Polarisation-dependent single-pulse ultrafast optical switching of an elementary ferromagnet

The ultimate control of magnetic states of matter at femtosecond (or even faster) timescales defines one of the most pursued paradigm shifts for future information technology. In this context, ultrafast laser pulses developed into extremely valuable stimuli for the all-optical magnetisation reversal in ferrimagnetic and ferromagnetic alloys and multilayers, while this remains elusive in elementary ferromagnets. Here we demonstrate that a single laser pulse with sub-picosecond duration can lead to the reversal of the magnetisation of bulk nickel, in tandem with the expected demagnetisation. As revealed by realistic time-dependent electronic structure simulations, the central mechanism is ultrafast light-induced torques acting on the magnetisation, which are only effective if the laser pulse is circularly polarised on a plane that contains the initial orientation of the magnetisation. We map the laser pulse parameter space enabling the magnetisation switching and unveil rich intra-atomic orbital-dependent magnetisation dynamics featuring transient inter-orbital non-collinear states. Our findings open further perspectives for the efficient implementation of optically-based spintronic devices.


Introduction
The manipulation and control of magnetic materials by ultrashort laser pulses has been extensively researched since the discovery of optically-driven ultrafast demagnetisation in nickel 1 . The technological potential of this discovery was quickly recognised, leading to proof-of-concept experiments in connection with information storage [2][3][4] . Such laser-driven magnetisation dynamics has also been explored in bulk rare-earth ferromagnets 5 , in ferrimagnets 2, 6-13 , and in ferromagnetic thin films 3,[14][15][16][17][18][19][20] . The underlying physical picture is not yet fully understood, given the diversity of mechanisms that can contribute to or influence the dynamics on distinct or overlapping time scales [21][22][23] .
For applications, the goal is not simply to change or demagnetise the material but to controllably reverse the magnetisation direction, which encodes an information bit. This has been successfully achieved in GdFeCo thanks to its ferrimagnetism 2,9 . For this material, the magnetisation switching is due to a laser-driven heating above the ferrimagnetic compensation point 9 , together with the different relaxation time scales of the two rare-earth and transition metal sublattices 6 , and only weakly depends on the polarisation of the laser 8 . Magnetisation switching has also been demonstrated for ferromagnetic Co/Pt multilayers, where the helicity of the laser is an important factor to achieve deterministic switching. While GdFeCo can be switched with a single pulse, for Co/Pt several long pulses 19 or hundreds of short pulses 24 are needed to achieve full switchingwhich is detrimental for technological applications due to the high energy consumption and the relative slowness of the whole process. It has been recently reported that only two laser pulses are enough to achieve complete helicity-dependent switching in Co/Pt 20 .
In order to make progress and to understand how to achieve full switching in a ferromagnet with a single laser pulse, the appropriate physical mechanism or combination of physical mechanisms have to be identified and simulated. The three-temperature model 1,22,25,26 describes the nonequilibrium thermodynamics of coupled electronic, magnetic and lattice subsystems, and provides a very good semi-phenomenological description of the demagnetisation of bulk Ni and of the switching in GdFeCo. Demagnetisation and switching due to the stochastic magnetisation dynamics driven by laser heating of the material were studied numerically using Landau-Lifshitz-Gilbert and Landau-Lifshitz-Bloch equations 6,9,27,28 . There are also several proposed microscopic pictures for how the electrons react to the laser and lead to ultrafast demagnetisation. The inverse Faraday effect was proposed as a direct mechanism for laser-induced demagnetisation 29 , which evolved into a more general picture of light-induced magnetic torques [30][31][32][33][34][35] . The superdiffusive spin transport model 36 introduces spin-polarised hot electrons that transfer angular momentum from the magnetic atoms to a non-magnetic material. Mechanisms for demagnetisation due to transfer of angular momentum from the spins to the lattice have also been extensively studied 5,[37][38][39][40] . Simulations considering electron-electron interactions 41 identified a three-step mechanism: the laser pulse creates electron-hole excitations, spin-orbit coupling converts the excited spin to orbital angular momentum, and the latter is then quickly quenched by the lattice. Lastly, time-dependent density functional theory simulations of the combined dynamics of the electrons and the magnetic moments that they form have yielded many microscopic insights into the ultrafast demagnetization in the sub-100 fs regime 17,18,39,[42][43][44][45] .
