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, $\mathbf{B}$ , oscillating in the $xz$ plane (see Figure 1(a)) given by the following:

(1) $$\begin{align}\mathbf{B}(t)=\mathbf{b}(t){e}^{i\omega t}+{\mathbf{b}}^{\ast }(t){e}^{- i\omega t},\end{align}$$

(2) $$\begin{align}\mathbf{b}(t)=\frac{B_0}{2}F(t)\left(\cos {\theta}_0{\widehat{\textbf{u}}}_x+\sin {\theta}_0{e}^{i{\phi}_0}{\widehat{\textbf{u}}}_z\right),\end{align}$$

where $\omega$ is the central angular frequency ( $\omega =2\pi f$ ), ${B}_0$ is the amplitude and ${\theta}_0$ and ${\phi}_0$ define the relative amplitude and phase between the $x$ and $z$ components, respectively. Here, $F(t)$ is the field envelope, given by $F(t)={\sin}^2\left(\pi t/{T}_{\mathrm{p}}\right)$ for $0\le t\le {T}_{\mathrm{p}}$ , with ${T}_{\mathrm{p}}=3{t}_{\mathrm{p}}/8$ being its full duration and ${t}_{\mathrm{p}}$ being the full-width-at-half-maximum (FWHM) pulse duration in intensity. A right-handed, RCP (left-handed, LCP) circularly polarized B-field in the $xz$ plane corresponds to ${\phi}_0=\pi /2$ ( ${\phi}_0=-\pi /2$ ) and ${\theta}_0=\pi /4$ , while a linearly polarized B-field corresponds to ${\phi}_0=0$ or $\pi$ .

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 $\times {10}^{13}$ W/cm² define a spatial region of radius $\simeq 100$ 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 $100\;\mathrm{MV}/\mathrm{m}$ , for which the demagnetization has been predicted to be less than $7\%$ ^[ Reference Bonetti, Hoffmann, Sher, Chen, Yang, Samant, Parkin and Dürr^14, Reference Hudl, d’Aquino, Pancaldi, Yang, Samant, Parkin, Dürr, Serpico, Hoffmann and Bonetti^48]. The conditions for which an intense circularly polarized B-field can be found spatially isolated from the E-field can be obtained by using two crossed azimuthally polarized laser beams, as sketched in Figure 1(b). We have performed PIC simulations using the OSIRIS 3D PIC code^[ Reference Fonseca, Silva, Tsung, Decyk, Lu, Ren, Mori, Deng, Lee, Katsouleas and Adam^49– Reference Fonseca, Vieira, Fiuza, Davidson, Tsung, Mori and Silva^51], in order to show how such isolated B-fields can be achieved with the state-of-the-art ultrafast laser technology. We have considered two orthogonal azimuthally polarized laser beams with waist ${w}_0=3.125\lambda =31.25$ μm, central wavelength of $\lambda =10$  μm (30 THz) and E-field amplitude of 12.5 GV/m (peak intensity of 2.1 $\times {10}^{13}$ W/cm²) at their radius of maximum intensity, ${w}_0/\sqrt{2}$ . The temporal envelope is modeled as a sin² function of 88.8 fs FWHM. Due to computational limitations the temporal envelope is much shorter than those considered in the μMag simulations presented in this work, which lies in the ps regime. However, we do not foresee any deviation in the results presented if longer pulses with similar amplitudes are considered.

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 $100\;\mathrm{MV}/\mathrm{m}$ , and thus the E-field can be neglected against the B-field. Thus, we can define a region of radius $\simeq 100$ nm in which the B-field exhibits a constant amplitude of 10.5 T and the E-field is maintained below $100$ MV/m. Although the use of additional currents, like in Refs. [Reference Blanco, Cambronero, Flores-Arias, Jarque, Plaja and Hernández-García39,Reference Sederberg, Kong, Hufnagel, Zhang, Karimi and Corkum44], could enhance the B-field amplitude, our simulations demonstrate that moderately intense laser beams can already reach the B-field amplitudes required to observe the nonlinear magnetization dynamics described below.

