Topological transformation of magnetic hopfion in confined geometries

Micromagnetic simulations

Micromagnetic simulations were performed using the MUMAX3 software package^[36], which is widely adopted for simulating various 3D topological magnetic structures. It is a GPU-accelerated operation and can significantly enhance the computational speed, particularly beneficial for handling larger volumes of data. We sandwiched the chiral magnet with a thickness of 78 nm by two 6 nm-thick layers with strong PMA (K_s = 1 MJ/m²). The initial state was computed from an ansatz^[22]. The materials parameters of the chiral magnet layer were taken from FeGe^[15,37], M_s = 384 kA/m, A = 4 pJ/m, and D = 0.718 mJ/m². The bulk PMA K_b is from 0 to 60 kJ/m³, and the perpendicular magnetic field H_z is from 0 to 0.44 T. The meshed cell size is 3 nm × 3 nm × 2 nm. For the current-driven dynamics, the spin dynamics of hopfion were simulated based on the LLG equation containing the spin-transfer torque (STT) term:

where γ is the gyromagnetic ratio, H_eff is the effective field. The second term denotes the Gilbert damping contribution where α is fixed at 0.05. The last two terms describe the coupling between spin and spin-polarized current according to the Zhang-Li STT model. β is the degree of non-adiabaticity. u = µ_BpJ/[eM_s (1 + β²)] is the velocity of the conduction electrons, in which p is the spin polarization of the magnet and J is the current density. Here, we set β = 0.05 and p = 0.3. The value of β/α can influence the STT-driven domain velocity^[38], and the parameter p affects spin transfer efficiency, consequently leading to the change of domain-motion velocity. In our study, we selected the α, β, and p values that align with the previous works to simulate the current-driven dynamic in FeGe^[37]. The size of the rectangular nanostripe is 900 × 600 nm², and the sample thickness is 100 nm. The boundary effect of the plate was considered without setting any periodic boundary conditions.

LTEM simulations

The LTEM images were calculated by using PyLorentz^[39], a software tool designed for mapping magnetic structures. The magnetic phase shifts (φ) based on the 3D spin textures were calculated by using the linear superposition method, which directly relates to the magnetization in Fourier space as

where k represents the position in Fourier space, $$k_{\perp}$$ is the position vector in the projection plane within Fourier space, while the z-axis corresponds to the direction of the incident beam. The summation of the phase shifts of each voxel yields the total phase shift, which can be related to the real-space position. To simulate the LTEM images, the magnetization file obtained from MUMAX3 is imported into PyLorentz. The SimTEM code within PyLorentz calculates the electron phase shift based on the input magnetization data, allowing for the simulation of LTEM images with dark and bright contrasts. The magnetic induction can also be reconstructed by using two through focal series via PyLorentz.

3D structures of hopfion and toron

Magnetic hopfions can be formed in a system consisting of a chiral magnet or asymmetric multilayers with one moderate PMA layer sandwiched by two magnetic layers with strong PMA. The free energy terms of this system are given by

where A is the Heisenberg exchange constant, ω_D is the DMI energy density depending on the symmetry of the system. We consider a chiral magnet with bulk DMI, where ω_D = Dm·[∇ × m]. K_b and K_s are bulk and the interfacial PMA constants, respectively. H_z is the magnetic field applied along the z-axis, M_s is the saturated magnetization and ε_d is the demagnetization energy density. First, we conduct simulations in a square geometry comprising a chiral magnet layer sandwiched by two layers with strong PMA. Consequently, a Bloch-type hopfion is permitted in the absence of an external magnetic field. Figure 1A shows the hopfion preimages of spins with zero perpendicular component (s_z), which represents the iso-surface configurations with zero s_z. The inset of Figure 1A shows the multiple cross-sectional spin texture images in the xy plane at different heights for the hopfion. The magnetization at the top and bottom surfaces is oriented upwards, which is due to the strong PMA in the outer layers. In the vicinity of the upper or lower layers, the hopfion exhibits a toroidal vortex structure with the central spins oriented upwards. The cross-sectional spin texture at the intermediate layers exhibits a Bloch-type skyrmionium^[40]. Unlike a skyrmion with a topological charge of one, a skyrmionium is a skyrmion surrounded by an annular domain wall with an opposite topological charge, resulting in a total topological charge of zero. When an external magnetic field H_z is applied, the core spins of the hopfion gradually contract, resulting in the transformation from hopfion into a toron state. Meanwhile, the midplane magnetization profile changes from the skyrmionium to a skyrmion. Figure 1B shows the preimages of a toron with zero s_z at H_z = 0.2 T. As shown in the inset of Figure 1B, the toron consists of skyrmion layers aligning across much of the xy plane with a monopole-antimonopole pair located at its upper and lower ends^[41]. In addition, the xz midplane cross-sectional spin texture of the hopfion [Figure 1C] is a skyrmion-antiskyrmion pair.

