Nonequilibrium sub–10 nm spin-wave soliton formation in FePt nanoparticles

Magnetic nanoparticles such as FePt in the L10 phase are the bedrock of our current data storage technology. As the grains become smaller to keep up with technological demands, the superparamagnetic limit calls for materials with higher magnetocrystalline anisotropy. This, in turn, reduces the magnetic exchange length to just a few nanometers, enabling magnetic structures to be induced within the nanoparticles. Here, we describe the existence of spin-wave solitons, dynamic localized bound states of spin-wave excitations, in FePt nanoparticles. We show with time-resolved x-ray diffraction and micromagnetic modeling that spin-wave solitons of sub–10 nm sizes form out of the demagnetized state following femtosecond laser excitation. The measured soliton spin precession frequency of 0.1 THz positions this system as a platform to develop novel miniature devices.

The PDF file includes: Supplementary Text Figs.S1 to S7 Table S1 Legend for movie S1 Other Supplementary Material for this manuscript includes the following:

Movie S1
A model for FePt magneto-elastic coupling This section describes in more detail the magneto-elastic coupling outlined in the methods section.The response of the FePt lattice structure to the presence of a spin-wave soliton in the nanoparticle was investigated using the magneto-elastic coupling formalism (36), that describes the influence of magnetization dynamics on lattice atom displacements, u, in the material.In Cartesian coordinates, x, the magneto-elastic energy density for the tetragonal FePt lattice is given by the eq.( 1) (37,38): where  &,',) are the direction cosines of the magnetization vector, M,  ./= 01 !0& " are components of the strain tensor and  %% ,  %( ,  (% ,  (( ,  * ,  * + ,  , are tetragonal magnetoelastic coupling constants.Using the magneto-elastic energy density it is possible to calculate the magneto-elastic force: For the tetragonal lattice eq. ( 2) leads to the following expression: In eq. ( 3) the first and the second terms are isotropic and the third term is the anisotropic component of the magneto-elastic force.
We note that all terms include spatial derivatives of the magnetization.This means that the force is nonzero only if the magnetization is not homogeneous over the nanoparticle volume.This implies that if the length of the magnetization vectors per unit volume does not change, the direction of the magnetization must change across the nanoparticle at any particular moment of time.As a consequence, homogeneous magnetic modes such as ferromagnetic resonance (FMR) cannot produce a considerable magneto-elastic coupling because the magnetic moments of all atoms in the nanoparticle precess as a whole with a similar amplitude and phase.Contrary to FMR, the spin-wave soliton has a well-defined profile that gives rise to strong magneto-elastic forces.These forces are especially large at the spin-wave soliton's perimeter.Here the Mz component reverses sign while Mx,y experiences a large-angle in-plane precession.
We calculate the details of the lattice dynamics in FePt nanoparticles by solving the equation of motion for the elastically strained solid with the magneto-elastic force resulting from the soliton magnetization dynamics as a driver for the atomic displacement: where  is the displacement per unit volume,  is the mass density,  is a damping time constant,  !"# is the magneto-elastic force and ∇ is the elastic force per unit volume which defines the elastic properties of material and the phonon spectrum.It is given by the gradient of the three-dimensional stress tensor: where  ./2#are the components of the fourth-rank tensor of elastic stiffness parameters.
We calculate the magneto-elastic force  !"# using the expression eq. ( 3) with the results of micromagnetic simulation for an 8 nm high and 22.5 nm wide cylindrical nanoparticle as input.This is among the nanoparticle sizes commonly found in the sample.The equation of motion ( 4) was solved numerically in the 3-dimensional Cartesian grid using the standard second order "leapfrog" algorithm from the central differences.We used the same grid spacing as in the micromagnetic simulation: 0.7 nm in x, y and 0.5 nm in z directions.The time interval of simulation was 100 ps with the time step 50 fs.The initial lattice displacements were assumed to be zero,  3 = 0.For simplicity we have used the Dirichlet boundary condition  4516789' = 0.The dissipation term with  = 5 ps was included in the eq.( 4) to address, in the generalized form, the damping of magnetoelastically induced lattice vibrations via transmission through the nanoparticle boundary into the carbon matrix and other possible mechanisms.
