Ultrafast imprinting of topologically protected magnetic textures via pulsed electrons

Tremendous progress has been made towards the realization of spatiotemporally controlled electron sources for probing the material local structural, electronic, and magnetic dynamics.^1–4 Working schemes rely on the electron emission from a laser-irradiated nanoscale apex^5–22 with the electron pulse duration being controllable with the laser pulse duration. The laser intensity dictates the electron number in the bunch. Electron pulse acceleration and control are achievable by intense THz fields.^23–25 Here, we explore the potential of very fast, relativistic electron bunches for a possible control of the magnetic dynamics in a thin film which is traversed by the electrons. Our focus is on the sample spin dynamics triggered by the electric and magnetic fields associated with the electron bunch.²⁶ In fact, a pioneering experiment²⁷ explored the ultimate speed limit for precessional magnetic dynamics of a CoCrPt film driven by the magnetic field $B(r,t)$ of short relativistic electron pulses (with a duration of $δ=2.3$ ps) passing a 14 nm thin film of granular CoCrPt ferromagentic samples with grain sizes of $20.6±4$ nm. The main experimental results are shown along with our simulations in Fig. 1. Prior to the electron-pulse, the sample was magnetized homogeneously in the z direction. The pulse induced ring pattern of the magnetic domains pointing either up or down (with respect to the easy direction of the magnetic films) is well captured by our micromagnetic simulations and can be interpreted by the analytical model enclosed in the supplementary material. As pointed out in Ref. 27, the critical precessional angle $ϕ≥π/2$ is determined by the local strength of the magnetic field and indicates the achieved angular velocity ω. The pulse duration δ plays a crucial role.²⁸ As discussed in Ref. 28, an appropriate sequence of ps pulses allows for an optimal control scheme, achieving a ballistic magnetic switching, even in the presence of high thermal fluctuations. Longer pulses might drive the system back to the initial state.²⁸ So, the critical precessional angle and δ are the two key parameters²⁷ for the established final precessional angle $ϕ=ωδ$⁠. Note that the demagnetization fields are also relevant, as inferred from Fig. 1, but they do not change the main picture (further details are in the supplementary material). Having substantiated our methods against the experiment, we turn to the main focus of our study, namely, the generation of topologically protected magnetic excitations such as skyrmions via the electron pulses. We consider that samples exhibiting Dzyaloshinskii-Moriya (DM) spin-orbital coupling are appropriate. A recent work²⁹ evidences that ultrathin nanodiscs of materials such as Co_70.5Fe_4.5Si₁₅B₁₀ (Ref. 30) sandwiched between Pt and Ru/Ta are well suited for our purpose. The magnetization's structure may nucleate spontaneously into skyrmionic configurations. We adapted the experimentally verified parameters for this sample and present here the result for the magnetic dynamics triggered by short electron beam pulses. Taking a nanodisc of a variable size, the ground state with a topological number $|N|=1$ is realized after propagating an initially homogeneous magnetization in the $±z$ direction according to the Landau-Lifshitz-Gilbert equation (LLG) including DM interactions.^35–38 The two possible ground states, depending on the initial magnetization's direction, are shown in Fig. 2 along with the material's parameters.

Our main focus is on how to efficiently and swiftly create skyrmions, an issue of relevance when it comes to practical applications. Previous theoretical predictions (e.g., Ref. 31) utilize a spin-polarized current for the skyrmion generation. However, large current densities and a finite spin polarization of injected currents are needed. Thus, it is of interest to investigate the creation and annihilation of skyrmions with current pulses similar to those discussed above using the surrounding magnetic field. Of interest is the skyrmion generation and modification via a nano-focussed relativistic electron pulse. While currently such pulses can be generated with micron size beam dimensions,³² future sources are expected to reach ficus sizes down to the few nm range.³³ In principle, the possibility of beam damage occurring in the beam focus as in the case of the experiment in Ref. 27 is present. However, ongoing experiments with relativistic electron beams³² indicate that the use of ultrathin freestanding films may alleviate damage concerns.