Figure 1. 3D topological spin textures of hopfions and torons. The set of preimages with s_z = 0 for hopfions at 0 T (A) and toron at 0.2 T (B) in a square domain with side length L = 150 nm and thickness t = 90 nm. The preimage refers to a mapping of the 3D space that contains spins with the same orientations. The left and right insets show the series of cross-sectional spin textures for the hopfion and the toron in the xy plane, respectively. The color sphere denotes the spin orientation in three dimensions. The color wheel denotes the spin orientation in the xy plane. (C) The xz midplane cross-section of the hopfion. (D) The dependence of the 2D topological charge (N_s) on the perpendicular position z for the hopfion and the toron. (E) The xz midplane cross-section of the toron.

The topological invariant of hopfions in real space, known as the Hopf index^[22], can be expressed as

where $$B_{i}=\varepsilon_{i j k} \mathbf{n} \cdot\left(\nabla_{j} \mathbf{n} \times \nabla_{k} \mathbf{n}\right) / 8 \pi_{k}$$ is the emergent magnetic field calculated from the unit vector field n, and A is the emergent vector potential that satisfies the equation ∇ × A = B. The Hopf index can also be calculated using the linking number of knots, determined by their toroidal (T) and poloidal (P) cycles. The Hopf index is therefore given by Q_H = TP^[21]. Accordingly, the Q_H in Figure 1A is calculated at approximately 1. However, the Hopf invariant is ill-defined for the toron due to the presence of two monopoles near two surfaces that induce the unsmooth distribution of B. In the presence of strong perpendicular boundary conditions in confined surfaces, the skyrmion tube ends with two monopoles, each carrying opposite charges^[42]. Moreover, within the planes between the confined surfaces, the cross-sectional spin texture of the toron exhibits an elementary skyrmion. Since the 3D hopfion and toron are constructed from multiple layers exhibiting an independent 2D spin texture, analyzing their layer-dependent 2D topological charge enables differentiation between these two structures. The 2D topological charge is defined as $$N_{s}=\frac{1}{4 \pi} \int \mathbf{n} \cdot\left(\partial_{x} \mathbf{n} \times \partial_{y} \mathbf{n}\right) d^{2} r$$. Figure 1D shows the layer dependence of the N_s for the hopfion and the toron, both of which have a zero topological charge on the top and bottom surfaces due to the surface pinning from strong PMA layers. The xz midplane cross-sectional spin texture of the toron is shown in Figure 1E. In the middle plane with weak PMA, the N_s of the hopfion remains 0, while the toron changes from 0 to 1. This result is consistent with the aforementioned 2D magnetization profiles, where the topological charges of skyrmionium and skyrmion are 0 and 1, respectively.

Hopfions in nanodisks with various outer boundary shapes

To gain more insight into how boundary geometry influences the topological transformation of the 3D magnetic structure, we analyze the spin configurations at zero fields in confined systems with various outer boundary shapes. The formation of hopfion stems from the competition terms in Eq. 3, where demagnetization energy is largely impacted by the confined geometric shape. When a hopfion is confined within a square geometry, it adopts a square shape [Figure 1A], while it turns into a circular shape in a circular nanodisk [Figure 2A]. Although the morphology of hopfions is altered by the shape of the geometric constraint, their topology remains unchanged with a linking number of knots that equals 1, which is consistent with a previous study^[29]. In order to facilitate experimental observation of hopfions, we simulate LTEM images that highlight their in-plane spin component. Their simulated phase-shift and under-focused LTEM images viewed along the z-axis in circular nanodisk are shown in Figure 2B and C, respectively. The LTEM images of a hopfion at different focus setting is shown in Supplementary Figure 1. Figure 2D shows the reconstruction of the magnetic component distribution of hopfions by solving the transport-of-intensity equation (TIE) via focal series, in line with the in-plane spin distribution depicted in Figure 2A. An under-focus image displays a ring-like pattern with a bright circular hola surrounding a dark spot, which is distinct from the LTEM contrasts of skyrmion tubes or bobbers, which only show a bright spot^[15]. However, the ring pattern observed in an under-focused hopfion image may resemble that of a skyrmionium tube with a ring pattern, and the comparison of phase intensity between the two textures is addressed in Supplementary Figure 2^[43].