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^[ Reference Vicario, Ruchert, Ardana-Lamas, Derlet, Tudu, Luning and Hauri^29, Reference Wegrowe and Ciornei^52]:

(3) $$\begin{align}\left(1+{\alpha}^2\right)\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma \mathbf{m}\times {\mathbf{B}}_{\mathrm{eff}}-\alpha \mathbf{m}\times \left(\mathbf{m}\times {\mathbf{B}}_{\mathrm{eff}}\right),\end{align}$$

where $\mathbf{m}$ is the normalized magnetization where both spatial and temporal dependencies are implicitly assumed, $\alpha$ is the Gilbert damping parameter and ${\mathbf{B}}_{\mathrm{eff}}$ is the effective magnetic field. We have performed μMag simulations using the well-known software MuMax^3[ Reference Vansteenkiste, Leliaert, Dvornik, Helsen, Garcia-Sanchez and Van Waeyenberge^53] to solve the LLG equation. The system under study is sketched in Figure 1(a), where we consider a circular nanodot with $1\;\mathrm{nm}$ thickness and $64\;\mathrm{nm}$ diameter discretized into $1\;\mathrm{nm}$ cubic cells. The material parameters correspond to CoFeB grown over a heavy metal layer: inhomogeneous exchange parameter $A=19\;\mathrm{pJ}/\mathrm{m}$ , saturation magnetization ${M}_{\mathrm{S}}=1\;\mathrm{MA}/\mathrm{m}$ , perpendicular uniaxial anisotropy (i.e., the anisotropy field is directed along the $z$ direction) parameter ${K}_{\mathrm{u}}=800\;\mathrm{kJ}/{\mathrm{m}}^3$ , Dzyaloshinskii–Moriya interaction (DMI) $D=1.8\;\mathrm{mJ}/{\mathrm{m}}^2$ and Gilbert damping $\alpha =0.015$ .

In Figure 2(a) we show the in-plane magnetization dynamics (perpendicular to the equilibrium configuration, ${m}_z=1$ ) induced by RCP and LCP B-fields lying in the $xz$ plane. Note that the equilibrium magnetization lies in the polarization plane. In both cases, ${B}_0=10\;\mathrm{T}$ , $f=30\;\mathrm{THz}$ and ${t}_{\mathrm{p}}=10\;\mathrm{ps}$ . We can observe a magnetization precession around the $z$ -axis triggered by a nonlinear chiral response to the B-field. While the RCP B-field induces a measurable negative $x$ component, the LCP leads to a positive one. After the pulse, the precession dynamics is dominated by the anisotropy field, and the system starts to precess around the z-axis. Note that the broad trace is due to the subsequent magnetization oscillations during the interaction with the pulse.

Figure 2 Micromagnetic simulation results of the temporal evolution (color code) of the magnetization components ( ${m}_x$ , ${m}_y$ ) of CoFeB excited by B-fields with different polarization states. (a) RCP (yellowish color scale) and LCP (greenish color scale) B-fields ( ${B}_0=10\;\mathrm{T}$ , $f=30$ THz, ${t}_{\mathrm{p}}=10$ ps). The RCP (LCP) B-field induces a measurable negative (positive) ${m}_x$ component. In both cases the anisotropy field induces a precession of $\mathbf{m}$ 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 x (yellowish trace) or y (greenish trace). (c) Circular polarization perpendicular to the equilibrium magnetization with RCP (yellowish trace) and LCP (greenish trace) helicities.

The nonlinear mechanism underlying such behavior can be understood as follows (see the bottom part of Figure 2(a)). At an initial time $t=0$ , in which $\mathbf{m}$ (black arrow) lies in the polarization plane of the circularly polarized B-field (red arrow), being perpendicular to it, a transverse torque $\tau$ (green arrow) drives $\mathbf{m}$ out-of-plane from this initial position. During the next quarter-period, $\tau$ decreases and rotates, inducing a precession of $\mathbf{m}$ around its initial axis. For the second quarter-period, $\tau$ increases again keeping its rotation but, at $t=T/2$ , it reverses its rotation direction, thus sweeping only half of the plane perpendicular to $\mathbf{m}$ . As a result, along a whole period, the torque component perpendicular to the polarization plane averages to zero, while a residual contribution along the intersection of the polarization plane and the plane perpendicular to $\mathbf{m}$ remains. With long lasting multicycle laser pulses it is then possible to accumulate the small torque along the polar coordinate on the polarization plane, $\theta$ , so as to promote the system to a targeted non-equilibrium state. This is reflected in Figure 2(a), where the magnetization components ${m}_x$ and ${m}_y$ are non-zero at the end of the pulse, and therefore the magnetization is not aligned along the anisotropy direction, $z$ . A more detailed scheme of the nonlinear mechanism is displayed in Supplementary Videos 1 and 2 for both RCP and LCP B-fields, revealing the chiral nature of the reported effect.

