Abstract
Subpicosecond magnetisation manipulation via femtosecond optical pumping has attracted wide attention ever since its original discovery in 1996. However, the spatial evolution of the magnetisation is not yet well understood, in part due to the difficulty in experimentally probing such rapid dynamics. Here, we find evidence of a universal rapid magnetic order recovery in ferrimagnets with perpendicular magnetic anisotropy via nonlinear magnon processes. We identify magnon localisation and coalescence processes, whereby localised magnetic textures nucleate and subsequently interact and grow in accordance with a power law formalism. A hydrodynamic representation of the numerical simulations indicates that the appearance of noncollinear magnetisation via optical pumping establishes exchangemediated spin currents with an equivalent 100% spin polarised charge current density of 10^{7} A cm^{−2}. Such large spin currents precipitate rapid recovery of magnetic order after optical pumping. The magnon processes discussed here provide new insights for the stabilization of desired metastable states.
Introduction
Spin dynamics upon femtosecond optical pumping^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} have been intensely studied during the last two decades both because of potential applications for information storage and the need to understand the fundamental physics involved^{16}. A variant of these dynamics is alloptical switching (AOS). While originally demonstrated for ferrimagnetic alloys with perpendicular magnetic anisotropy (PMA)^{2}, AOS has been reported to occur in ferromagnetic PMA materials subject to optical pumping^{9,10,11,12} or by use of ultrafast hot electrons^{14,15,17}. Whereas the picosecond magnetisation dynamics, even for nonuniform states^{5,6}, has been successfully modelled with spatially averaged, quasiequilibrium models^{1,3,4,18}, there is a growing understanding of the important role played by spatiallyvarying magnetisation. For example, the chemical inhomogeneity of amorphous ferrimagnetic GdFeCo alloys results in the picosecond transfer of angular momentum that both drives magnetisation switching^{8} and influences the equilibrium state after pumping with a single laser pulse^{13}. More recently, the effective domain size during cooling has been identified as a criterion to predict whether macroscopic AOS can occur^{12}.
To further investigate the fundamental physics involved in the evolution of spatially varying magnetisation after ultrafast optical pumping, and to elucidate which physical mechanisms are most important for the recovery of local magnetic order at picosecond timescales, we study the space and timedependent magnetisation dynamics in ferrimagnetic Gd_{0.24}Fe_{0.665}Co_{0.095} alloys with timeresolved resonant Xray scattering. We then compare our data with a multiscale model that utilises both atomistic and largescale micromagnetic components to simulate the spatiotemporal evolution of the magnetisation. We identify two distinct dynamic processes, which we term magnon localisation and magnon coalescence. These processes describe the nucleation and subsequent dynamics of localised textures that arise from attractive nonlinear interactions between thermalized magnons^{19}. This is in contrast to theories that predict the order parameter recovery of the spatially averaged magnetisation, as described by the damping of a heated spinwave distribution^{20}.
Magnon localisation is the process by which small, nonequilibrium, localised magnetic textures nucleate and grow. The textures are necessarily longterm unstable transient features that are not to be confused with magnetic domains, which are equilibrium or metastable states. In the context of conservative dynamics, localised textures can be described as dynamical magnon bound states^{21} known as magnon drops^{22}. Magnon localisation can be detected by the appearance of a broad ring centred at low q in the Xray scattering pattern with a rapid radius expansion in reciprocal space and simultaneously decreasing ring width. A subsequent shrinking of the ring radius accompanied by the continual decrease in the ring width indicates a stage of magnon drop growth we term magnon coalescence. Microscopically, magnon coalescence is driven by the ongoing nonlinear attraction between magnon drops and unbound magnons that depletes the thermal magnon population as the magnon drops continue to grow, on average. The substantial, highly turbulent flux of angular momentum in the vicinity of magnon drops during the coalescence stage can be estimated by use of numerical simulations accompanied by a hydrodynamic formulation of magnetisation dynamics that show the presence of strong exchange flow spin currents (EFSCs)^{23,24}, which are equivalent to a 100% spin polarised charge current density on the order of 10^{7} A cm^{−2}. These simulation results suggest that magnon drop dynamics driven by such large EFSCs expedite magnon coalescence via magnon drop growth, breakup, and merger^{25}.
Our study suggests that the picosecond evolution of the spatially varying magnetisation can be understood from a phase kinetics approach^{26,27}. When the magnetisation quenching upon femtosecond optical pumping is almost 100%, the initial condition of the system can be described as a nonequilibrium distribution of randomised spins that undergo rapid restoration of the magnetic order parameter, subject to a multiplicity of final equilibrium (or quasiequilibrium) states. In other words, the subsequent rapid passage from a nearly paramagnetic to a magnetically ordered state will generally do so via pathways of unstable magnondrop growth, i.e., phaseordering kinetics. Such dynamics are in contrast to the critical behaviour expected from an adiabatic evolution through a phase transition^{28}. Because of the large degeneracy of the equilibrium states, unstable growth necessarily leads to pattern formation, examples of which include domains in magnetic materials^{29} and metallic alloys^{26}, phase separation in binary fluids and superfluids^{30}, and optical solitons^{31}. In addition, it has been argued that rapid quenching of a randomised state can dynamically stabilise topological defects via the Kibble–Zurek mechanism^{32,33}, as seen in superfluids^{30,34}, ferroelectrics^{35}, magnetic vortices^{36}, and bubble domain lattices^{37}. Therefore, the magnon processes identified here shed light upon the physical mechanisms that are important in the initial stages of unstable growth and pattern formation triggered by ultrafast optical pumping.