Figure S4 illustrates the spin-wave soliton magnetization dynamics obtained from micromagnetic simulations described in the main paper (see methods section).The 22.5 nm nanoparticle displays a specific soliton mode with a frequency of 0.05 THz (Fig. S4F) for the in-plane magnetization components Mx and My.This mode is best observed as a precession of the magnetization along the soliton's perimeter (marked in white in Figs.S4A-E).At the same time the dynamics of the out-of-plane Mz component shows a lower-frequency peak near 0.10 THz that is related to the soliton breathing, i.e. periodic changes of its size and shape.In Figs.S4A-E, snapshots of the magnetization state demonstrate the spin-wave soliton precession and breathing dynamics.While the magnetization at the perimeter makes one full 360° rotation, the soliton shape changes about twice as fast.The soliton breathing can be clearly seen as the squeezed and elongated (blue) shapes in Figs.S4A-E (see also Movie S1).
Figure S5A shows the spectra of the magneto-elastic force calculated as the timefrequency Fourier transform of the x, y, z force components in eq.(3) averaged over the particle's volume.The used FePt magneto-elastic coupling parameters are shown in Table 1 in the methods and materials section of the main paper.We see a strong peak at 0.10 THz and a weaker broader band between 0.2-0.3THz.The former is caused mainly by the precession of the magnetization along the soliton's perimeter, while the latter corresponds to the magneto-elastic frequency mixing between the frequency-doubled inplane magnetization precession and the spin-wave soliton breathing.
This assignment can be explained by considering a simplified model of the soliton magnetization precession and breathing modes.The x, y, and z components of magnetization near the soliton perimeter have the following form: where  is the frequency of the soliton mode observed in Mx and My (Fig. S4F), that corresponds to the magnetization precession; while  3 $(, )& is the spatial dependence of the polar magnetization angle for a spherical spin-wave soliton described in ref. (7). 3 is zero at the soliton perimeter but its detailed shape is only accessible to numerical calculations (7).We will use  3 here to parametrize the soliton shape.To address the spin-wave soliton breathing and it's coupling to phonons, the substitution can be made: where  + is the soliton breathing frequency observed mainly in the Mz component (Fig. S4F) and Dx describes the breathing motion of the soliton perimeter which is "small" compared to the soliton size.Substituting the magnetization components calculated using eqs.( 6) and ( 7) into eq.( 3) the following approximate terms of the magneto-elastic force x component can be derived: In eqs.(8) we can notice the terms which contain the phases 2 and  + .Since the micromagnetic simulations of a 22.5 nm particle give  ≈ 0.05 THz and  + ≈ 0.10 THz (Fig. S4F), these terms apparently cause the peak in the force frequency spectrum at ~ 0.10 THz (Fig. S5A) as the second harmonic of  and the fundamental  + frequency.Eqs. ( 8) also contain cross-terms with the product of 2 and  + , that will lead to the addition (and subtraction) of these frequencies together resulting in the frequency band between 0.2-0.3THz (and very low frequencies contributions that are not resolved experimentally).Thus, two frequency bands in the magneto-elastic force in Fig. S5A can be explained by the terms containing the doubled spin-wave soliton precession and the breathing frequency with 2 ≈  + ≈ 0.10 THz as well as the cross-terms between them.
In other words, the interference between the magneto-elastic interactions caused by the soliton magnetization precession near the soliton perimeter and the soliton breathing produces two bands which differ approximately by a factor of two in frequency.Furthermore,  &%.% in eq. ( 8) contain a time-independent term causing a constant force component near the domain wall, which is indeed observed in the full simulation.Finally, the term  &* may produce a linear frequency band near  ≈ 0.05 THz.However, it is not observed in the experimental result because the soliton polar angle  3 at the perimeter likely does not change along the z direction, so that the derivative is close to zero.Also, the magneto-elastic constant  , responsible for this coupling is the smallest of all.
Figure S5B shows the calculated frequency spectrum of in-plane atomic displacements averaged over the particle volume.It shows a distinct peak at 0.10 THz and a weaker, broader band between 0.2-0.3THz.The former corresponds well to the peak of the magneto-elastic force at the same frequency and is caused by the forced harmonic oscillations of the atoms with this frequency.The band at 0.2-0.3THz overlaps with the band of the magneto-elastic force at the same frequency range and thus it is of induced nature as well.On the whole, the dominating lattice vibration bands at 0.10 and 0.2-0.3THz, calculated for the 22.5 nm particle, are in good agreement with the soliton scattering contribution in experimental result in Fig. 4A of the main paper.S1.Frequency (THz) ) Table S1.The fitted parameters of the ultrafast magnetization dynamics in FePt of the form, / 3 = 1 + [>?$%$ # & '( @ − A?