Figure 2. Hopfion under geometric constriction. (A) Mid-plane cross-sections of hopfions in circular nanodisk with d = 150 nm. (B) Simulated phase-shift image and (C) under-focused LTEM image of hopfion shown in (A) for viewing directions along the z-axis. (D) In-plane magnetic component distribution reconstructed by solving the transport-of-intensity equation. (E-H) Magnetization structures stabilized in elliptical nanodisks with different short axis lengths at the mid-plane. (I-L) Results of simulated under-focused LTEM images in elliptical nanodisks corresponding to hopfions in (E-H).

Recently, several studies have delved into the existence and stability of elliptical skyrmions, which showcase exceptional dynamics and resonance modes through the implementation of anisotropic DMI and anisotropic magnetic anisotropy^[44-46]. Here, we find that the elliptical hopfion can be formed via geometric constraints. Figure 2E-H presents an overview of the relaxed magnetic configurations in elliptical nanodisks with varying axial ratios, where the long axis (LA) length is fixed at 150 nm and the short axis (SA) length varies at 130 nm [Figure 2E], 100 nm [Figure 2F], 90 nm [Figure 2G], and 70 nm [Figure 2H]. When the SA length is greater than 100 nm, the elliptical hopfion remains stable. However, as it decreases below 100 nm, the hopfion undergoes a gradual transformation into a toron state, which is attributed to the boundary effect. It should be noted that the demagnetizing fields play a crucial role in determining the stability of 3D textures with diverse geometric shapes, which is further demonstrated in Supplementary Figure 3. Figure 2I-L presents the simulated under-focused LTEM images in elliptical nanodisks corresponding to the magnetic configuration in Figure 2E-H. LTEM images with distinct contrast can reproduce hopfion transformation. The dark central spots in Figure 2I and J represent the core spin arrangement of hopfions, while the central shiny point in Figure 2K and L indicates the position of the Bloch points of the toron with a deeper magnetic length. Thus, careful observation of LTEM contrast can reveal detailed information about these topological textures. For the real-space observation of these topological textures, a promising system involves combining chiral magnets with magnetic multilayers. Specifically, multilayer systems, such as Ta/CoFeB/MgO^[47], Pt/Co/Ta^[48], and Pt/Co/Ir^[32], allow for tunable PMA by adjusting the ferromagnetic layer thickness and stack configuration. The fabrication of heterostructures by incorporating chiral magnets such as FeGe^[49], Co₉Zn₉Mn₂^[50], and Cu₃OSeO₃^[13] sandwiched between multilayers possessing high PMA holds great promise for this purpose.

Current-driven motion of hopfion in a nonostripe

In comparison with the nanodisk, understanding hopfion behavior and stability within a nanostripe contributes to their application in racetrack memory. Figure 3 illustrates typical metastable states in a finite-size nanostripe with different values of K_b and H_z. A wide range of K_b (0-60 KJ/m³) and H_z (0.20-0.40 T) allows for the stabilization of hopfion, which indicates that the formation of the topological structure is not parameter-sensitive and appears to be widespread. Low K_b values result in an in-plane magnetic background, and small H_z values lead to the formation of stripe domains. This occurs because the enlarged hopfions reach the boundary and knots break. With an increase in H_z, the radius of a hopfion gradually shrinks, and it changes into a toron structure and eventually transitions to a ferromagnetic uniform state.

Figure 3. Phase diagram of different topological states with respect to PMA constant K_b and perpendicular magnetic field H_z. The metastable states are obtained by relaxing the magnetic nanostripe with an initial hopfion state at the center. The midplane (z = 50 nm) cross-sections of m are presented. The size of the nanostripe is 900 × 600 × 100 nm³.