Results
Xray scattering
The evolution of the scattered intensity is measured by timeresolved, resonant magnetic soft Xray scattering, a pumpprobe technique schematically shown in Fig. 1 (see details in Methods). A 0.5 T field is applied perpendicular to the film plane during the measurement. As such, the magnetisation is always reset into the saturated state prior to each pump pulse. The elementspecific, spatiallyaveraged dynamics are simultaneously measured by Xray magnetic circular dichroism (XMCD) of the unscattered beam.
From the scattered intensity measurements, we directly obtain the timevarying spin–spin correlation function, ΔS^{2}(q,t) (see Methods). This quantity provides information related only to the magnetisation’s spatial profile, in contrast to the spin correlations projected onto the sample’s chemical nanostructure studied in ref. ^{8}. The background spin–spin correlation signal prior to timezero is an order of magnitude smaller than the features observed at t > 0.
For illustrative purposes, we show two schematic examples in Fig. 2 as to the expected scattering patterns correlated to representative spatial patterns. A broad peak centred at q = 0 corresponds to a low density of randomly located textures of variable size^{38}, as schematically shown in the top row. However, if these azimuthally disordered textures are sufficiently closepacked so as to have a welldefined averaged spatial separation, they will exhibit a longrange correlation length^{39}, i.e., a ring structure develops, as shown in the bottom row.
We measured the magnetisation dynamics for two cases where the XMCD data within 20 ps exhibits partial or full quench of the Gd and Fe moments. We refer to nonAOS for partial quench and AOS for full quench. As shown below, the dynamic scattering data are qualitatively similar between these cases. We reiterate that the applied field saturates the sample so that any large amplitude inhomogeneity in the spatial spin distribution is not stable at long times. NonAOS was obtained with a 30 nm thick sample and an absorbed 800 nm pump fluence of 3.91 mJ cm^{−2}. In Fig. 3a, the corresponding XMCD response for both Gd and Fe exhibits a partial quench of the magnetisation for t < 3 ps, followed by an approximately constant state of demagnetisation up to the longest delay time of 20 ps. AOS was not achieved with the available pump fluences in this sample. Using a 20 nm thick sample and an absorbed 800 nm pump fluence of 4.39 mJ cm^{−2}, AOS was achieved. The XMCD data in this case shows that the magnetic moments are fully quenched and switch at ≈3 ps, as presented in Fig. 3b. However, similar to the nonAOS case, the spatially averaged magnetisation remains approximately constant for as long as 20 ps after timezero. The extremely slow time dependence of the XMCD data in both cases indicates that the average magnetisation is essentially constant for 3 ps < t < 20 ps. A critical implication is that the quasithermal redistribution of magnon occupation caused by either damping or other inelastic interactions that eventually drives the magnetisation towards a saturated state is not important at these timescales.
The azimuthally averaged spin–spin correlation function for Gd in the nonAOS case is shown by contours in Fig. 3c. Spin–spin correlation profiles at selected time instances are shown in Fig. 3d by solid black curves that have been shifted vertically for clarity. These lineouts have two ringlike spectral features: one with a radius close to or below the smallest resolved wavevectors and one with a radius in the range 0.4 nm^{−1} < q < 0.8 nm^{−1}. Fits to the data shown by the dashed red curves are obtained by using a Gaussian lineshape for the highq feature (with a ring radius indicated by black circles) and a Lorentzian lineshape for the lowq feature. The fitted Gaussian lineshape indicates the appearance of a ring and therefore a shortrange correlated magnetisation pattern at subpicosecond timescales. After ≈5 ps, reliable fits were obtained by use of only a Lorentzian lineshape.
For the AOS case, the azimuthally averaged spin–spin correlation shown in Fig. 3e exhibits a peak at low q that appears in a fraction of a picosecond. In this measurement, the maximum measured q ≈ 0.46 nm^{−1} was insufficient to determine the appearance of a Gaussian peak at higher wavevectors. Spin–spin correlation profiles at selected time instances are shown in Fig. 3f, with vertically shifted curves for the sake of clarity. Reliable fits were obtained solely by use of a Lorentzian lineshape, shown with the dashed red curves in Fig. 3f.
Numerical simulations
To better understand the physical mechanisms that are most important in driving spatially inhomogeneous magnetisation dynamics after pumping, we performed atomistic simulations^{40,41}. The amorphous alloy is modelled as a polycrystalline Gd and Fe–Co thin film with elemental inhomogeneity with a characteristic length of 7 nm, guided by recent experimental results^{8}. The spatially averaged magnetic moments for Gd and Fe obtained with atomistic simulations are shown in Fig. 4a for the nonAOS case utilising an absorbed fluence of 10.7 mJ cm^{−2} and Fig. 4b for the AOS case utilising a similar absorbed fluence of 11 mJ cm^{−2}. The atomistic simulations assume uniform heating across the thickness, and the utilised fluences are tuned to qualitatively reproduce the experimental XMCD data, cf. to Fig. 3a, b.
