Binding a Hopfion in Chiral Magnet Nanodisk
BBinding a Hopfion in Chiral Magnet Nanodisk
Yizhou Liu, ∗ Roger K. Lake, † and Jiadong Zang ‡ Department of Electrical and Computer Engineering,University of California, Riverside, California 92521, USA Department of Physics and Materials Science Program,University of New Hampshire, Durham, New Hampshire 03824, USA
Hopfions are three-dimensional (3D) topological textures characterized by the integer Hopf invari-ant Q H . Here, we present the realization of a zero–field, stable hopfion spin texture in a magneticsystem consisting of a chiral magnet nanodisk sandwiched by two films with perpendicular magneticanisotropy. The preimages of the spin texture and numerical calculations of Q H show that the hop-fion has Q H = 1. Furthermore, another non-trivial state that includes a monopole–antimonopolepair (MAP) is also stabilized in this system. By applying an external magnetic field, hopfion andMAP states with the same polarization can be switched between each other. The topological tran-sition between the hopfion and the MAP state involves a creation (annihilation) of the MAP andtwist of the preimages. Our work paves the way to study non-trivial 3D topological spin texturesand stimulates more investigations in the field of 3D spintronics. A topological soliton carries an integer topological in-dex that cannot be changed by a continuous deforma-tion [1]. A celebrated example is the skyrmion, a two-dimensional (2D) topological soliton originated from theSkyrme model [2], which can be characterized by theskyrmion number (or winding number) [3]. The addi-tion of a third spatial dimension brings more diverse andcomplicated topological solitons, such as rings, links andknots [4–6]. Some of these three-dimensional (3D) topo-logical solitons are “hopfions”, since they can be classi-fied by the Hopf invariant ( Q H ) [7], a topological indexof the homotopy group Π ( S ) that can be interpretedas the linking number [8]. Due to their complex struc-tures and models, the detailed study of the hopfion wasproperly established not long ago in terms of toroidalcoordinates[9, 10]. Hopfions have been observed in avariety of physical systems including fluids, optics, liq-uid crystals, Bose-Einstein condensates, etc. [11–17] Buttheir observation in magnetic materials remains elusive.In magnetic systems, topological solitons in one di-mension and two dimensions such as domain walls andvortices have been extensively studied over the past fewdecades. Much of the recent attention is attracted bythe magnetic skyrmions residing in magnetic materialswith the antisymmetric Dzyaloshinskii-Moriya interac-tion (DMI) [18–20]. The spins of a magnetic skyrmionwind around the unit sphere once, which results in theunit winding number of a skyrmion. Skyrmions areproposed to be promising candidate for spintronic ap-plications due to their prominent features such as thenanoscale size and low driving current density [21, 22].Although numerous studies have been made on the ∗ [email protected] Address: Beijing National Laboratory for CondensedMatter Physics, Institute of Physics, Chinese Academy of Sci-ences, Beijing 100190, China † [email protected] ‡ [email protected] low-dimensional topological solitons, 3D topological soli-tons like hopfions have still not been well explored innanomagnetism. Understanding the static and dynam-ical properties of these 3D topological solitons are notonly of fundamental interest, but may also enable futureapplications. Only a few theoretical proposals predictthe existence of hopfions in ferromagnets, but only inthe dynamical regime [23–25]. It has been recently pro-posed that a higher order exchange interaction and anexternal magnetic field will stabilize a metastable hop-fion in a frustrated magnet [26], but how to create suchmetastable state is not clear.In this Letter, we show that a Q H = 1 hopfioncan be enabled in a chiral magnet nanodisk in the ab-sence of external magnetic fields. The nanodisk is sand-wiched by two magnetic layers with perpendicular mag-netic anisotropy (PMA) to nucleate the hopfion in be-tween. The hopfion is identified by both the preimagesand the numerical calculations of Q H . Associated withthe hopfion, another non-trivial state that includes amonopole-antimonopole pair (MAP) is also stabilized atzero fields in this structure. Furthermore, the hopfioncan be switched into a MAP state with the same po-larization by an applied magnetic field, and vice versa.The topological transition between the hopfion state andthe MAP state involves the creation (annihilation) of themonopole-antimonopole pair and a twist of the preim-ages.We consider a chiral magnet nanodisk with radius100 nm and thickness 70 nm sandwiched by two PMAmagnetic thin layers with 10 nm thickness, as shown inFig. 