According to the above results, the shape confinement effect in small-scale nanodisks enables the stable existence of hopfion at zero fields. In nanostripes or thin plates, hopfions exist as a metastable state and require an assistant magnetic field. Given the potential application of hopfions in racetrack memory, understanding their stability and current-induced dynamics within a nanostripe is crucial. Figure 4A shows the relaxed state at K_b = 40 kJ/m³ and H_z = 0.34 T, with the hopfion initially located at the left side of the nanostripe. The hopfion eventually stabilizes near the center of the nanostripe due to the repulsive force from the surrounding borders^[51]. It was found that as the length of the nanostripe increases, the hopfion can also stabilize on the left side of the stripe. Detailed information regarding the relationship between the hopfion and its distance from the boundary as the length varied is available in Supplementary Figure 4. When applying a current with J = -8 × 10¹¹ A·m^-2 along the x direction, the hopfion first moves against the spin-polarized current direction for a short distance via STT. Subsequently, the hopfion slightly bounces back and eventually comes to a halt at a position approximately 300 nm away from the right boundary [Supplementary Movie 1]. It fails to reach the boundary owing to the large repulsive force from the edge, a phenomenon similar to that observed in skyrmion-host systems^[38]. For a larger current density J = -10 × 10¹¹ A·m^-2, hopfions can overcome the potential barrier and reach the boundary, as demonstrated in Figure 4A-D. The threshold current density J_c to overcome the boundary barrier is -8.9 × 10¹¹ A·m^-2 in the nanostripe at 0.34 T. Hopfions move straight initially [Figure 4B] and gradually deform [Figure 4C] as they approach the right boundary. Thereafter, the deformed hopfion transforms into the toron structure [Figure 4D] with the collapse of core spins, ultimately leading to its annihilation at the sample edge. The corresponding motion and transition processes are shown in Supplementary Movie 2. Magnetic contrasts of LTEM images for the current-driven motion of hopfion are presented in Figure 4E-H. The conversion of the magnetic contrasts from a ring-like pattern [Figure 4E-G] to a bright spot [Figure 4H] identifies the topological transition process. Different from the direct annihilation of skyrmion at the edge under an electric current, we find that hopfion undergoes a topological transition into toron before annihilation due to the combined effect of STT and the repulsive potential from the boundary. The driving force from STT and the repulsive force from the boundary exert a compression effect on the hopfion, thereby causing its distortion into a toron.

Figure 4. Current-driven hopfion motion by STT in a nanostripe. (A-D) Current-driven motions of a hopfion at J = -10 × 10¹¹ A·m^-2 under a magnetic field of 0.34 T. The size of the simulated region is 900 × 600 × 100 nm³. The midplane (z = 50nm) cross-sections of m are shown. (E-H) Simulated under-focused LTEM images corresponding to structures in (A-D). (I) Trajectory of the spin texture driven by STT during a period of 16 ns. (J) The hopfion motion velocity along the x-axis v_x and 2D topological number N_s as a function of time t. The v_x and N_s are calculated based on spin textures in the midplane. (K) The demagnetizing energy density in the z = 50 nm plane at 0 ns.

The motion of hopfion can be qualitatively described within the framework of Thiele’s equation

The first term denotes the Magnus force, G = (0, 0, 4πN_s) is the gyrovector, v is the velocity of the hopfion, and u is the STT-related electron driving velocity. The second term represents the dissipation force related to the tensor D defined as $$\boldsymbol{D}_{i j}=\int \partial_{i} \mathbf{m} \cdot \partial_{j} \mathbf{m} d V $$. The third term F is the force acting on the hopfion, which mainly comes from the interaction from the surrounding boundaries in our case. Due to a vanishing gyrovector, hopfions can move along the current via STT according to Eq. 3. However, for intermediate layers of toron, N_s are non-zero, which means that the Magnus force cannot be neglected and may induce a transversal motion of the toron. Figure 4I displays the trajectory of the local spin texture induced by current during a period of 15 ns; the position is determined by calculating the guiding center R(X, Y) of spin textures, expressed as $$\mathrm{R}(\mathrm{X}, \mathrm{Y})=1 / 4 \pi N_{s} \int i \mathbf{n} \cdot\left(\partial_{x} \mathbf{n} \times \partial_{y} \mathbf{n}\right) \mathrm{d} x \mathrm{d} y$$, i = x, y. The hopfion exhibits straight motion along the x direction, whereas the toron shifts to the -y direction, consistent with the analysis presented above. Note that the slight +y direction deviation of the hopfion at about 630 nm is caused by its deformation. Figure 4J shows the longitudinal component of the soliton velocity v_x and 2D topological charge N_s in the midplane as a function of time t. Owning to the repulsive force from the edge, hopfion shows a gradual decrease in v_x before the transformation (1-10 ns). During the period of current-driven transition from a hopfion to toron (10-12 ns), the N_s of the midplane changes from 0 to 1 and v_x increases significantly and maintains a relatively large value until it is annihilated at the edge (12-16 ns). Besides, the toron structure can be formed by only increasing the applied magnetic field from 0.34 to 0.38 T. More details about the current-driven toron in the nanostripe are shown in Supplementary Figure 5 and Supplementary Movie 3. The repulsive potential from the boundary is mainly induced by the chiral edge state, which represents the twist of the spin and nearly-in-plane spins at the edge of the nanostripe. Such edge state has been quantitatively verified in chiral magnets^[52]. To further illustrate this, Figure 4K presents the distribution of the demagnetization energy densities for the spin configurations depicted in Figure 4A. The energy in the interior zone is stronger than that at the edge, coinciding with the perpendicular spin alignment at the intermediate region and in-plane tilted spin distribution at the edge.

