Bloch point dynamics in exchange-spring heterostructures

Topologically non-trivial magnetization textures^1,2 have shown great promise for the development of non-volatile logic and storage devices because the topological constraints provide stability to the texture, thus making them attractive as information carriers.^3–7 Recent breakthroughs have shed light on the physics of magnetic skyrmions^8–10 and chiral bobbers^11,12, with particular emphasis on materials that exhibit the Dzyaloshinskii–Moriya interaction (DMi), either in chiral crystals such as FeGe and MnSi^13,14 or in multilayers with heavy metals.^15–17 Skyrmions and other topologically non-trivial textures can also be stabilized in a wide range of materials without DMi by dipolar interactions,¹⁸ and even in the absence of both DMi and dipolar interactions, skyrmions can be created dynamically.¹⁹ In this work, a simple geometry combining soft and hard ferromagnets is proposed to enable control over the creation and dynamics of topologically non-trivial magnetization textures.

The ability to create and control magnetization textures in simple ferromagnets is important in overcoming the material limitations of chiral crystals, and opens the way to investigating dynamic phenomena in nanostructured ferromagnets. In nanoparticles with azimuthal symmetry, such as cylindrical nanowires, the magnetization reversal mechanism depends on the diameter of the particle.²⁰ In ultrafine single-domain particles, the reversal follows the single-energy barrier Néel–Brown model,²¹ whereas for larger particles reversal occurs via the motion of domain walls. When a nanowire has diameter comparable to the domain wall width, however, topological constraints lead to a different reversal mechanism: the magnetization switching is initiated via curling and the formation of a skyrmion tube²² (see Fig. 1). The onset of irreversibility occurs once the skyrmion tube breaks to form Bloch-points, i.e., point singularities where the local magnetization goes to zero.²³ Recent micromagnetic simulations have predicted that these Bloch points move at extremely high speeds (⁠$∼103$ m/s), both in DMi materials²⁴ and in soft ferromagnets,²⁵ and, as a result, emergent electric fields with a magnitude of the order of megavolts per meter are induced. This is highly promising for the development of nanomagnetic devices for energy conversion, but a deeper understanding and precise control of the magnetization reversal must be obtained in order to harness these phenomena.

Here, we take a step further toward functionalizing magnetization-texture dynamics and investigate a geometry that provides control and tunability of the Bloch-point mediated magnetization switching. Specifically, a hard–soft heterostructure is proposed, where the soft ferromagnet undergoes magnetization reversal upon the application of an external field, but the exchange with the hard ferromagnetic layer restores the system to the original configuration. The combination of hard and soft ferromagnetic materials, so called exchange-spring magnets, has been extensively investigated for the design of advanced high-performance magnets with high energy products,^26–28 where the strong anisotropy of the hard component stabilizes the high saturation magnetization of the soft component. In this work, the hard component is a ferromagnetic layer with perpendicular magnetic anisotropy (PMA), and it is exchange-coupled to a cylindrical soft ferromagnetic nanowire, as shown in Fig. 1. As discussed below, micromagnetic simulations predict that an external magnetic field can switch the magnetization in the soft cylinder, which proceeds via the breaking of a skyrmion tube and the motion of Bloch points. When the external field is removed, the interaction with the PMA layer will drive a Bloch point in a reverse motion and restore the magnetization to its original configuration, thus making the process fully reversible. The magnetization reversal process in the soft ferromagnet can be highly tuned, and this geometry enables the forward and backward motion of Bloch points in soft ferromagnets. Furthermore, the emergent electrodynamics of the rapidly moving Bloch points makes this system promising for generating localized ultrashort electromagnetic pulses in sub-micron devices.

Micromagnetic simulations were performed for hard–soft structures by taking into account contributions from the ferromagnetic exchange interaction with exchange stiffness A_ex, perpendicular magnetic anisotropy with K_u (first-order) for the hard layer, long-range dipole–dipole interactions incorporated with the local dipolar field B_dip, exchange interaction between the layers incorporated with a local interaction field B_exc, and interaction of both layers with the external field B. The total energy density is

where k = 1 is for the soft ferromagnet and k = 2 is for the PMA layer, and m_i is the ith component of the magnetization unit vector m = M/M_s with M_s the saturation magnetization.

For the soft ferromagnet, several materials were tested (see Table I), but for the discussion here the archetypal soft ferromagnet permalloy Fe₂₀Ni₈₀ (Py) was chosen. For the PMA layer, two different materials were considered: (i) PtCo with A_ex = 15 pJ/m, M_s = 580 kA/m, uniaxial anisotropy of K_u = 600 kJ/m³, taken as representative values from Refs. 29–31, and (ii) a − TbCo with A_ex = 20 pJ/m, M_s = 400 kA/m, and K_u = 10⁶ J/m³, adopted from Ceballos et al.³² Simulations were performed with both PMA materials and yielded similar results. Therefore the discussion here is limited to the Py–PtCo system.