1(a). An isotropic bulk type DMI is employed tomodel the chiral magnet. The Hamiltonian of this sys-tem is given by H = (cid:90) dr [ − A ( ∇ s ) − (1 − p) D s · ( ∇× s ) − p K u ( s z ) + E d ] , (1)where A and D are the exchange and DMI constant, re-spectively, K u is the PMA constant, and p is 0 in the a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l FIG. 1. (a) Schematic of the proposed structure. The thindisks at the top and bottom represent the magnetic films withPMA. The transparent region in the middle is the chiral mag-net nanodisk. The color ring at the center represents the setof preimages with s z = 0 of a Q H = 1 hopfion. (b), (c) Thecross-sectional spin textures in the x-y plane (z=0) for thehopfion (b) and MAP (c). (d), (e) The cross-sectional spintextures in the y-z plane (x=0) for the hopfion (d) and MAP(e). In the color scheme, black indicates s z = − s z = 1. The color wheel is for s z = 0. chiral magnet nanodisk and 1 in two PMA layers. E d isthe magnetic dipole-dipole interaction (DDI). It dependson the exact shape of the system. When the systemsize goes down to nanoscale, the DDI becomes impor-tant in determining the corresponding spin textures. Forexample, the DDI favors the stabilization of magneticskyrmion at zero-field in confined geometries. It leadsto the formation of the so-called target skyrmion, whichhas been theoretically proposed and recently experimen-tally observed in magnetic nanodisks without any exter-nal fields [27–30]. Thus, the effect of DDI is essential inconfined systems and cannot be ignored.We minimize the Hamiltonian (1) in the nanodiskstructure with different initial states (for details of thesimulation methods and parameters, see SupplementalMaterials [31]). After minimizing the energy, we findtwo stable non-trivial states at zero-field, the hopfionstate and the MAP state. The hopfion state includesa Q H = 1 hopfion, and the MAP state includes a monopole-antimonopole pair.To present the detailed spin textures of the hopfion andMAP, cross-sections of both states are plotted in Fig. 1.For the hopfion, the cross-section in the x-y plane (z=0)shown in Fig. 1(b), has a skyrmion at the center sur-rounded by two concentric spin helical rings. This is typ-ically a target skyrmion configuration recently observedin an FeGe nanodisk [30]. A conventional skyrmion iswrapped by a concentric helical ring. From the centerto periphery, spin rotates by an angle of 2 π instead of π in typical skyrmions. The outmost spin helical ring isnot part of the hopfion but an edge state induced by theDDI from the circular shape and the DMI of the chiralmagnet. The lateral cross-section in the y-z plane (x=0)shown in Fig. 1(d) includes a skyrmion–antiskyrmionpair. The cross-section taken at any plane containingthe z -axis always contains a skyrmion–antiskyrmion pair.This is a result of the hopfion spin texture that consistsof a 2 π twisted skyrmion tube with its two ends gluedtogether as shown in Fig. 1(a). For the MAP state, thecross-section in the x-y plane (z=0) shown in Fig. 1(c)is a typical skyrmion. The cross-section in the y-z plane(x=0) shown in Fig. 1(e), has only one spin up region, incontrast to the skyrmion–antiskyrmion pair of the hop-fion. Instead, a monopole (antimonopole) is formed nearthe top (bottom) surface. This originates from the re-stricted spin polarization of the PMA layers on the topand bottom.To further visualize and understand the spin config-urations of the hopfion and MAP in 3D, we plot theirpreimages using Spirit [32]. A preimage is the regionin 3D real space that contains spins with the same ori-entations. It is a Hopf map of a point on the S unitsphere to 3D space. We first plot the set of preimagesof all spins with s z = 0 for the hopfion (Fig 2(a)) andMAP (Fig. 2(c)), which corresponds to a Hopf map fromthe equator of the S unit sphere to the 3D space. Twopreimages are topologically distinct as characterized bydifferent genus g , i.e., the number of holes. The preim-age of the hopfion forms a torus with g = 1, whereasthe preimage of the MAP is a trivial surface with g = 0,which satisfies the Poincar´e-Hopf theorem [33].The Hopf invariant, also called the linking number,counts the number of links between two arbitrary closed-loop preimages. Therefore, preimages of two arbitraryspins must form closed loops that are linked with eachother. These features can be identified by the preim-ages of s = (1 , ,