So far all the simulations based on a realistic description of the electronic structure were limited to ultrafast demagnetisation processes. Here we address all-optical magnetisation reversal and the possibility of inducing it with a single laser pulse in an elementary ferromagnet such as fcc bulk Ni. We employ a recently developed time-dependent tight-binding framework parameterized from DFT calculations, with a specific algorithm enabling to monitor the non-linear magnetisation dynamics up to a few picoseconds. We show that a single laser pulse can trigger the magnetisation reversal of Ni, for which we identified the pulse parameter space enabling magnetisation switching summarized in Fig. 1. We identify ultrafast light-induced torques as the underlying mechanism, which act on the magnetisation if the polarisation of the pulse obeys specific conditions. We found strong non-collinear, ferromagnetic and antiferromagnetic intra-atomic transient states that are shaped by the interplay of optical inter-orbital electronic transitions and spin-orbit induced spin-flip processes.

Results
All-optical magnetisation reversal We perform tight-binding simulations parameterized from Our method enables us to use larger pulse widths to investigate the effect of both linearly and circularly polarised pulses over a long time scale (see Methods section for more details).
In the ground state of bulk Ni, the spin moment is found to be 0.51µ B and prefers to point along the cubic axes. Here, we assume it to point along the [001] direction, which we choose to be the z cartesian axis. We then systematically apply single optical pulses while tracking the time-dependent magnetisation dynamics of the system. The pulses have a fixed frequency ω = 1.55 eVh −1 and varying widths and intensities of the electric field E 0 , and we consider both linear and circular polarised light (see schematic Fig. 1a and Methods section for more details). Up to 70 fs, we recover the usual demagnetisation pattern characterizing Ni. A reduction of about 30% is found when the absorbed laser fluence reaches 0.41 mJ cm −2 at the end of the applied pulse. This defines an initial demagnetisation region, which can be followed by a slight "remagnetisation" (i.e., an increase in the z-magnetisation), before entering another regime on longer time scales that, in this case, continues to demagnetise Ni up to 5 ps, as illustrated in Fig. 2b. On these longer time scales one can also identify mild oscillations in the magnetisation while it continues decreasing until it reaches about 25% of the ground state moment at 5 ps.
To characterize the effects of the laser field within the different regimes, we start by applying  Time (ps) initial demagnetisation regime, increasing the laser intensity leads to a stronger reduction of the spin moment, while an oscillatory behavior emerges for larger times after a clear threshold around E 0 = 6.5E * − 6.8E * . These oscillations decrease in amplitude when the intensity is further increased, and the largest responses are then limited to a relatively small range of E 0 values.
By increasing the laser width to 100 fs, as shown in Fig. 2d, the laser pulse switches the sign of the z-component of the magnetisation. We also see that increasing the magnitude of the laser electric field leads to a stronger initial demagnetisation regime and also stronger oscillation amplitudes with longer periods, the latter two becoming smaller again for E 0 = 11E * . For smaller laser field intensities, the associated switching point (M z = 0) remarkably moves to earlier times. However, the oscillatory behavior and magnetisation reversal identified in panels c and d of Fig. 2 were only found for simulations where the laser electric field is circularly polarised and rotates in a plane that contains the initial orientation of the magnetic moment. Therefore, unless explicitly mentioned, in the following discussions we focus on the results obtained with pulses polarised in the yz-plane.
By extending the parameter space with systematic simulations of various pulses, we map all the switching and no-switching cases into the phase diagram shown in Fig. 1b, where the horizontal and vertical axes represent the pulse width and the absorbed laser fluence, respectively. The shaded green region illustrates the switching region, where the probability of spin reversal is high. One can see from the diagram that there exists a critical minimum width for switching, which for fcc bulk Ni is about 60 fs. Once the pulse width is larger than that value, we find that there is a laser fluence window for the switching to occur where the lower-bound slightly decreases and the upper-bound increases as the pulse gets wider. The requirement of a minimum threshold for switching is a reasonable and expected condition; however, a critical pulse energy to induce magnetisation reversal is surprising taking into consideration the demagnetization and precession induced by the laser (and illustrated in Fig. 2c,d). In this sense, one would naively expect that higher energies would further excite the system leading to more spin moment loss and larger precession amplitude.