Snapshots of the simulated perpendiculartoplane magnetisation evolution are shown in Fig. 4c, d for nonAOS and AOS, respectively. In both cases, coarsening of the perpendiculartoplane magnetisation from a finegrained randomised state is observed. The similar spatial evolution for both cases suggests that the same dynamic processes take place after ultrafast optical pumping, insofar as the magnetic moments are substantially quenched. The coarsening of the spatially varying magnetisation at such short timescales is necessarily the result of spinconserving nonlinear magnon interactions, whereby spatial localisation of textures rapidly minimises magnon energy^{21,22} while maintaining a quenched, average magnetisation. This is in contrast to the simple picture of fielddriven growth of domains in an applied field that is operative on much longer timescales, on the order of hundreds of picoseconds^{42}.
To directly compare with the experimental results, the simulated spin–spin correlation function is calculated via Fourier analysis of the spatiallydependent perpendiculartoplane magnetisation. Contours of the azimuthally averaged spin–spin correlation function in the nonAOS case are shown in Fig. 5a. Lineouts at selected time instances are shown in Fig. 5b in addition to fits of the relevant diffraction rings by a linear combination of Lorentzian and Gaussian functions centred at q > 0 with radius positions indicated by black circles. While the appearance of the Gaussian peak is less apparent than in the case for the data in Fig. 3d, the precise fitting of the ring radius and width was still possible, as we further demonstrate below. For the case of AOS, contours of the azimuthally averaged spin–spin correlation function are shown in Fig. 5c while selected lineouts and Lorentzian fits are shown in Fig. 5d by solid black and dashed red curves, respectively. Both cases qualitative agree with the experimental data.
To further identify the role of exchange coupling between the rare earth (Gd) and transition metal (Fe) lattices, we performed multiscale micromagnetic simulations based on the Landau–Lifshitz (LL) equation^{43} that consider an effective, homogeneous exchange stiffness, see Supplementary Note 1. The ferrimagnetic GdFeCo is modelled as a singlespecies ferromagnet, with an initial condition provided by the atomistic simulations at a specified delay t_{c} ≥ 3 ps after optical pumping. By use of this multiscale approach, we can isolate the role of the atomicscale exchange interactions, which dominate at short times, from the longerrange exchange stiffness. Because the micromagnetic model approximates exchange dispersion as q^{2}, spatial fluctuations should be sufficiently concentrated on small wavenumbers. However, the choice of t_{c} within 10 ps does not significantly change the qualitative features of the magnetisation’s coarsening (see Supplementary Note 2). As such, we only show a representative example at the shortest delay, t_{c} = 3 ps, at the limit of the micromagnetic approximation.
For micromagnetic simulations in the nonAOS case, the azimuthally averaged spin–spin correlation function is shown in Fig. 5e. The black area indicates the temporal range in which atomistic simulations are used to calculate the initial conditions for the micromagnetic simulations. Corresponding lineouts, along with fits by the previously described sum of Lorentzian and Gaussian functions, are shown in Fig. 5f by solid black and dashed red curves, respectively. A striking feature is that the Gaussian profile diffraction ring, identified by black circles, persists as long as 10 ps, suggesting that slower dynamics are at play in the micromagnetic approximation. After 10 ps, fits of the lowq diffraction ring with a Lorentzian function break down. Better fits are achieved by use of a squared Lorentzian function that exhibits a q^{−4}like decay. This is an artefact associated with the approximation that the magnetisation in each cell is uniform, with sharp magnetic interfaces between cells. Such a form factor is an artificial constraint in the finitedifference micromagnetic simulations that is avoided in atomistic simulations. For the case of AOS, the azimuthally averaged spin–spin correlation function is shown in Fig. 5g. Lineouts and corresponding squared Lorentzian fits are shown in Fig. 5h. The qualitative agreement to both experimental data and atomistic simulations suggests that atomicscale exchange interactions have a limited influence on the dynamics when only a single lineshape can be fitted.
Imprinted demagnetisation and dissociation
The highq diffraction ring in the nonAOS case appears during the opticallyinduced quench of the magnetic moments, indicating shortrange correlations. To conclusively elucidate the physical mechanisms that drive the magnetisation dynamics at such short timescales, we analyse the fitted parameters obtained from experiments and atomistic simulations. The fitted Gaussian lineshape with ring radius q_{max}, ring width σ_{q}, and normalised ring amplitude are shown in Fig. 6a–c, respectively. Detailed fits to the experimental data are shown in Supplementary Note 3.