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 ( ${m}_x$ , ${m}_y$ ) obtained from full micromagnetic simulations for a linearly polarized B-field, being perpendicular to the equilibrium magnetization, and for a circularly polarized B-field (either RCP or LCP), where the polarization plane is perpendicular to the equilibrium magnetization. Whereas in both cases a small magnetization deflection is observed, the net torque exerted by the field on the magnetization over a period is null, and the magnetization recovers its equilibrium state after the B-field pulse. In addition, at frequencies larger than a few tens of THz, the response is not enough to promote a significant change in the magnetization even for a B-field as high as $B=10\;\mathrm{T}$ . Consequently, in the cases presented in Figures 2(b) and 2(c), the response is completely linear and the magnetization comes back to the initial configuration at the end of the B-field pulse. Nonetheless, for a circularly polarized B-field (either RCP or LCP) with the equilibrium magnetization lying in the polarization plane, the nonlinear chiral phenomenon described above triggers the magnetization out of equilibrium, as shown Figure 2(a). This dragging, being a nonlinear effect, is sensitive to the B-field envelope and does not cancel out at the end of the pulse.

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^[ Reference Vicario, Ruchert, Ardana-Lamas, Derlet, Tudu, Luning and Hauri^29]. With these approximations in Equation (3), the magnetization dynamics out of the polarization plane reads as follows:

(4) $$\begin{align}\frac{\mathrm{d}m_y}{\mathrm{d}t}{\mathbf{u}}_y=-{\gamma}^{\prime }{\mathbf{m}}_{\parallel}\times \mathbf{B},\end{align}$$

with ${\mathbf{m}}_{\parallel }$ being the magnetization in the polarization plane and ${\gamma}^{\prime}=\gamma /\left(1+{\alpha}^2\right)$ . Considering the initial magnetization in the $z$ direction, ${m}_y$ at any time $t$ is given by the following:

(5) $$\begin{align}{m}_y(t){\mathbf{u}}_y=-{\gamma}^{\prime }{\int}_0^t{\mathbf{m}}_{\parallel}\left(\tau \right)\times \mathbf{B}\; \mathrm{d}\tau.\end{align}$$

The Cartesian components of the magnetization can be decomposed at each point in its Fourier components:

(6) $$\begin{align}{m}_j(t)=\sum \limits_q{m}_q^j(t){e}^{iq\omega t},\quad j=x, y, z.\end{align}$$

Using Equations (5) and (6) in the simplified LLG equation, we obtain the following:

(7) $$\begin{align}\begin{array}{l}{\sum}_q\left[\frac{\mathrm{d}{\mathbf{m}}_q^{\parallel }(t)}{\mathrm{d}t}+ iq\omega {\mathbf{m}}_q^{\parallel }(t)\right]{e}^{iq\omega t}=\\ {}+{\gamma^{\prime}}^2\left[{\sum}_q{\int}_0^t{\mathbf{m}}_{q-1}^{\parallel}\left(\tau \right)\times \mathbf{b}\left(\tau \right){e}^{iq\omega \tau} \mathrm{d}\tau \right]\times \mathbf{B}(t)\\ {}+{\gamma^{\prime}}^2\left[\sum \limits_q{\int}_0^t{\mathbf{m}}_{q+1}^{\parallel}\left(\tau \right)\times {\mathbf{b}}^{\ast}\left(\tau \right){e}^{iq\omega \tau} \mathrm{d}\tau \right]\times \mathbf{B}(t).\nonumber\end{array}\\\end{align}$$

Assuming that the magnetization components in the polarization plane, ${\mathbf{m}}_{q\pm 1}^{\parallel }$ , and the B-field envelope, $\mathbf{b}\left({t}\right)$ , evolve slowly, considering $\mathbf{b}(0)=0$ , and selecting only the slowly varying terms ( $q=0$ ), Equation (7) transforms into the following:

(8) $$\begin{align}\frac{\mathrm{d}{\mathbf{m}}_0^{\parallel }(t)}{\mathrm{d}t}=-\frac{2i{\gamma^{\prime}}^2}{\omega }{\mathbf{m}}_0^{\parallel }(t)\times \left[\mathbf{b}(t)\times {\mathbf{b}}^{\ast }(t)\right].\end{align}$$

It is well known that the effective field dependence on the magnetization can lead to nonlinear effects^[ Reference Hudl, d’Aquino, Pancaldi, Yang, Samant, Parkin, Dürr, Serpico, Hoffmann and Bonetti^48, Reference Bertotti, Mayergoyz and Serpico^54, Reference d’Aquino, Serpico, Bertotti, Mayergoyz and Bonin^55]. However, it must be noticed that, differently from those cases, the described effect is nonlinear on the external field, not on the effective field. Moreover, it is proportional to the gyromagnetic ratio and the inverse of the frequency, being equivalent to a drift magnetic field ${\mathbf{B}_{\mathrm{d}}}$ , given by the following:

(9) $$\begin{align}{\mathbf{B}}_{\mathrm{d}}=\frac{\gamma^{\prime }}{2\omega}\sin {\phi}_0\left({\mathbf{B}}_x\times {\mathbf{B}}_z\right).\end{align}$$

Using this definition, Equation (8) describes the slowly varying LLG dynamics in terms of the drift field, ${\mathbf{B}}_{\mathrm{d}}$ . Equations (8) and (9) constitute the main contribution of the present work, as they reveal a second-order dependency of the magnetization dynamics with the external B-field. From Equation (9) we can already infer that ${\mathbf{B}}_{\mathrm{d}}$ is maximal for circular polarization, decreases with the ellipticity and is zero for a linearly polarized B-field ( ${\phi}_0=0$ or $\pi$ ). Note also the chiral nature of the presented mechanism, as the direction of ${\mathbf{B}}_{\mathrm{d}}$ is helicity dependent. Finally, we stress the purely precessional nature of ${\mathbf{B}}_{\mathrm{d}}$ – being linear with the gyromagnetic ratio – and its inverse proportionality with the driving frequency. It is worth noting that a small misalignment of the azimuthally polarized laser beams would convert the circularly polarized magnetic field into an elliptically polarized magnetic field, and/or would introduce a small angle between the initial magnetization and the polarization plane. Nonetheless, it is possible to decompose the total magnetic field into a circularly polarized magnetic field in the $xz$ plane and a linearly polarized magnetic field in the $y$ direction. However, this latter component would not affect the slow dynamics presented here.

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, ${\mathbf{B}}_{\mathrm{d}}$ , which can be computed as follows:

(10) $$\begin{align}\Delta \theta ={\gamma}^{\prime}\left[\frac{\gamma^{\prime }}{2\omega}\sin {\phi}_0\left({B}_x{B}_z\right)\right]{t}_{\mathrm{p}}.\end{align}$$

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 ${\phi}_0$ ) of an external B-field of ${t}_{\mathrm{p}}=3$ ps, for amplitudes of 60 T (blue), 100 T (red) and 140 T (black). Our simulations confirm no rotation for a linearly polarized B-field, and a maximum rotation for circular polarization. The chiral character of the phenomenon is also evidenced.

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 ${\phi}_0$ , and using ${\theta}_0=\pi /4$ ) 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 $f=50\;\mathrm{THz}$ are represented. (c) Total magnetization rotation as a function of the circularly polarized B-field amplitude, with three different central frequencies ( $f=50\;\mathrm{THz}$ blue, $f=100\;\mathrm{THz}$ red and $f=250\;\mathrm{THz}$ black). In (a), (b) and (c), the B-field pulse duration is ${t}_{\mathrm{p}}=3\;\mathrm{ps}$ . (d) Total magnetization rotation as a function of the circularly polarized B-field pulse duration, ${t}_{\mathrm{p}}$ , with three different B-field amplitudes ( $60\;\mathrm{T}$ blue, $100\;\mathrm{T}$ red and $140\;\mathrm{T}$ black) and a central frequency of $f=50\;\mathrm{THz}$ . 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 $50\;\mathrm{THz}$ as a function of the pulse duration, ${t}_{\mathrm{p}}$ . This latter result confirms that the nonlinear chiral effect presented in this work is cumulative in time, as predicted from Equation (10).

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 ${m}_x$ , ${m}_y$ , ${m}_z$ are represented with blue, yellow and black, respectively. The first case makes use of a short, 1 ps, 60 THz, 275 T B-field pulse, whereas the second case uses a 10 ps B-field pulse of 60 T and 30 THz. In both cases the ${m}_z$ component reverses its direction along the course of the pulse, showing that complete switching at the fs or ps timescale can be achieved, depending on the strength, pulse duration and frequency of the B-field.

Figure 4 Micromagnetic simulation results of the temporal evolution of the magnetization components ( ${m}_x$ blue, ${m}_y$ yellow, ${m}_z$ black) of CoFeB excited by an RCP B-field. The normalized B-field envelope is shown in dashed red. While a B-field of ${B}_0=60\;\mathrm{T}$ , $f=30$ THz and ${t}_{\mathrm{p}}=10$ ps shows switching at the ps timescale, a B-field of ${B}_0=275\;\mathrm{T}$ , $f=60$ THz and ${t}_{\mathrm{p}}=1$  ps achieves it at the femtosecond timescale.