Topologically protected magnetic configurations, such as magnetic skyrmions, are well-defined quasiparticles. They can be characterized mathematically by the topological or winding number $N=14π∫m·(∂m∂x×∂m∂y)dxdy$ (Ref. 34), which counts simply how often the unit vector of the magnetization wraps the unit sphere when integrated over the two-dimensional sample. Therefore, skyrmions are typically a quasiparticle in thin (mono)layers. The topological number adopts integer values, indicating the magnetic configuration to be skyrmionic (⁠$N=±1$⁠) or skyrmion multiplexes (⁠$|N|>1$⁠). If the topological number is not an integer, the topological protection is lifted and the magnetic texture is unstable upon small perturbations. The topological stability of skyrmionic states stem from the necessity of flipping at least one single magnetic moment by $180°$⁠, to overcome the barrier and transfer the state into a “trivial” state, like a single domain or vortex state. In the following, we will attempt to overcome this energy barrier with the previous methods so that the magnetization will be converted into a state with a different topological invariant. Advantageous is the spatial structure of the magnetic field curling around the beam's center, which gives a good point of action in order to manipulate topologically protected configurations.^35–38 

Using the short electron pulses, one may overcome the topological energy barrier with a magnetic “kick” and the magnetization relaxes afterwards, possessing a different winding number. In contrast to the mesoscopic system studied in Fig. 1, in the following, not only the far field but also the near magnetic field of the Gaussian pulses will be treated. To do this, the Biot-Savart law is solved numerically and fitted with a model function. For magnetic systems, a minimum time of exposure is necessary, whereas the spatial focus of the beam is limited. To overcome this conflict, the pulse duration is fixed at 2.3 ps as before, when nothing different is mentioned. Details on the resulting magnetic field can be seen in the supplementary material. Starting from such an electron beam, two main parameters can be adjusted to achieve the favored reaction of the nanodiscs. Those are the pulse width and the number of electrons, which will be treated independently. In Fig. 3, the final topological charges after a single Gaussian electron pulse irradiating a nanodisc are plotted as a function of the number of electrons and the width of the Gaussian distributed electrons. The results do not show the transient time evolution of the sample but only the final steady-state values of the winding number. They are obtained by applying an electron pulse, propagating the magnetization during the pulse, and then relaxing the magnetic configuration as to approach a local minimum of the free energy's hypersurface.

We note the strong correlation between the change of the topological charge and the number of electrons or accordingly the beam width. Relatively large intervals of both parameters lead to the same final values for N. We note that not only the variation of these control parameters but also of the duration of the pulse is experimentally accessible, particularly in a nanoapex ultrafast transmission electron microscope. It is noteworthy that the graphs for opposite initial configurations [see Fig. 3(a)] are axially symmetric with respect to the x axis. This can be explained by the coinciding symmetry centers of the pulse and the skyrmionic structure. This symmetric and robust behavior can be exploited to switch between accessible different values for the topological charges which are quite close to the ideal integer values that would be realized in an infinitely small discretization.

Interestingly, the switching between the two stable states occurs repetitively for an increase in the number of electrons, whereas the spatial manipulation of the beam leads to one regime only in which the fields are sufficient to switch the topological number. The first observation can be explained with the schematics shown in Fig. 2(c). Depending on the strength of the pulse, the magnetic moments are forced to rotate multiple times around the $êφ$ vector in a collective manner, as each moment of equal distance to the center experiences the same torque. The final position of the surrounding moments couples backwards to the center and determines the new topological charge. The electron number linearly translates to the peak magnetic field, whereas the beam width has a more complicated influence. When the width is increased, the spatial profile in the xy-plane is manipulated, as the maximum magnetic field is shifted towards the disc's rim and beyond. How the system reacts on this change depends crucially on the exact profile of the beam, especially on the point of maximum magnetic field strength, as can be seen in Fig. 3(a). This leads to the question of the optimum parameter regime, to manipulate the system reliably, which cannot be finally answered as it strongly depends on the experimentally available capabilities. Hence, this work focuses on an exemplary study on the effect.