The blue circles are obtained from fits to experiments. For the first ≈3 ps, the XMCD data in Fig. 3a indicates that the spatially averaged Gd magnetisation is 75% quenched. At the same time, both the ring radius and the ring width are approximately constant at q_{max} = 0.57 ± 0.014 nm^{−1} and σ_{q} = 0.24 ± 0.002 nm^{−1}, though the ring amplitude continues to increase. The ring radius is consistent with a magnetisation pattern of 2π/q_{max} ≈ 11 nm characteristic correlation length, similar to the ≈10 nm average chemical inhomogeneity characteristic of such amorphous GdFeCo alloys^{8}. Fits to atomistic simulations shown by the red circles in Fig. 6 exhibit a similar behaviour. For the simulated fluence, the demagnetisation is almost 100% during the first 2 ps, and the average fitted ring radius and ring width are q_{max} = 0.88 ± 0.012 nm^{−1} and σ_{q} = 0.18 ± 0.04 nm^{−1}, respectively. The corresponding correlation length of 2π/q_{max} ≈ 7.1 nm agrees well with the chemical correlation length used for the simulation. The diffraction ring amplitude also increases with time for the first 2 ps during the fastest part of the demagnetisation.
Both experiments and simulations provide evidence for an opticallyinduced spatial demagnetisation pattern that imprints on the material’s chemical inhomogeneities. The short spatial fluctuations require an atomistic description. This imprinted demagnetisation is supported by the previously identified subps transfer of angular momentum between chemical species^{8}.
Between ≈3 ps and ≈4.5 ps, the fitted ring radius from experiments shifts towards lower q. Unlike the experimental data, the highq diffraction ring amplitude from atomistic simulations collapses after 3 ps, making further comparison between experiments and simulations impossible. Regardless, the shift in the ring radius for both data and simulations indicates that the magnetic system dissociates from the sample’s chemical inhomogeneity, and transitions into a state where any correlations are emergent features of the magnetisation energetics itself. We interpret these features as a collection of randomly located localised spin textures. Such a rapid dissociation is facilitated by the nonlinear attraction of high energy magnons to each other^{21}. This effect is a consequence of the focusing nature of the effective nonlinear anisotropy and is wellknown in other nonlinear systems such as photonics^{44,45} and Bose–Einstein condensates^{46}. For this reason, we generically refer to the resulting localised textures as magnon drops^{22}.
In the case of AOS, we surmise that a similar imprinted demagnetisation and subsequent dissociation processes must occur at short time scales, as suggested in ref. ^{8}. However, our experimental data was not collected at the relevant wavevectors and atomistic simulations did not exhibit strong enough features to be reliably fitted.
Magnon localisation and coalescence
The subsequent evolution of the magnetisation is quantified from the Lorentzian fits to the lowq diffraction ring. Details of the fitting procedure are discussed in Supplementary Note 4. The two fitted quantities of interest are the ring radius and the ring width. These quantities provide information on the size of and spatial spacing between magnon drops. A collection of randomly located magnon drops, regardless of their spatial spacing, constitute random telegraph noise that results in a Lorentzian lineshape centred at q = 0 whose ring width is inversely proportional to the average magnon drop size. However, because overlapping magnon drops compose a single feature, a finite ring radius inversely proportional to the magnon drops’ spatial spacing ensues. These spectral features are similar to those observed in Xray scattering experiments of a molecular liquid–liquid transition where hardcorelike repulsive interactions between socalled locally favoured structures is invoked^{47}.
The absence of harmonics indicates that the magnon drops’ size distribution dominates the scattering: a periodic array of identically sized magnon drops would consist of harmonic rings by virtue of a Fourier series decomposition whose harmonicdependent coefficients would encode the magnon drops’ size and profile. Deviations of the lattice periodicity would result only in the rings’ spectral broadening.
The evolution of the fitted ring radius is shown in Fig. 7a, b for nonAOS and AOS cases, respectively. The evolution of the average magnon drop diameter L = 2π/Δq, where Δq is the ring width, is shown in Fig. 7c, d for nonAOS and AOS cases. Blue, red, and black circles correspond to, respectively, experiments, atomistic simulations, and micromagnetic simulations.
In the experimental nonAOS case, a ring with a nonzero radius appears after 5 ps. The ring radius increases for the first 3 ps, indicating that the average spatial spacing is decreasing as magnon drops continue to nucleate. A maximum ring radius of 0.0626 ± 0.0011 nm^{−1} is observed at 8 ps, corresponding to an average spatial spacing of ≈100 ± 11 nm. During the same temporal window, the average magnon drop diameter increases from ≈10 to ≈20 nm. The observation of an expanding ring radius accompanied by the growth of magnon drop diameters defines the magnon localisation process. During this stage, an initially sparse collection of magnon drops becomes closepacked due to the continual nucleation of magnon drops.
At longer times, the ring radius drops and appears to reach a plateau at 0.0334 ± 0.0016 nm^{−1} that corresponds to an average spacing of ≈188 ± 9 nm. However, L continues to increase approximately following a power law growth. This behaviour is consistent with the growth of the already present magnon drops via merger and breakup as well as nonlinear attraction of thermal magnons, so that the average spatial spacing increases. We refer to this dynamical process as magnon coalescence and is characterised by the shrinking of the scattered ring radius and its width.