0) and s = ( − , ,
0) for the hopfion(Fig. 2(b)) and MAP (Fig. 2(d)). For the hopfion, twoclosed-loop preimages are formed and linked with eachother once. Q H = 1 in this case, and the topology of thehopfion state in this system is confirmed. In contrast, theMAP does not have closed-loop preimages and thus nolinks. Monopole and antimonopole are source and drainof all preimages. The two MAP preimages of s = (1 , , s = ( − , ,
0) join at the monopole and antimonopoleindicating their singular natures. The MAP is considereda defect state, while the hopfion is a smooth spin texture (a) (c)(b) (d) m x m y FIG. 2. (a), (c) The set of preimages with s z = 0 for thehopfion and MAP, respectively. (b), (d) The preimages of s =(-1,0,0) (cyan) and s =(1,0,0) (red) for the hopfion (b)and MAP (d). In the color scheme, black indicates s z = − s z = 1. The color wheel is for s z = 0. with no singularity. These preimages successfully reflectthe topological natures of the two states.Other than the linking number of preimages, topologyof the hopfion can also be confirmed by directly calcu-lating the Hopf invariant. The integral form of the Hopfinvariant in real space can be expressed as [34, 35] Q H = − (cid:90) B · A d r , (2)where B i = π (cid:15) ijk n · ( ∇ j n × ∇ k n ) is the emergent mag-netic field associated with the spin textures, and A is anyvector potential that satisfies the magnetostatic equation ∇ × A = B . The Hopf number is invariant under a gaugetransformation A → A + ∇ χ only when the emergentfield B is free of singularities, i.e. , ∇ · B = 0. Cross-sections in the y-z plane of the emergent magnetic fields B of the hopfion and MAP states are shown in Fig. 3.The emergent B field of the hopfion shown in Fig. 3(a)flows smoothly and streams intensively near the centerof the nanodisk. In contrast, the emergent B field ofthe MAP shown in Fig. 3(b) clearly presents two mag-netic monopoles with opposite charges near the top andbottom surface. The Hopf invariant is thus ill–definedfor the MAP state, and it is well defined for the hopfiontexture.A gauge field solution A must also be solved in or-der to directly calculate Q H in real space. To this end,we solve for the vector potential A in momentum spacewith the Coulomb gauge k · A = 0, and then compute Q H in momentum space [36]. To carry out the numericalintegral, discrete grids in the momentum space are em-ployed. As shown in Fig. 3(c), as the grid number ( N tot )increases, Q H rapidly converges to 1. We thus obtain aHopf invariant of Q H = 0 .
96 for the hopfion spin textureunder investigation. Here Q H is slightly deviated froman integer due to the finite size and open boundary con-dition. The manifold is not compact, as indicated by the
15 25 35 45 55 (a)(b)(c) z yx Q H N
10 20 30 40
Column 1 C o l u m n Untitled Data 2 | B | ( a . u . )
10 20 30 40
Column 1 C o l u m n Untitled Data 2 | B | ( a . u . ) FIG. 3. (a), (b) The emergent magnetic field B in the y-z plane (x=0) for hopfion (a) and MAP (b). (c) Numericalcalculations of the Hopf invariant Q H for different meshes.The total number of grid points N tot = N . edge state around the disk boundary. Nevertheless, theHopf invariant is close to 1, and the topological natureof the hopfion is further confirmed.At zero external magnetic field, two states with op-posite spins share the same energy. Therefore, stablehopfion and MAP states each have two polarizations,i.e. spin points up or down at their cores. As shownin Fig. 4(a), the MAP state has lower energy than thehopfion state at zero magnetic field. But they can beswitched between each other by sweeping an externalmagnetic field. When applying a magnetic field in thesame (opposite) direction with the MAP (hopfion) po-larization, the MAP (hopfion) can be switched into ahopfion (MAP) with the same polarization. Thus, de-spite the MAP state having lower energy, the hopfionstate can still be realized by using an applied field. InFig. 4 (a), we only show the switching between the hop-fion and MAP with the same polarization, but it is alsopossible to switch between MAP states with opposite po-larizations using a large field to saturating spins in theopposite direction.Since the hopfion is topologically protected by thenonzero Hopf invariant, a topological transition musttake place in the switching between the hopfion and MAPstates. To investigate this topological transition, we per-formed a minimal energy path (MEP) calculation be-tween these two states [37–39]. The MEP calculation –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 –3–2–10 Reaction Coordinate E n e r gy ( - J ) (b)(c) (f) m y m x c d(d) (e)e fd e -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15–3–2–10 hopf_downmap_downhopf_upmap_uphopf_downhopf_upmap_downmap_up hopfion downhopfion upMAP downMAP up E n e r gy D e n s it y ( J / m ) (a) Magnetic field (T) –0.05 0.0 0.05–1.0–0.50.0 –0.12 –0.06 0.00–0.4–0.20.0