The same switching phenomenon as discussed before can also be observed for different setups. Weaker pulses, as long as they are able to overcome the internal fields to excite the system, can be used as well, but obviously, the field's amplitude translates to the strength of the resulting torque. This implies a longer radiation time needed for pulses of lower intensity to be capable of switching the system. In the case of different materials or geometries, the accessible topological states have to be investigated, before they can be utilized. Otherwise undesired, interstitial states might be achieved by accident, and the switching is not deterministic anymore.

Aside from the manipulation of the topological charge of a given nanodisc system, the creation of skyrmions on extended thin layers is an open challenge. To treat this, we start from a quadratic region with a size of $(800×800)$nm² and periodic boundary conditions in x and y directions to avoid finite size effects. To overcome the homogeneously magnetized state, the peak intensity has to be increased further, as well as a focussing down to the scale of the desired skyrmion. The Gaussian profile in the xy-plane has a standard deviation of 30 nm, whereas the pulse duration is reduced to 500 fs and a single pulse includes 10⁸ electrons. Even though this is experimentally challenging, it is necessary to create skyrmions on an extended film. If the beam size is too large, only trivial domain rings are built up. On the other hand, the duration has to be long enough to allow the magnetic texture to react to the pulse. A well-known characteristic of magnetic skyrmions is a tendency to have a blow-up behavior to minimize the exchange energy, when no external stabilization is present. In nanodiscs, the stabilizing factor is the geometric confinement. Several experimental works incorporate an additional external magnetic field perpendicular to the surface to amplify the uniaxial anisotropy in an attempt to block the blow-up effect. As we are mainly interested in the generation of topological defects, we focus on this aspect keeping in mind that a stabilization of the induced skyrmions is necessary to maintain their localized structure. As can be seen in Fig. 4, the irradiation by the electron beam leads to the injection of a topological charge. The used pulse has the peak intensity at $5σt=2.5$ps to account for its finite rising time. After short oscillations, the system relaxes to a skyrmionic state. On the time scale of the electron pulse, the state's topological nature is perfectly stable, but the excitations tend to expand when no further stabilizations are present. Therefore, the diameter of the skyrmion increases, the type changes from Néel to Bloch, and the topological charge is still conserved. The situation changes markedly for longer pulses (⁠$σt≳20$ps). Just like before the pulse leads to topological excitations. On top of this, a domain-wall ring similar to the results presented above is induced, which shields the included skyrmions in two different ways (see Fig. 5). The blow-up behavior is blocked due to the domain wall. The spin waves reflected from the open boundaries are absorbed by the domain ring so that the inside is not affected. In the supplementary animation “(Multimedia view),” the formation of the structure shown in Fig. 5 can be seen. In the first 220 ps, the time step between the single frames is 4.4 ps. Then, the movie is accelerated to 22 ps/frame which can be easily recognized by the arising spin-waves outside the domain ring. Applying another pulse, the ring structure can be opened and the topological charge changed with the same process.

Summarizing, we demonstrated the usefulness of ultrashort electron pulses for generating and steering the magnetization dynamics due to the electromagnetic fields associated with the electron pulses. In particular, topologically protected magnetic textures such as skyrmions can be imprinted and manipulated in a controllable spatiotemporal way.

See supplementary material for the information on the analytical macrospin approach towards the results shown in Fig. 1, further details on the numerical calculations, and the magnetic fields induced by ultrafast electron pulses.

A.F.S. and J.B. are supported by the German Research Foundation (No. SFB 762) and the Priority Programme 1840. H.A.D. acknowledges support by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-76SF00515.