The fits to the atomistic data in the nonAOS case return a qualitatively similar behaviour for both the ring radius and L. Micromagnetic simulations exhibit delayed development of the lowq diffraction ring, with an onset of a nonzero ring radius at ≈10 ps. Such delayed dynamics result from the micromagnetic magnon dispersion proportional to q^{2}, an approximation that is only valid at long wavelengths, as opposed to the more accurate 1cos(qa) form associated with quantum mechanical exchange in a periodic lattice with lattice constant a. Consequently, upon transference of the spatial magnetisation distribution from the atomistic to the micromagnetic simulations, magnons with wavenumbers approaching the Brillouin zone boundary convey an artificially inflated amount of thermal energy into the spin system. This artificially inflated spin temperature commensurately increases the time required to form magnon drops of a similar size to those obtained via the more accurate atomistic simulations. Despite this caveat, the qualitative features of the lowq ring obtained micromagnetically agree with the occurrence of magnon localisation and magnon coalescence.
For the AOS case, Fig. 7b, the lowq ring radius for the experimental data is nonzero after ≈2.5 ps. The ring radius monotonically increases to a maximum of ≈0.047 ± 0.001 nm^{−1} at the longest experimental delay time of 20 ps, corresponding to a minimum average spacing of ≈133 ± 2 nm. The associated evolution of L, Fig. 7d, exhibits a fixed value of ≈100 nm for ≈7 ps, after which L gradually increases to ≈250 nm out to the longest delay times. Viewed together with the evolution of the average spatial spacing, the process of AOS appears to be one in which switching is mediated by an ever decreasing spacing between drops, due to monotonically increasing density of magnon drops. In other words, only magnon localisation is operative for AOS, in a manner consistent with the eventual switching of the macroscopic magnetisation. This is in clear contrast to the case of nonAOS, where the magnon localisation is arrested and gives way to magnon coalescence.
Atomistic simulations in the AOS case yield a much more rapid increase in the ring radius on a time scale of 4 ps, followed by a slow reduction until the radius is close to that of the experimental data at the longest delay. Micromagnetic simulations exhibit a similar increase in the ring radius between 4 and 10 ps, at which point the maximum ring radius is ≈0.09 nm^{−1}. However, there is a rapid collapse of the ring radius after 10 ps, such that the spatial spacing completely diverges at 11 ps. The failure of both the atomistic and micromagnetic models to quantitatively reproduce the evolution of the ring radius in the AOS case suggests that there remains important nonequilibrium physics that affect the rapid magnetisation dynamics in amorphous alloys, which are not contained in either our atomistic or micromagnetic simulations.
Despite differences in ring radii for both AOS and nonAOS, the similar monotonic increase of L for both experiments and simulations with time after 4 ps, indicates that the necessary physics to describe the growth of magnon drops at ps time scales are properly captured by the models. More importantly, the fact that micromagnetic simulations can qualitatively describe the evolution of L beyond 10 ps indicates that nothing more than exchange stiffness and uniaxial anisotropy are necessary to drive the growth of magnon drops.
Power law fits to L are shown in Fig. 7c, d by colour coded dashed lines that utilise the fitting function \(L\left( t \right) = bt^a\). The resultant fitting parameters are listed in Table 1. We find exponents in the range 0.71 ≤ a ≤ 1.14 for all cases. Fits to experimental data obtained at different fluences for both Gd and Fe yield exponents of similar values (see Supplementary Note 5). Taking into account exponents obtained from experiments and simulations, we obtain an average exponent of a = 0.82 ± 0.04.
Domain growth in 2nd order phase kinetics in a nonconservative system is typically modelled with the Lifshitz–Cahn–Allen (LCA) theory, which postulates that domain wall velocity is linearly proportional to the local curvature of the phase interface. This leads to powerlaw growth of phaseordered domains with an exponent of 1/2^{26,48,49}, confirmed in 2D simulations, e.g., refs. ^{50}^{,}^{51}. Solid magenta lines in Fig. 7c, d show exemplary LCA behaviour. However, LCA assumes domain growth proceeds by progression through a continuous series of intermediate, energetically favoured metastable states via shortrange interactions. As such, the applicability of LCA is questionable in the case of magnon drops, where spins at the magnon drop perimeter are necessarily dynamic, and where longrange spin interactions can be mediated by both nonlocal dipole fields^{29} and the hydrodynamic flow of angular momentum via exchange^{23,24,52,53}, all of which would tend to decouple the domain growth rate from the phase boundary local curvature.
Magnon drop dynamics via exchange flows of spin current
Analysis of the magnetisation dynamics from a micromagnetic perspective can shed light onto the origins of how L grows during magnon coalescence. Interactions between magnon drops can be quantified and visualised in terms of the longrange transport of angular momentum arising from the noncollinear magnetisation^{54}. Utilising a hydrodynamic formulation^{23,24,52,53}, the transfer of angular momentum is represented by EFSCs. EFSCs can be expressed as an equivalent charge current density that is 100% spin polarised in the perpendiculartoplane direction^{23,24}.
