Abstract
1 Introduction
The pioneering work on ultrafast demagnetization in Ni[1] paved the way towards a large number of theoretical and experimental studies on magnetization dynamics at femtosecond (fs) timescales induced by ultrashort laser pulses[2–25]. In these studies the dynamics is mediated primarily by the electric field (E-field), which can excite non-equilibrium states[5–9], demagnetize the sample[1,14–21], generate localized charge currents[24,25] or induce the inverse Faraday effect[22,23]. While most of the techniques are mediated mainly by the E-field, other techniques, such as the excitation of phononic modes[26], have recently provided routes for non-thermal magnetization manipulation.
An appealing alternative to induce coherent magnetization dynamics consists of the use of magnetic fields (B-field). The role of the B-field in ultrafast magnetization dynamics has been extensively studied, especially in the regime of linear response to THz fields[27–32]. At this picosecond timescale, few teslas (T) are required to introduce small deflections from the equilibrium magnetization direction, while tens of teslas are needed for achieving complete switching. Higher driving frequencies, which could break into the femtosecond timescale, would require very high B-field amplitudes. Although intense magnetic fields can be achieved, for example, using plasmonic antennas[33], in such a regime, the associated E-field would potentially demagnetize the sample[34] or even damage it. In addition, although substantial advances have been made towards the generation of electromagnetic fields in the range of THz (0.1–30 THz), their intensity is still small as compared with the infrared case[35–37].
In this work we introduce an appealing alternative to drive magnetization dynamics at the sub-picosecond timescale, by using isolated ultrafast intense B-fields. Recent developments in structured laser sources have demonstrated the possibility to spatially decouple the B-field from the E-field of an ultrafast laser pulse. For instance, azimuthally polarized laser beams present a longitudinal B-field at the beam axis, where the E-field is zero[38]. Depending on the laser beam parameters, the contrast between the longitudinal B-field and the radial B-field and the E-field can be adjusted, so as to design a local region in which the longitudinal B-field can be considered to be isolated from both the radial B-field and E-field[39]. In such a region, the stochastic processes driven by the E-field could be avoided, and the coherent precession induced by the B-field can be exploited. Besides, only the longitudinal component of the B-field is present and, consequently, the magnetic field is linearly polarized. Indeed, azimuthally polarized laser beams have been shown to induce isolated millitesla static B-fields[40], with applications in nanoscale magnetic excitations and photoinduced force microscopy[41,42]. More recently, ultrafast time-resolved magnetic circular dichroism has been proposed[43]. In addition, theoretical proposals[39,44] and experiments[45,46] have raised the possibility to generate isolated tesla-scale fs magnetic fields by the induction of large oscillating currents through azimuthally polarized fs laser beams.
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Our theoretical study unveils the nonlinear, chiral, precessional magnetization response of a standard ferromagnet to a tesla-scale circularly polarized ultrafast magnetic field whose polarization plane contains the initial equilibrium magnetization. First, we show in Section 2 the feasibility to use state-of-the-art structured laser beams to create a macroscopic region in which such B-fields are found to be isolated from the E-field by particle-in-cell (PIC) simulations. It is worth mentioning that this circularly polarized B-field is a non-propagating solution that should not be confused with circularly polarized structured laser beams, such as those considered in Ref. [47]. Then, we present our micromagnetic (μMag) simulations for moderate fields in Section 3, showing the presence of measurable magnetization dynamics in CoFeB when a circularly polarized
2 Spatially isolated circularly polarized B-fields out of structured laser beams
In order to study the interaction of an isolated circularly polarized B-field with a standard ferromagnet (CoFeB), we consider a B-field,
Figure 1.(a) Sketch of the system under consideration. A circularly polarized magnetic field illuminates a magnetic sample whose dimensions are smaller than the region for which the E-field can be considered negligible. This field can trigger ultrafast magnetization dynamics. (b) Two crossed azimuthally polarized beams of 30 THz and peak intensity 2.1 W/cm2 define a spatial region of radius nm in which the E-field is lower than 100 MV/m, as depicted in the panel. In such a region, a constant circularly polarized B-field of amplitude 10.5 T and central frequency 30 THz is found.