FIG. 4. (a) Energy density plot of the hopfion and MAP stateas a function of external magnetic field. Black arrows indicatethe field sweeping directions and the switching events. Insetsshow the enlarged details of the plot. (b) Minimal energy pathbetween the hopfion and MAP state. Points c and f representthe hopfion and the MAP, respectively. The hopfion is nearlyannihilated at saddle point d, and the MAP is created ate. Insets show the half-plane view preimages of s z = 0 forspin textures at d and e. (c)-(f) The preimages of s =(-1,0,0)(cyan) and s =(1,0,0) (red) corresponding to points c–f in (a). is carried out using the geodesic nudged elastic band(GNEB) method associated with the Hamiltonian in Eq.(1). The stable spin textures from the energy minimiza-tions are employed as the initial states in the MEP cal-culation.Results from the MEP calculation are shown in Fig.4(b). There exists an energy barrier between the hop-fion and the MAP state. Thus, an activation energy is required to enable the transition from the hopfion(MAP) to MAP (hopfion) state. To capture details of thetopological transition, we plot preimages of s =(1,0,0)and s =(-1,0,0) at the initial hopfion state, the barrierpeak, the intermediate MEP state and the final MEPstate (Fig. 4(c)-(f)). Transitioning from the hopfionstate in (c) to the intermediate state (e), the two linkedpreimages break and reconnect generating the monopole–antimonopole pair with a 2 π rotation. The two preimagesare then topologically equivalent to those of the MAPstate in Fig. 4(f), although they are twisted by 2 π . Re-laxing from point (e) to to the MAP state of point (f),the preimages untwist to π , while the monopole and an-timonopole move towards the top and bottom surface,respectively. Videos of the transition also capture thetransformation from a torus ( g = 1) to a trivial sur-face ( g = 0) for the preimages of s z = 0 (see movies inthe Supplemental Materials). To create a hopfion from aMAP state, the reverse process is applied. The preimagesfirst rotate from π to 2 π . The monopole–antimonopolepair move towards each other until they eliminate eachother. Then each preimage becomes close-looped andlinked with the other preimage.To conclude, a Q H = 1 hopfion can be stabilized ina chiral magnet nanodisk sandwiched by two magneticlayers with PMA at zero external magnetic fields. Thehopfion is identified by its preimages and the Hopf invari-ant. A MAP state is also stabilized at zero field in theproposed structure. The hopfion (MAP) can be switchedinto a MAP (hopfion) state by applying a magnetic field.The minimal energy path calculation reveals the topolog-ical transition between the hopfion and the MAP state.3D magnetic imaging techniques such as the X-ray vectornanotomography could be a powerful tool for visualizingthe spin texture of hopfion in real space [40]. The hopfionmay exhibit fascinating electronic transport and dynam-ical properties due to its novel topology. This work pavesa way in the development of 3D spintronics and high di-mensional memory architectures [41]. Acknowledgements:
JZ acknowledges stimulating discus-sions with Jeffrey Teo. Conception and analytical worksin this work were supported by the U.S. Departmentof Energy (DOE), Office of Science, Basic Energy Sci-ences (BES) under Award No. de-sc0016424. Numericalsimulations and part of the analytical work were sup-ported as part of the Spins and Heat in Nanoscale Elec-tronic Systems (SHINES) an Energy Frontier ResearchCenter funded by the U.S. Department of Energy, Of-fice of Science, Basic Energy Sciences under Award No.de-sc0012670. Numerical simulations and collaborativetravel between UCR and UNH were also supported bythe NSF ECCS-1408168. [1] N. Manton and P. Sutcliffe,
Topological Solitons (Cam-bridge University Press, Cambridge, England, 2004). [2] T. H. R. Skyrme, Nucl. Phys. , 556 (1962).[3] R. Rajaraman, Solitons and Instantons (North-holland,