A narrow window in the same laser parameter space was experimentally identified for Co/Pt 19 . An interesting feature in the obtained demagnetisation curves is that the switching may occur while the pulse is still effective (see Supplemental Figure 2), when the system is pumped with more energetic pulses of widths larger than 100 fs.
Laser-induced torque The dynamics of the magnetisation in each scenario is actually more complicated than its z-component in Fig. 2 can display-the transverse components also change dramatically depending on the laser characteristics. In Fig. 3, we show the 3-dimensional (3D) magnetisation response to a 100 fs laser pulse with two different intensities. The lower intensity case in The identified magnetisation precession results from a torque induced by the laser. As we The effective field that generates the torque has two distinct contributions: B IFE is due to an inverse Faraday-like effect (IFE) and acts only during the laser pulse, while B MAE is due to the timedependent magnetic anisotropy of the non-equilibrium electronic system and so also acts after the laser pulse is over. The last term is a longitudinal contribution that represents the demagnetisation driven by the laser with rate χ L and is also contained in T SOC .
We now consider a 300 fs laser pulse, in order to enhance the torque contribution which is driven directly by the laser. B IFE is expected to point perpendicular to the polarisation plane of the circular laser pulse, i.e. along the x-direction, given that the electric field rotates in the yz-plane as shown in Fig. 4a. The resulting torque points along y and enforces the observed rotation within the yz-plane. Inverting the polarisation of the laser pulse changes the direction of the torque and so also the sense of rotation of the magnetisation. This can be identified in plane of the circular polarisation. After the laser pulse is over, the second torque shown in Eq. (1) kicks in and rotates the moment out of the yz-plane (sketched in Fig. 4f) at a time scale of the order of picoseconds, settled by the non-equilibrium magnetic anisotropy energy. The magnetisation dynamics on this longer time scale is shown in Fig. 4g, where we can also identify oscillations in the magnitude of the magnetisation. These hint at internal dynamics that we now discuss.
Orbital-dependent magnetisation dynamics. Our method enables us to study not only the timedependent amplitude of the total magnetisation but also the internal dynamic contributions from different electronic states, as we now discuss for the same circular right-handed pulse of 300 fs We can identify three main regions with distinct dynamical behavior while the laser is acting on the material. In the first region, up to about 60 fs, there is a small reduction of the magnetisation of the d states, while the one contributed by the sp states falls to zero at the end of this region. The dynamics of all the orbitals belonging to the respective s, p and d groups closely follow each other, with the d yz orbital starting to split from the other d orbitals (Fig. 5b), while their occupations change little (Fig. 5c). As is well known, the sp and d orbitals are antiferromagnetically coupled to each other in the ground state, but surprisingly the transition to the next dynamical region is accompanied by a strong noncollinearity of M sp and M d (Fig. 5a). In the second dynamical region, from about 60 to 165 fs, there is a strong collapse of the magnetisation of the d states accompanied by a strong increase of the magnetisation of the sp states (Fig. 5a). The angle between the two magnetisation vectors varies in a complex way and their coupling becomes ferromagnetic-like, with an accompanying rotational motion of the total magnetisation vector in the yz-plane. The switching of the z component of the magnetisation occurs due to the large transfer of spin angular momentum from the d to the sp states (Fig. 5a), which is now also accompanied by a large transfer of orbital population (Fig. 5c). At the end of this region, the d magnetisation is minimal and is in the process of rotating from being parallel to being antiparallel to the larger magnetisation now displayed by the sp states. In the third dynamical region, from about 165 fs to essentially the end of the laser pulse, the d magnetisation partly recovers and assumes an almost antiparallel alignment to the sp one (Fig. 5a). The orbital occupations stabilize (Fig. 5c), but the d orbitals develop internal oscillations with a short period of tens of femtoseconds, which continue after the laser is over (Fig. 5b).