Examples of EFSCs mediating magnon drop interactions in micromagnetic simulations are shown in Fig. 8 by three snapshots spanning a 2 ps time interval. The magnon drops’ perimeters, where the perpendiculartoplane magnetisation is less than 0.2, are represented as solid black areas. The white and grey background indicate areas where the perpendiculartoplane magnetisation is preferentially parallel (m_{z} > 0.2) or antiparallel (m_{z} < 0.2) to the applied field, respectively. The grey areas are nascent magnon drops in the early evolution of the phase coarsening process. The pinkshaded streamlines represent the flow of angular momentum quantified by EFSCs. We find that EFSCs with an equivalent charge current density on the order of 10^{7} A cm^{−2} can persist for many 10’s of ps after optical pumping. Such magnitudes are similar to those used for magnetisation switching via spin transfer torque^{55}. While the EFSCs are spatially nonuniform and highly turbulent, they can effectively deform the magnon drops’ perimeters, causing both breathing and rotating^{25.56}. Furthermore, the EFSCs mediate longrange interactions between magnon drops that result in both mergers and breakups^{25}. An example of a merger is observed between the leftmost and central magnon drops, labelled A and B, respectively. At 25 ps, largemagnitude EFSCs flow between the magnon drops, so that a torque is exerted at the perimeters. At 26 and 27 ps, the perimeters merge into a single drop B and continue to transfer angular momentum. Examples of breakup are observed at the top of the central magnon drop, where EFSCs transfer angular momentum away from the magnon drop. As a result, the topleft and top magnon drops, labelled C and D, break up at 26 and 27 ps, respectively.
Discussion
While our experimental and numerical results focus on the first 20 ps evolution of the magnetisation in GdFeCo alloys, these dynamics shed light onto the nonlinear magnon processes that drive coarsening while the spin system equilibrates.
The qualitative agreement between experiments and multiscale simulations demonstrates that current models incorporate the most important physical effects responsible for the nonlinear magnon processes identified here. From a theoretical perspective, this agreement implies that appropriate scaling of the equation of motion can be used to describe other magnetic materials insofar as they exhibit PMA and a net magnetisation at equilibrium. In other words, the magnon processes described here are universal for PMA ferromagnets and ferrimagnets. To substantiate this claim, we performed additional atomistic simulations for a chemically homogeneous GdFeCo (see Supplementary Note 6). The evolution of the spatially varying magnetisation is consistent with a magnon coalescence process at times greater than 5 ps. Because there are no inhomogeneities to seed a magnetisation pattern during demagnetisation, we conclude that magnon localisation and coalescence in PMA magnets are indeed general processes that are independent of the material’s structure, although the details of the spatial evolution process can be affected by the presence of chemical inhomogeneities. In addition, micromagnetic simulations in the magnon coalescence regime exhibit qualitatively similar evolution of both the ring radius and the average magnon drop diameter L for different sample thicknesses, initial state of spatial magnetisation, and variation of other micromagnetic parameters (see Supplementary Note 7).
The hydrodynamic formulation of magnetisation dynamics provides a valuable tool to understand the longtime magnetisation evolution. For example, the nucleation of topological defects after fast quench^{36,37} represents an interesting possibility in the context of opticallyinduced applications. In the hydrodynamic formulation, topological defects are evidenced by curved trajectories in the EFSCs^{23}. However, an accurate calculation of the topological number for a spatially localised defect requires a uniform magnetisation surrounding the defect^{57}. In other words, defects must be sparsely located. We surmise that understanding the evolution of EFSCs from tens to hundreds of picoseconds timescales, where dissipative processes are operable, will lead to a better understanding of the dynamic evolution of topological defects upon ultrafast optical pumping.
Based on our study, we speculate that desired magnetisation states may be stabilised by nanopatterning magnetic materials to take advantage of both subpicosecond seeded magnetisation states and EFSCs, even for crystalline materials. For example, a closepacked spatially periodic patterning may favour a likewise closepacked magnetisation pattern during localisation to induce AOS. Conversely, sparse engineered defects may serve as pinning potentials to stabilise topological defects.
Methods
Experiments
The GdFeCo samples were fabricated on 100 nm thick Si_{3}N_{4} membranes by magnetron sputtering. A 5 nm seed layer of Si_{3}N_{4} was first grown on the membrane followed by the Gd_{0.24}Fe_{0.665}Co_{0.095} film, which was then capped with 20 nm of Si_{3}N_{4}. Xray measurements were conducted at the SXR hutch of the Linac Coherent Light Source^{58}. The Xray energy was selected to be resonant with the Fe L_{3} resonance edge at 707 eV or the Gd M_{5} resonance edge at 1185 eV with a 0.5 eV bandwidth and a pulse duration of 80 fs. The Xray pulses were circularly polarised at the Fe L_{3} and Gd M_{5} edges by using the XMCD in magnetised Fe and GdFe films, respectively placed upstream of the experiment. A degree of polarisation was 85% at the Fe L_{3} edge and 79% at the Gd M_{5} edge. Measurements were made in transmission geometry with Xrays incident along the sample normal. An invacuum electromagnet was used to apply a field of 0.5 T perpendicular to the GdFeCo film. The diffracted Xrays were collected with a p–n chargecoupled device (pnCCD) twodimensional detector placed behind the sample. A hole in the centre of the detector allowed the transmitted beam to propagate to a second detector used to collect the transmitted Xray beam. The experiment was conducted in an optical pump—Xray probe geometry. Optical pulses of 1.55 eV and 50 fs duration were incident on the sample in a near collinear geometry. The delay between the optical and Xray pulses was achieved using a mechanical delay line, where the delay was continuously varied. Xrayoptical jitter was monitored and removed from the experimental data using an upstream crosscorrelation arrival monitor^{59}.