Current-driven hopfion in artificially designed nanostructures

Finally, we study the current-driven dynamics of hopfions in artificially designed nanostructures, including a nanostripe with a single square notch, “H-like” shaped configurations, and stepped nanostructures. Figure 5A illustrates the motion of the hopfion in the nanostripe with a single-notch structure. The size of the notch is 60 nm × 150 nm and the hopfion trajectory is depicted by the black dashed line. Due to the extra repulsive force introduced by the notch, the hopfion stabilizes at the left side instead of the center of the nanostripe. When applying a current with J = -10 × 10¹¹ A·m^-2, the hopfion initially moves straightly along the +x direction and then shifts away from the notch. As the hopfion traverses the notch, it shifts back due to the diminished influence from the notch. The dynamical response of hopfion when approaching the boundary is similar to what is discussed in the case of the nanostripe. For the “H-like” nanostripe featuring two symmetrical notches of dimensions 150 nm × 150 nm, the hopfion moves almost entirely along the x direction with minimal deviation [Figure 5B]. This result is anticipated, given the canceled repulsive forces from the top and bottom notches. However, the hopfion experiences compression when traveling between the notches, leading to a reduction in its size. This indicates that introducing specific shapes of nanostructures can significantly alter the current-driven trajectory and size of the hopfions.

Figure 5. Current-driven dynamics of hopfions in designed nanostructures. Trajectory of hopfions in (A) nanostripe with a single notch. (B) “H-like” shaped nanostructures at J = -10 × 10¹¹ A·m^-2. (C) Current-driven transition from hopfion to toron in a stepped geometry at J = -10 × 10¹¹ A·m^-2. The cross-sectional spin textures in the xz plane show (D) a hopfion, (E) a distorted hopfion, and (F) a toron structure at different times.

Taking inspiration from previous work on the transformation between a skyrmion tube and bobber in a stepped nanostructure^[37], we investigate the current-driven dynamics of hopfion in a nanostripe containing a vertical step, where a topological transformation potentially takes place. Figure 5C schematically shows the motion of the hopfion in a stepped geometry, and the cross-sectional spin textures in the xz plane are shown in Figure 5D-F, depicting the transformation from hopfion to toron in the stepped nanostructure. As the hopfion approaches the step edge, the potential barrier induces the hopfion to distort [Figure 5E], and finally, a toron forms in the thicker region [Figure 5F]. Unlike the toron state that occurs after a topological transition near the boundary or notch, each topological state in the stepped nanostructure can be easily generated and is capable of traveling long distances.

Studying the interaction between 3D topological spin textures and the sample geometry presents an effective strategy for addressing their stability and tunability^[53,54]. Additionally, topological transformations hold immense potential for spintronic applications and could lead to the emergence of novel practical uses based on diverse 3D topological structures. The polarity of vortices^[55] , topological charge^[56], and other topological quantities^[57] are found to be altered by various external stimuli, including magnetic field pulses, in-plane magnetic fields, and thermal energy. In comparison to the topological phase transitions induced by the aforementioned methods, the transitions of 3D topological spin textures caused by geometric shapes and currents discovered in this study make it easier to integrate them into spintronic devices, which can then be used to control the size and motion trajectory of topological spin textures. The reliable transformation from hopfion to toron makes the 3D topological magnetic structures act as reliable information carriers that can be applied to (i) hopfion racetrack memory enabling binary bit states^[37]; (ii) magnetic memories for temporal computing^[58]; and (iii) artificial synapses and neurons^[59]. However, the nucleation of hopfions within nanostripes or nanoflakes remains challenging in experiments. For the formation of hopfion, two potential methods could be considered: Injecting an annular spin polarized current to reverse the local magnetization and facilitate hopfion creation; Applying positive and negative magnetic fields to induce hopfion formation. Moreover, for practical applications in devices, the hopfion-hopfion interaction and their interaction with defects should be further investigated.