The previous observations lead us to propose the following physical picture for the different dynamical processes actively driven by the laser pulse, as illustrated in Fig. 5d-f. In the first demagnetisation region up to 60 fs, intra-orbital spin-flip processes, i.e. within each orbital channel (d-d), (sp-sp) are responsible for the initial reduction of the magnetisation until the spin moment of the sp-electrons is fully quenched (Fig. 5d). Both the mechanism and the time scale are explained by spin-orbit coupling. In the second region starting after 60 fs, inter-orbital optical transitions become important while maintaining strong intra-atomic noncollinearity with the sp and d moments entering a transient ferromagnetic coupling and reaching similar magnitudes (Fig. 5e). Here the time scale is set by effective inter-orbital exchange interactions. The nature of the effective interorbital exchange coupling changes due to the orbital repopulation. After 165 fs, a new equilibrium between the occupations of the orbitals is reached, which recovers the initial inter-orbital antiferromagnetic coupling, enforcing the weaker moment (which is now the d-magnetisation) to point in the opposite direction to the stronger one originating from the sp states (Fig. 5g). There is also a significant remagnetisation of the d orbitals which could be driven by the coupling to the larger magnetisation of the sp orbitals and is assisted by the laser (Fig. 5a). To summarize, the demagnetisation rate of both types of electrons is not the same since the strength of the matrix elements responsible for the spin-orbit driven spin-flip processes is orbital dependent. When both families of orbitals are demagnetized, the strong population switching of the electronic states in favor of the sp-type forces the d-electrons to have their moments growing in the direction opposite than that of the sp-electrons when their natural inter-orbital antiferromagnetic coupling is restored.

Discussion
In conclusion, we predict via time-dependent electronic structure simulations that the so far elusive magnetisation switching in an elementary ferromagnet such as bulk fcc Ni is possible with a single laser pulse. We mapped the laser-pulse parameter regime enabling the reversal of the magnetisation and found that a minimum pulse width of 60 fs is required, while increasing the pulse width widens the laser fluence range allowing all-optical manipulation of the direction of the magnetisation. The

Methods
Theory We utilize a multi-orbital tight-binding Hamiltonian that takes into account the electronelectron interaction through a Hubbard like term and the spin-orbit interaction, as implemented on the TITAN code to investigate dynamics of transport and angular momentum properties in nanostructures [46][47][48][49] . To describe the interaction of a laser pulse with the system, we include a timedependent electric field described by a vector potential A(t) = − E(t)dt. The full Hamiltonian is given by More details on each term can be found in Supplementary Note 1. The dipole approximation was used in the implementation of the vector potential, meaning that the spatial dependency is not included since the wavelength of the used light (hω = 1.55 eV → λ = 800 nm) is much larger than the lattice constant, and that the quadratic term as well as the other higher terms are zero 50 .
We approximate the absorbed laser fluence by the change in the energy of the system divided by its cross section, where a is the lattice constant. This approaches a stable value after the end of the laser pulse.
Pulse shape For the right-handed circular pulse (σ + ) polarised in the yz-plane, for example, the pulse shape is described using a vector potential of the following form 51 , where E 0 is the electric field intensity, τ is the pulse width, ω is the laser central frequency which is set to 1.55 eVh −1 . The magnetic field of the laser is neglected since it is much smaller than the electric field.
For the linear pulse (π) of a propagation direction along the diretionû, using the same central frequency, the vector potential is described as Computational details Calculations were performed on bulk face centered cubic Nickel using the theoretical lattice constant of 3.46 Å given in Ref. 52, one atom in the unit cell for all calculations, a uniform k-point grid of (22 × 22 × 22) and a temperature of 496 K in the Fermi-Dirac distribution.
The initial step size for the time propagation is ∆t = 1 a.u. which changes in the subsequent steps to a new predicted value such that a relative and an absolute error in the calculated wave functions stay smaller than 10 −3 53 .
We tested the results for accuracy by increasing the number of k-points and decreasing the tolerance for the relative and absolute errors. The method was also tested for stability by changing one of the laser parameters by a very small number while keeping the other parameters fixed, for one case that we already have results for. The results then were not very different 54 .
Code availability The tight-binding code that supports the findings of this study, TITAN, is available from the corresponding author on request.
Data availability The data that support the findings of this study are available from the corre-sponding authors on request.