Fig. S1 .
Fig. S1.Nanoparticle size distribution of FePt samples: (A), transmission electron microscopy (TEM) image of FePt nanoparticles embedded in a C matrix (white).The different greyscales visible in the FePt nanoparticles is caused by a spread in crystallographic alignment of the individual nanoparticles.(B), distribution of nanoparticle sizes, D. The average particle size evaluated from the TEM image is Dave = 15.8 ± 5.6 nm.The grain size distribution and average grain size were extracted from plane-view bright-field TEM images.More than 1,500 grains were randomly selected for the statistical analysis.The selected grains were covered with masks in the bright-field TEM image and the diameter information was acquired through analysis with the Gatan Microscopy Suite software.The collected data were fitted by a lognormal function to determine the average grain size and its standard deviation (error) to Dave = 15.8 ± 5.6 nm.

Fig. S3 .
Fig. S3.Ultrafast demagnetization of FePt for the indicated pump fluences.Following refs.(23,33) circularly polarized X-rays with energy tuned to the Fe 2p -3d core-valence resonance (708 eV) were scattered from the FePt sample in the geometry shown in Fig. 3A.The sample magnetization was oriented (anti)parallel to the X-ray incidence direction by applying a magnetic field of ±350mT.The difference in the scattering yield, integrated over all accessible wavevectors, is a measure of the sample magnetization, M, and is plotted relative to the ground-state magnetization, M0, measured before arrival of the pump pulse.The measured data (symbols) are fitted with an exponential decay followed by an exponential recovery (lines) of the form, / 3 = 1 + [>?$%$ # & '( @ − A?$%$ # & )( B] (23,33).The fit parameters are compiled in TableS1.

Fig S4 .
Fig S4.Micromagnetic simulations of spin-wave solitons in nanoparticles.(A-E), Snapshots of the micromagnetic simulation showing the correspondence between the two phases of soliton distortions (breathing and motion) and perimeter precession.The colour scale denotes the out-of-plane Mz component.The black arrows show the magnetization in the perimeter that is predominantly oriented in-plane.(F), The Fourier transform amplitudes for the Mx, My and Mz magnetization components.The frequency of the Mz precession corresponds to the spin-wave soliton breathing and motion.

Fig. S5 .
Fig. S5.Results of the numerical magneto-elastic calculations: amplitudes of the Fourier transform of (A), magneto-elastic force and (B), atomic displacement.Both quantities show two dominating frequency bands at 0.10 and 0.2-0.3THz.

Fig. S6 .
Fig. S6.Calculated FePt spin-waves.Ab initio calculated spin-wave dispersion of bulk FePt in the simple tetragonal Brillouin zone with corresponding high-symmetry points as indicated.