Micromagnetic simulations were performed with the GPU-accelerated finite-difference software Mumax3³⁵ by numerically integrating the Landau–Lifshitz–Gilbert equation of motion ∂_tm = −γ(m ×B_eff) + α(m × ∂_tm), where γ/2π = 28 GHz/T is the gyromagnetic ratio, α the unitless Gilbert damping parameter, and B_eff = −∂_mE/M_s is the effective field. The volume of the PMA layer was kept constant, with side lengths l = 100 nm and height h = 50 nm. The Py cylinder had a diameter d = 60 nm and simulations with different heights h = 50−150 nm were tested. The system was initialized in the state m = (0, 0, 1), the equilibrium state was achieved by running the simulation at B = 0, and then the external field was applied opposing the remanent state B = (0, 0, −B) for 250 ps and then removed to allow the system to return to its original configuration (see the supplementary material for a sample simulation script). The strength of the external field was chosen to be larger than the anisotropy field of the soft ferromagnet (B > μ₀NM_s/2, with the demagnetizing factor N).

The spatial resolution of the simulation is important for the modeling of nanoscopic magnetization textures, particularly for Bloch points where the local magnetization exhibits a steep gradient and vanishes at its center. Notable works have developed hybrid techniques,^36,37 where micromagnetic simulations were combined with atomistic simulations in order to overcome the micromagnetic breakdown of the exchange-energy gradient in Eq. (1). However, recent studies have suggested that micromagnetics and atomistic simulations converge when the micromagnetic cell is comparable to the atomistic lattice unit-cell size,²⁴ and that micromagnetics performs well in modeling the magnetic texture of a substantial volume around the Bloch point,³⁸ which is what affects the overall state of the structure. Considering that with micromagnetics we can simulate geometries much larger than what is accessible with atomistic methods, thus being able to model device-scale structures, this work focused on micromagnetic simulations with a small cell size. Different simulation cell sizes were used to test numerical stability, with the high-resolution simulations having a cell size of 1 × 1 × 1 nm³, which is smaller than the exchange length of any of the investigated materials (⁠$δexPtCo=8.4$ nm, $δexTbCo=14$ nm) and comparable to the crystal unit cell dimensions.

The application of an external field opposing the remanent state causes the moments along the edge of the Py cylinder to curl, which leads to the formation of a skyrmion tube as shown in Fig. 1. Figure 2 shows the magnetization configuration inside the Py cylinder at the time before the skyrmion tube breaks. Taking a disk cutout from the cylinder, the z component of the magnetization unit vector follows the profile of a 2π domain wall²² $θ(x)=tancosh(R0)/sinh(x/δ)$⁠, where the parameters depend on the external field and R₀ = 1/4 defines the structure and δ = 10.5(5) nm is the size of the skyrmion. The skyrmion structure shown in Fig. 2(b) is associated with a topological charge on the surface of the disk D, which is Q_D = 1. With increasing time under the external field, the skyrmion tube diameter becomes smaller as an increasing volume fraction of the magnetic moments turn toward the external field, but with Q_D not changing. As the local magnetization fluctuates, a bottleneck forms and the tube breaks, creating a pair of Bloch points, as shown in Fig. 2(b). This marks a topological transition and the onset of irreversibility because the Bloch points drive the magnetization reversal by propagating along the cylinder, leaving behind a trivial ferromagnetic state: in the region between the two Bloch points, the topological charge is zero. This reversal mechanism was originally proposed by Arrott et al.,²³ who identified that topology requires the pair creation of point singularities for complete magnetization reversal, but as seen here and in Ref. 25, the Bloch points are pair-created inside the cylinder and not at the surface, as it was originally suggested.²³ 

Note that the duration of the external magnetic field pulse (250 ps) was chosen so that there is enough time for the formation and breaking of the skyrmion tube. A pulse with smaller field strength would require longer time.³⁹ Furthermore, if the external field is tilted, the process persists, but the skyrmion tube is skewed and the Bloch points are unstable and escape through the surface of the soft ferromagnet (see the supplementary material for examples with tilted and with stronger external field).