In our simulations, we do not include any E-field coupling, as the B-field is assumed to be isolated. Such an assumption is valid for CoFeB in spatial regions where the E-field is lower than
In Figure 1(b) we also show the spatial distribution of the B-field (color background) and the E-field (contour lines) at the overlapping region. We have highlighted the region in which the E-field is lower than
3 Nonlinear magnetization response to ultrafast B-fields
The interaction between the oscillating B-field and the magnetization is given by the Landau–Lifshitz–Gilbert (LLG) equation[29,52]:
where
In Figure 2(a) we show the in-plane magnetization dynamics (perpendicular to the equilibrium configuration,
Figure 2.Micromagnetic simulation results of the temporal evolution (color code) of the magnetization components (, ) of CoFeB excited by B-fields with different polarization states. (a) RCP (yellowish color scale) and LCP (greenish color scale) B-fields (, THz, ps). The RCP (LCP) B-field induces a measurable negative (positive) component. In both cases the anisotropy field induces a precession of around the equilibrium configuration. The bottom part sketches the mechanism during a B-field period of constant amplitude. The B-field (red), magnetization (black) and torque (green) vector representations at four different times reveal the magnetization dynamics mechanism over one period. (b) Linear polarization along
The nonlinear mechanism underlying such behavior can be understood as follows (see the bottom part of Figure 2(a)). At an initial time
To highlight the importance of the polarization state and orientation to get the nonlinear response, Figures 2(b) and 2(c) depict the temporal evolution of the magnetization components (
4 Analytical model
To give insight into the nonlinear mechanism introduced in the previous section and sketched in Figure 2(a), we derive an approximated analytical model. The exchange field is not included in the model because we assume that the sample remains uniformly magnetized. In addition, we neglect the anisotropy and DMI fields – which are small if compared with the external one – and the damping term. Similar assumptions have been proven reasonable at this timescale in previous studies[29]. With these approximations in Equation (3), the magnetization dynamics out of the polarization plane reads as follows:
Assuming that the magnetization components in the polarization plane,
We now analyze the dependency of the magnetization dynamics on the B-field, both with the analytical model represented by Equation (8) and the full micromagnetic simulations, where all the interactions on the effective field, as well as the damping, are included. To highlight the accuracy of our model based on the equivalent drift field, we compare the total rotation of the magnetization from our simulations with the magnetization rotation induced by the drift B-field,
Figure 3 presents the induced magnetization rotation as derived from the analytical model (solid lines) and the micromagnetic simulations (dots). The excellent agreement allows us to validate our model and demonstrate the reported nonlinear chiral effect. Firstly, Figure 3(a) shows the total rotation of the magnetization as a function of the polarization state (characterized by
Figure 3.Analysis of the nonlinear effect dependencies. Total magnetization rotation as a function of (a) the polarization state of the B-field (characterized by , and using ) and (b) the inverse of the frequency of a circularly polarized B-field. In both (a) and (b), three different B-field amplitudes (60 T blue, 100 T red and 140 T black) oscillating at are represented. (c) Total magnetization rotation as a function of the circularly polarized B-field amplitude, with three different central frequencies ( blue, red and black). In (a), (b) and (c), the B-field pulse duration is . (d) Total magnetization rotation as a function of the circularly polarized B-field pulse duration, , with three different B-field amplitudes ( blue, red and black) and a central frequency of . In all panels, symbols indicate results from micromagnetic simulations while lines correspond to Equation (10).
Figure 3(b) depicts the inverse dependency of the magnetization rotation with the B-field frequency. This frequency scaling suggests that the nonlinear induced rotation is particularly relevant for external B-fields at THz frequencies. However, note that the linear dynamics (with the external field) would also contribute at those frequencies. Figure 3(c) shows the second-order scaling of the magnetization dynamics with the external B-field amplitude for central frequencies of 250 THz (blue), 100 THz (red) and 50 THz (black). As expected, the total rotation increases with the B-field amplitude, being already measurable at tens of teslas. Finally, Figure 3(d) depicts the total rotation of the magnetization for a B-field pulse of frequency
One of the most appealing opportunities of this nonlinear effect is the possibility to achieve non-thermal ultrafast all-optical switching driven solely by an external circularly polarized B-field. Based on the dependencies presented in Figure 3, we show in Figure 4 two different micromagnetic simulation results in which switching is achieved through the use of an RCP B-field pulse. The B-field envelopes of each case are represented with dashed red lines, whereas the magnetization components
Figure 4.Micromagnetic simulation results of the temporal evolution of the magnetization components ( blue, yellow, black) of CoFeB excited by an RCP B-field. The normalized B-field envelope is shown in dashed red. While a B-field of , THz and ps shows switching at the ps timescale, a B-field of , THz and ps achieves it at the femtosecond timescale.
5 Discussion
Our results unveil a nonlinear chiral magnetic effect driven by ultrafast circularly (or elliptically) polarized B-field pulses, lying in the plane containing the initial magnetization. This purely precessional effect is quadratic in the external B-field, and proportional to the inverse of the frequency, being equivalent to a drift field that depends linearly on the gyromagnetic ratio. This nonlinearity is proved to be essential at this timescale, since a linear response would follow adiabatically the magnetic field and, consequently, would restore the magnetization to its initial state after the pulse is gone. Conversely, the reported drift field plays a significant role in the magnetization dynamics driven by moderately intense circularly-polarized B-fields – tens of teslas at the ps timescale, and hundreds of teslas at the fs timescale. Although we have studied the magnetization dynamics in CoFeB, this effect is a general feature of the LLG equation, thus being present in all ferromagnets, but also in ferrimagnets and antiferromagnets. In addition, this rectification effect may be exploited to generate THz electric currents via the inverse spin Hall effect, which would emit electromagnetic THz radiation[37] when illuminated with infrared light.
In addition, it should be stressed that, even when the E-field is non-negligible, the reported nonlinear mechanism on the B-field may play a role, so a complete study of the ultrafast magnetization dynamics would require taking into account this effect. We note that recent works pointed out the need to include nutation in the dynamical equation of the magnetization[52,56,57]. This term could also lead to second-order effects. Thus, our work serves as a first step towards the investigation of higher-order phenomena induced by magnetic inertia, potentially leading to even shorter timescale magnetization switching.
Finally, our work demonstrates that the recently developed scenario of spatially isolated fs B-fields[39,44–46] opens the path to the ultrafast manipulation of magnetization dynamics by purely precessional effects, avoiding thermal effects due to the E-field or magnetization damping. Although the spatial decoupling of the intense B-field from the E-field using fs structured pulses is technologically challenging, it is granted by the rapid development of intense ultrafast laser sources, from the infrared (800 nm, 375 THz) to the mid-infrared (4–40 μm, 75–77 THz)[58–60]. Thinking forward, we believe that our work paves the way towards induced all-optical magnetization dynamics at even shorter timescales, towards the sub-femtosecond regime. Recent works in the generation of ultrafast structured pulses in high-order harmonic generation[61–63] may open the route towards such ultrafast control.
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