Atomistic simulations
A model system of a GdFe ferrimagnet was developed to perform numerical simulations of the atomistic spin dynamics after femtosecond laser excitation. The inhomogeneous microstructure is generated by specifying random seed points representing areas of segregation of the Gd from the alloy, leading to 15–30% higher local Gd concentration. These regions are interpolated using a Gaussian with a standard deviation of 5 nm, representing the scale of the segregation. Due to low packing of the seed points, the characteristic length of the spatial variations is approximately 7 nm. An atomistic level simulation model is used to properly describe the ferrimagnetic ordering of the atomic moments with Heisenberg exchange^{40}. The energy of the system is described by the spin Hamiltonian
where the spin S_{i} is a unit vector describing the local spin direction. J_{ij} is the exchange integral, which we limit to nearest neighbour interactions, and k_{u} is the anisotropy constant. Timedependent spin dynamics are governed by the Landau–Lifshitz equation at the atomistic level
where γ is the gyromagnetic ratio and α = 0.01 is the Gilbert damping factor that can be used in the LandauLifshitz form when α << 1. The onsite effective induction can be derived from the spin Hamiltonian with the local field augmented by a random field to model the interactions between the spin and the heat bath
where the second term \(\varsigma _i\) is a stochastic thermal field due to the interaction of the conduction electrons with the local spins, and μ_{i} is the local (atomic) spin magnetic moment. The stochastic thermal field is assumed to have Gaussian statistics and satisfies
where k_{B} is the Boltzmann constant and T is the temperature. We incorporate the rapid change in thermal energy of a system under the influence of a femtosecond laser pulse. The spin system is coupled to the electron temperature, T_{e}, which is calculated using the twotemperature model^{60} with the free electron approximation for the electrons
where C_{e} = 225 J m^{−3} K^{−1}, C_{1} = 3.1 × 10^{6} J m^{−3} K^{−1}, G_{el} = 2.5 × 10^{17} W m^{−3} K^{−1}, and P(t) models the temperature from a single Gaussian pulse into the electronic system. The pulse has a width of 50 fs.
We use Heun numerical integration scheme to integrate the stochastic equation of motion with timevarying temperature^{41}. We use \(\mu _{{\mathrm{F}}_{\mathrm{e}}} = 1.92\mu _B\) as an effective magnetic moment containing the contribution of Fe and Co and we set \(\mu _{{\mathrm{G}}_{\mathrm{d}}} = 7.63\mu _B\) for the Gd sites, where μ_{B} is Bohr’s magneton. The standard parameters of the exchange coupling constants are used: \(J_{{\mathrm{Fe  Fe}}} = 4.526 \times 10^{  21}\) J per link, \(J_{{\mathrm{Gd  Gd}}} = 1.26 \times 10^{  21}\) J per link, and \(J_{{\mathrm{Fe}}  {\mathrm{Gd}}} =  1.09 \times 10^{  21}\) J per link. We assume a uniaxial anisotropy energy of 8.07246 × 10^{−24} J per atom. The numerical simulations are conducted using the VAMPIRE software package^{41}. The simulation volumes were 200 nm × 200 nm × 2 nm and 1000 nm × 1000 nm × 2 nm.
Multiscale micromagnetic simulations
Micromagnetic simulations were performed with the graphic processing unit (GPU) package MuMax3^{61} that solves the Landau–Lifshitz equation for a ferromagnet
where μ_{0} is the vacuum permeability, m is the magnetisation vector normalised to the saturation magnetisation, and B_{eff} is an effective induction that includes the required physical terms to model a ferromagnetic material. Here, we included exchange, nonlocal dipole, uniaxial anisotropy, and external fields. The exchange interaction in the micromagnetic approximation takes the form of a Laplacian scaled by the exchange length, λ_{ex}. In MuMax3, the Laplacian is numerically resolved by a 4th order central finite difference scheme, i.e., each micromagnetic cell is subject to exchange interaction due to itself and two neighbouring cells in each dimension. We ran our simulations on NVIDIA GPU units K20M, K40, K80, and P100. Due to the coarse resolution of micromagnetic simulations, we utilise approximately cubic cells of size 2 nm × 2 nm × δ, where δ = D/2 ^{N} and the factor N is chosen to take advantage of the GPU spectral calculations such that δ < λ_{ex} ≈ 5 nm and D is the physical thicknesses equal to 30 or 20 nm for the nonAOS or AOS cases, respectively. The lateral simulation area was determined from atomistic simulations and the full thickness for each case was achieved upon the atomistic observation that the magnetisation is approximately homogeneous across the thickness at t ≥ 3 ps. The coarse micromagnetic discretization allows for a significant speed up in the computations. Note that the size of the cells only impacts the stability and accuracy of the numerical algorithm while the physics can only be interpreted in the framework of the continuum Landau–Lifshitz equation, i.e., longwavelength features relative to the exchange length. We set the software to solve Eq. (8) with an adaptivestep, 4th order Runge–Kutta time integration method. Periodic boundary conditions (PBCs) were imposed along the film’s plane. For both dynamical behaviours we used the equilibrium magnetic parameters: saturation magnetisation M_{S} = 47170.6 A m^{−1}, anisotropy constant k_{u} = 31127.228 J m^{−2}, exchange constant A = 1 pJ m^{−1}, and α = 0.01. The value for A was numerically found to best match the atomistic, average perpendicular magnetisation evolution (see Supplementary Note 1).