Turning to the structure of the Bloch point, the magnetization profile can be fitted with the Bloch π domain-wall³³ θ(z) = 2 arctan e^z/δ, where δ = 1 nm is equal to one simulation cell. The topological charge can be found by integrating the local magnetization over a spherical surface S² around the center of the Bloch point, as indicated in Fig. 2(b). The magnetization unit vector in spherical coordinates is parameterized with $m=sin⁡θ⁡cos⁡ϕ,sin⁡θ⁡sin⁡ϕ,cos⁡θ$⁠, and the spherical surface S² is parameterized with $r=r⁡sin⁡ϑ⁡cos⁡φ,r⁡sin⁡ϑ⁡sin⁡φ,r⁡cos⁡ϑ$⁠. With that, the topological charge is⁴⁰ 

For the top Bloch point, the magnetization can be written as $θϑ=ϑ$ and $ϕφ=π+φ$⁠, which yields $QS2=+1$⁠, whereas, for the bottom Bloch point, it is $θϑ=π−ϑ$ and $ϕφ=π+φ$⁠, and therefore, $QS2=−1$⁠.

These results are similar to those published in Ref. 25, which showed simulations of free-standing Py nanowires. In a cylinder with two free edges, a skyrmion tube forms at each edge. In contrast, here the switching becomes selective and only one skyrmion tube forms at the free edge of the cylinder. Once the skyrmion tube breaks, the top Bloch point propagates toward the free end and escapes through the surface. The bottom Bloch point propagates downward, toward the PMA layer, and becomes trapped, i.e., it does not cross the interface between Py and PtCo, as shown in Fig. 3, which shows contour plots of the z component of the magnetization at different time steps.

As soon as the external field is removed, the interaction with the PMA layer turns the circumferential surface moments of the Py cylinder upward [Fig. 3(e)], and re-forms the Bloch point that is pushed upward and eventually escapes through the free top surface. The PMA layer, thus, reverses the entire process and restores the magnetization in the Py cylinder to its original configuration. Hence, with this geometry, it is possible to create a skyrmion tube and a Bloch-point pair with the application of an external field, and then expel the Bloch point in the absence of an external field. While the stray field of the PMA layer contributes to the recovery of the original state, the main driver for the magnetization recoil is the exchange coupling with the PMA layer. Simulations with different inter-layer exchange strengths show that the process persists even for a non-perfect exchange. When the soft ferromagnet and the PMA layer are weakly or fully decoupled, however, the Bloch point at the interface with the PMA layer breaks and the magnetization in the soft ferromagnet does not recover the original configuration (see Fig. 4).

Turning to Bloch point dynamics, Fig. 5 shows (a) the z component of the Py magnetization and (b) the position and speed of the Bloch point as a function of time, with and without the external field; for example, with h = 50 nm. The onset of irreversibility is marked by an arrow in Fig. 5(a) at t = 128 ps when the Bloch point pair is created. In this simulation with a short Py cylinder (h = 50 nm), the Bloch point pair was created close to the free surface, and the top Bloch point vanishes rapidly. Therefore, only the trajectory of the bottom Bloch point is analyzed here. During the initial phase, the Bloch point travels a distance of ~50 nm in under 100 ps, i.e., with an average speed of $∼500$ m/s, but as seen in Fig. 5(b), the instantaneous speed reaches values as high as $∼3000$ m/s. After a strong fluctuation of speed around 150 ps, the Bloch point reaches the interface, and its speed fluctuates with frequencies in the range 0.05–0.5 THz, which is comparable to the frequencies reported in Ref. 24 for moving Bloch points in a chiral crystal. As soon as the external field is removed at t = 250 ps, after an initial fluctuation phase, the Bloch point accelerates upward, reaching a speed of $∼1000$ m/s until it reaches the free surface of the Py cylinder and escapes, leaving behind a saturated ferromagnetic state. This is marked by a second arrow in Fig. 5(a) at t = 334 ps.

These findings show that the Bloch points in the proposed exchange-spring system achieve speeds that greatly surpass those of conventional domain walls. The simulation images shown here are for a cylinder that is only 50 nm tall, but the effects remain similar for longer cylinders, whereas the Bloch point travels at similar speeds through the cylinder and it takes the magnetization respectively longer to return to the original configuration. This can be seen in the inset to Fig. 5(a), which shows the z component of the magnetization of the soft ferromagnet as a function of time for three different Py cylinder heights; with increasing cylinder height, the restoring of the magnetization takes longer.

Importantly, the rapid motion of these textures and, in turn, the rapid change of flux around the Bloch point, will induce an electric field. The order of magnitude of the field can be estimated from the force on the conduction electrons, which, following Ref. 41, gives F ≈ ℏv/2λ², where ℏ is the reduced Planck’s constant, v is the speed, and λ ≈ 1 nm is the size of the Bloch point, as seen from the fit in Fig. 2(f). With a top speed of v = 3.0³⁰ × 10³ m/s, the force is F ≈ 1.56 × 10⁻¹³ N, which corresponds to the force from an electric field of magnitude E = F/e ≈ 9.6 × 10⁵ V/m, which is comparable to the values found in Ref. 25 for free-standing Py cylinders. Note that this magnitude is at the location of the Bloch point, but the field will extend beyond the Py cylinder, and taking E ∝ 1/R², analogous to B ∝ 1/R² for a moving charge, at the edge of the cylinder, the electric field will be $∼1000$ V/m, which is a substantial magnitude, and it is the field produced by only one moving Bloch point.

