.[12] J. Shibata, K. Shigeto, and Y. Otani, Phys. Rev. B ,224404 (2003); J. Shibata and Y. Otani, Phys. Rev. B , 012404 (2004); A. Y. Galkin, B. A. Ivanov, andC. E. Zaspel, Phys. Rev. B , 144419 (2006); K.-S. Lee,H. Jung, D.-S. Han, and S.-K. Kim, J. Appl. Phys. ,113903 (2011); O. V. Sukhostavets, J. Gonz´alez, andK. Y. Guslienko, Phys. Rev. B , 094402 (2013).[13] A. Barman, S. Barman, T. Kimura, Y. Fukuma,and Y. Otani, J. Phys. D: Appl. Phys. , 422001(2010); S. Barman, A. Barman, and Y. Otani, IEEETrans. Magn. , 1342 (2010); A. Vogel, A. Drews,T. Kamionka, M. Bolte, and G. Meier, Phys. Rev. Lett. , 037201 (2010); H. Jung, K.-S. Lee, D.-E. Jeong,Y.-S. Choi, Y.-S. Yu, D.-S. Han, A. Vogel, L. Bocklage,G. Meier, M.-Y. Im, P. Fischer, and S.-K. Kim, Sci. Rep. , 59 (2011); R. Streubel, F. Kronast, U. K. R¨oßler, O. G.Schmidt, and D. Makarov, Phys. Rev. B , 104431(2015); C. F. Adolff, M. H¨anze, M. Pues, M. Weigand,and G. Meier, Phys. Rev. B , 024426 (2015).[14] S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Bar-man, and Y. Otani, Phys. Rev. Lett. , 197203 (2011).[15] D.-S. Han, A. Vogel, H. Jung, K.-S. Lee, M. Weigand,H. Stoll, G. Sch¨utz, P. Fischer, G. Meier, and S.-K.Kim, Sci. Rep. , 2262 EP (2013).[16] C. Behncke, M. H¨anze, C. F. Adolff, M. Weigand, andG. Meier, Phys. Rev. B , 224417 (2015).[17] T. Taniuchi, M. Oshima, H. Akinaga, and K. Ono, J.Appl. Phys. , 10J904 (2005); B. C. Choi, J. Rudge,E. Girgis, J. Kolthammer, Y. K. Hong, and A. Lyle,Appl. Phys. Lett. , 022501 (2007); Y.-S. Choi, M.-W.Yoo, K.-S. Lee, Y.-S. Yu, H. Jung, and S.-K. Kim, Appl.Phys. Lett. , 072507 (2010).[18] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[19] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod.Phys. , 2731 (2010); Rep. Prog. Phys. , 026501(2013).[20] T. Gilbert, IEEE Trans. Magn. , 3443 (2004).[21] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E.Welland, and D. M. Tricker, Phys. Rev. Lett. , 1042(1999); C. L. Chien, F. Q. Zhu, and J.-G. Zhu, PhysicsToday , 40 (2007).[22] K. L. Metlov and K. Y. Guslienko, J. Magn. Magn.Mater. , 1015 (2002); K. Y. Guslienko, X. F. Han,D. J. Keavney, R. Divan, and S. D. Bader, Phys. Rev.Lett. , 067205 (2006); K. L. Metlov, Phys. Rev. Lett. , 107201 (2010).[23] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005).[24] A. H. Bobeck and E. D. Torre, Magnetic Bubbles (North-Holland, Amsterdam, 1975); A. P. Malozemoff and J. C.Slonzewski,
Magnetic Domain Walls in Bubble Materi-als (Academic, New York, 1979); A. H. Eschenfelder,
Magnetic Bubble Technology (Springer, Berlin, 1981);M. Hehn, K. Ounadjela, J.-P. Bucher, F. Rousseaux,D. Decanini, B. Bartenlian, and C. Chappert, Sci-ence , 1782 (1996); T. Fukumura, H. Sugawara,T. Hasegawa, K. Tanaka, H. Sakaki, T. Kimura, andY. Tokura, Science , 1969 (1999).[25] J. Kim, J. Yang, Y.-J. Cho, B. Kim, and S.-K. Kim, Sci.Rep. , 45185 EP (2017).[26] S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B , 060404(2013); D. Pinna, F. Abreu Araujo, J.-V. Kim, V. Cros,D. Querlioz, P. Bessiere, J. Droulez, and J. Grollier,arXiv:1701.07750.[27] S. K. Kim and Y. Tserkovnyak, Appl. Phys. Lett.111