Change in the spin–spin correlation function
Experimentally, the change in the spin–spin correlation function, ΔS^{2}(q,t), was obtained from the scattered intensities of circularly polarised Xrays. For this, the scattering intensities are added to obtain
where I_{+}(q,t) and I_{−}(q,t) are the timedependent scattered intensities obtained with righthanded and lefthanded circularly polarised light, S^{2}(q,t) is the spin contribution to the intensity, and C^{2}(q) is the charge contribution to the intensity. Because the charge contribution is timeindependent for the used pump fluences, the spin–spin correlation function can be isolated as
where the background was subtracted by averaging the data collected at times before the optical pulse irradiated the sample.
To compare the data with simulations, the spin–spin correlation function for both atomistic and micromagnetic simulations was determined by computing a twodimensional fast Fourier transform (FFT) of the perpendicular magnetisation for each layer as a function of time. To minimise error, the FFTs obtained for each layer at a given time were averaged. Since PBCs were used for simulations, a window function was not necessary.
Exchange flow spin currents
In the dispersive hydrodynamic formulation of magnetisation dynamics^{23,24}, the normalised magnetisation vector m = (m_{x}, m_{y}, m_{z}) in Eq. (8) can be cast in hydrodynamic variables by the canonical transformation
where n is the spin density and u is the fluid velocity. For the case of conservative dynamics, α = 0 in Eq. (8), the dispersive hydrodynamic equations are
expressed in dimensionless space, time, and field scaled by, respectively \(\sqrt {\left {H_{\mathrm{k}}/M_{\mathrm{S}}  1} \right} \lambda _{{\mathrm{ex}}}^{  1}\), \(\gamma \mu _o\left {H_{\mathrm{k}}  M_{\mathrm{S}}} \right\), and \(M_{\mathrm{S}}^{  1}\), where the anisotropy field is given by \(H_{\mathrm{k}} = 2k_{\mathrm{u}}/\left( {\mu _oM_{\mathrm{S}}} \right)\), and h_{0} is a dimensionless field applied normal to the plane. The spin density flux in Eq. (13) is identified as the EFSC in hydrodynamic variables. To establish a clear comparison to spin currents obtained by chargetospin transduction, the EFSC are expressed as a 100% spin polarised charge current density in units of A m^{−2} by^{24}
We note that the factor (1 − n^{2}) leads to maximum EFSC for a given u when the magnetisation is in the plane. For this reason, the magnon drop perimeters are primarily subject to EFSCs.
Data availability
The data that supports the findings of this study are available from the corresponding author upon reasonable request.
Code availability
MuMax3 is open source software and available from http://mumax.github.io. VAMPIRE 5 is open source software and available from https://vampire.york.ac.uk. The data fitting procedure and calculation of EFSC were custom written in MATLAB and are available from the corresponding and leading author upon reasonable request.
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Acknowledgements
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number 0000231415 and DESC0017643 and is partly supported by the European Research Council (ERC) Grant Agreement No. 339813 (Exchange) and the Netherlands Organisation for Scientific Research (NWO). Operation of LCLS is supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Contract No. DEAC0276SF00515. The atomistic simulations in this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) and used code enhancements implemented and funded under the ARCHER embedded CSE programme (eCSE0709). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 737093 (FEMTOTERABYTE). This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (http://www.csd3.cam.ac.uk/), provided by Dell EMC and Intel using Tier2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk). E.I. acknowledges support from the Swedish Research Council, Reg. No. 63720146863. M.A.H. was partially supported by NSF CAREER DMS1255422. L.L.G. would like to thank the VolkswagenStiftung for the financial support through the PeterPaulEwald Fellowship. R.W.C. is grateful to Radboud University, Nijmegen, for hospitality under the Radboud excellence scheme. E.I. thanks Leo Radzihovsky for fruitful discussions.
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E.I. performed micromagnetic simulations. A.T. prepared the samples. A.H.R., T.M.L., P.W.G., E.J., A.X.G., S.B., C.E.G., R.K., Z.C., D.J.H., T.C., L.L.G., K.H., H.O., W.F.S., G.L.D., G.C., M.C.H., S.C., A.K., A.V.K., T.R., J.S., and H.A.D. performed experiments. Z.F. and R.F.L.E. performed atomistic simulations. R.F.L.E. developed the numerical representation of Fe and Gd inhomogeneities in the atomistic model. Z.F., S.R., R.F.L.E., T.O., and R.W.C. analysed the atomistic data. T.M.L. and D.J.H. analysed the experimental data. E.I., M.A.H., and T.J.S. fitted the data and analysed the micromagnetic simulations. All authors contributed to discussions, data analysis, and writing the manuscript.
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Iacocca, E., Liu, TM., Reid, A.H. et al. Spincurrentmediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys. Nat Commun 10, 1756 (2019). https://doi.org/10.1038/s41467019095770
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