The linear trajectory of the Bloch points will create a solenoidal E^em, analogous to the magnetic field produced by a moving charge, in full accordance with Maxwell’s equations. This can be determined by the magnetic vector potential using the convolution integral⁴² 

During the field-induced reversal process, two Bloch points are moving in the cylinder and produce an electric field (E^em = −∂_tA) with the same handedness because they have opposite charge and move in opposite direction. During the PMA-induced reversal, only one Bloch point is moving and produces a field of opposite handedness. Figure 6 shows the electric field for the bottom Bloch point during the two segments of its trajectory.

Given the above, the emergent electric field will rise to $Eem=+E0emφ̂$ during the initial reversal stage and then reverse to $Eem=−E0emφ̂$ during the second stage, i.e., within a time window of 150 ps. Therefore, this structure can, in theory, produce an ultrafast oscillating electromagnetic pulse, and the electric field can be amplified by combining many elements in a matrix.

The phenomena described here can be predicted for other combinations of materials, where the two important conditions are as follows: (i) the diameter of the soft ferromagnet has to be comparable to 2πδ_ex, so that it can fit a 2π domain wall,^22,25 and (ii) the anisotropy of the PMA layer needs to be larger than the shape anisotropy of the soft ferromagnet: $4Ku>2μ0Ms,s2+μ0Ms,h2$⁠. Similarly, the lower limit for the length of the soft ferromagnetic cylinder is also 2πδ_ex, whereas the lower limit for the thickness of the PMA layer is set by the superparamagnetic limit: the volume of the PMA layer needs to be V_h ≥ 25k_BT/K_u. Furthermore, the anisotropy of the PMA layer also sets the upper limit for the strength of the external field, which should not exceed B < 2K_u/M_s,h − μ₀M_s,h, because the magnetization of the PMA layer will become unstable.

Table I lists some soft ferromagnets that were tested with micromagnetic simulations and found to reproduce the phenomena described here. Figure 7 shows a comparison between the four soft ferromagnets that exhibit the same behavior, and the reversal follows the same process. The arrow in each panel in Fig. 7 indicates the time at which the skyrmion tube breaks and the Bloch-point pair is created. (The magnetocrystalline anisotropy of Fe and Ni was not taken into account in the simulations because it is 1 to–2 orders of magnitude smaller than the magnetostatic energy.) Moreover, choosing a soft ferromagnet with a higher M_s can maximize the electric field, because the magnetic vector potential and, in turn, E^em are proportional to M_s. Additionally, with increasing M_s/A_ex ratio the dimensionality decreases, and the materials can be chosen such as to optimize the performance of the exchange-spring system.

There are many different factors that are important for the effects described above and many ideas need to be investigated, e.g., switching with currents or including thermal effects. There is a vast parameter space to be explored, and this can potentially reveal new insights into the formation and behavior of nanoscopic three-dimensional magnetization textures and their promise for the development of novel energy efficient magnetic technologies.

The micromagnetic simulations presented here demonstrate how a combination of hard and soft ferromagnetic components can be used to create and manipulate topologically non-trivial magnetization textures. Application of an external field reverses the magnetization of the soft component via a curling process that leads to the formation of a skyrmion tube and a subsequent pair-creation of Bloch points that complete the magnetization reversal. One Bloch point exits through the free end of the cylinder, while the second Bloch point remains trapped at the soft/hard interface. Once the external field is removed, the exchange coupling with the hard layer drives the Bloch point toward the free end of the soft cylinder and restores the magnetization to its original configuration. The Bloch points move at speeds of the order of kilometers per second, both with and without an external field. This gives rise to an alternating emergent solenoidal electric field with a magnitude of the order of 10⁵ V/m. The speed of the Bloch points and the duration of their motion can be tuned by adjusting the size of the soft ferromagnetic component. Therefore, the proposed architecture can be used to create ultrashort electromagnetic pulses, which, in turn, can serve in a wide range of energy-conversion applications.

The supplementary material contains several examples of the magnetization process described here under different conditions and includes a movie showing the process for PtCo(50 nm)/FeNi(50 nm).

The author gratefully acknowledges funding from the Louisiana Board of Regents [Contract No. LEQSF(2020-23)-RD-A-32] and wishes to thank David Slay, Tarikul Milon, and Leonardo Pierobon for their fruitful discussions.

The author has no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.