Element-specific ultrafast lattice dynamics in FePt nanoparticles

Future magnetic data storage media will require magnetic nanoparticles with stable ferromagnetic order at diameters of only 10 nm and smaller.¹ In this respect, granular thin films of the L1₀-ordered phase of FePt displaying perpendicular magnetic anisotropy (along the out-of-plane c-axis) are one of the most suitable storage media. The FePt nanoparticles composing such granular materials remain ferromagnetic as a result of the strong magnetocrystalline anisotropy needed to overcome the superparamagnetic limit.^2–5 However, a byproduct of strong magnetocrystalline anisotropy is the large magnetic field required to reverse the nanoparticle magnetization. Applications strive to reduce the magnetic switching field by locally heating the nanoparticles above their Curie temperature with a laser in order to thermally assist the switching, a technique known as heat-assisted magnetic recording.⁶ 

The magnetization dynamics of FePt nanoparticles following optical femtosecond (fs) laser excitation has been the subject of various studies resulting in the observation of sub-picosecond demagnetization,^7,8 element-specific spin dynamics,⁹ and even all-optical magnetic switching.¹⁰ However, much less is known about the ultrafast lattice response, which has been a topic of attention from theoretical work¹¹ and could only recently be experimentally addressed using ultrafast x-ray and electron scattering. Reid et al.¹² showed that the response of suspended 13-nm FePt nanoparticles is characterized by a lattice expansion along the Fe and Pt layers (a, b directions in Fig. 1) accompanied by a contraction of the lattice spacing in the perpendicular direction (c direction in Fig. 1). This reflects a magnetostrictive stress on the lattice due to the laser-induced quenching of the magnetic order.¹² Key of such studies is that the nanoparticle lattice is free to follow the intrinsic stress buildup within the particles. For instance, FePt nanoparticles with their lattice spacing along the Fe/Pt layers locked into that of a supporting substrate still react to magnetostrictive stress via a lattice contraction along the perpendicular direction.¹³ 

Here we address the question if the observed changes of the FePt nanoparticle lattice are uniform for the Fe and Pt sublattices. Utilizing the constructive and destructive interference of scattering from both atomic sublattices for Bragg peaks with even and odd sums of Miller indices, respectively, we show that the Pt sublattice expands faster than the Fe sublattice. We correlate this observation with element-specific measurements of the temporal variations observed in mean square vibrations and Brillouin zone boundary phonon occupations.

Single crystalline L1₀ FePt was grown epitaxially onto a single-crystal MgO(001) substrate by co-sputtering of Fe, Pt, and C.¹⁴ This resulted in FePt nanoparticles of approximately cylindrical shape with heights of 6 nm and diameters in the 4–12 nm range with an average of 7.1 ± 1.8 nm as corroborated with transmission electron microscopy [Fig. 1(c)]. Due to the single-crystalline substrate, the FePt nanoparticles are identically oriented with the a and b crystallographic directions along the MgO surface and the c-axis perpendicular to it. The volume in-between nanoparticles is filled with glassy carbon at 30% volume fraction. Subsequently, the MgO substrate was chemically removed and the FePt-C films were floated onto copper wire mesh grids with 100 μm wide openings.

The dynamic lattice response of FePt was measured by ultrafast electron diffraction in a transmission geometry [see Fig. 1(a)] with 3.6 MeV electrons from the SLAC ultrafast electron diffraction source.¹⁵ The pump-probe experiments described below were carried out at room temperature with 1.5 eV/50 fs laser excitation at a nominal pump fluence at a normal incidence of 4 mJ/cm². To keep the deposited energy density constant, the incident laser pulse energy was adjusted to account for the change in beam spot size and sample reflectivity when changing the sample tilt angle.

To meet the Bragg condition for different lattice reflections, the film was rotated around axes normal to the probe beam. Due to geometrical reasons, rotation angles were limited to 45° from normal incidence. Measurements made at normal incidence, with the [001] c-axis parallel to the electron beam, showed changes in the diffraction pattern displayed in Fig. 1(b) as the difference of the pattern at 1 ps pump-probe time delay with respect to that obtained before time zero, i.e., when the optical pump pulses arrive after the electron probe pulses. The Bragg peak positions were determined using a fit of two-dimensional Gaussian profiles to the experiments.¹² While at normal electron incidence, we probe Bragg peaks along the a,b-crystal axes [see inset of Fig. 1(d)], also c-axis Bragg peaks are accessible when the lattice was tilted away from [001] normal incidence direction. Following Ref. 12, we collected different Bragg reflection position and intensity data after a rotation of the film about the [100] a-axis to a point where the [111] reflections were easily visible. The observed time evolution data of Bragg peaks with different projections along the out-of-plane direction are used to reconstruct the [001] c-axis Bragg intensity following Ref. 12. The results are shown in Fig. 1(e) together with the determination of the average unit cell size variation in Fig. 1(d).

The data displayed in Fig. 1 resemble those obtained for larger FePt nanoparticles.¹² They show an initial lattice expansion along the a- and b-axes as well as a concomitant c-axis reduction of the lattice spacing due to the reduction of the magnetostrictive stress following a laser-induced ultrafast quenching of the ferromagnetic order.¹² This leads to a long-lasting average increase in the FePt unit cell during approximately 1 ps after laser excitation followed by a volume-conserving breathing mode at a frequency given by the time it takes an acoustic lattice strain wave to move through the nanoparticles.¹² 

Mean square displacements are a way to measure the vibrations of atoms around their equilibrium positions in a crystal, unveiling the energy stored in the lattice. Within the context of ultrafast lattice dynamics, they serve as an ultrafast proxy for the energy stored in the lattice, making them an ideal tool for tracking the energy flow in an excited, out-of-equilibrium system. The mean square displacements of atoms affect the scattering intensity through the Debye–Waller factor, M. In the case of FePt, the integrated intensity of the Bragg peaks is dependent on the Debye–Waller factors of both iron, M_Fe, and platinum, M_Pt. The Debye–Waller factor for diatomic species like FePt has been widely studied in a static regime,^16–20 but their analysis of Debye–Waller factors in a pump-probe scheme remains constrained for either mono-atomic species^21,22 or by using lattice temperatures without a clear distinction of the different chemical species.^23–25 

Figure 2(c) shows the determined changes of Fe (blue symbols) and Pt (orange symbols) mean square displacements, $Δεhkl·u2$⁠, projected onto the direction, $εhkl$⁠, of the respective reciprocal lattice vectors, $qhkl=qhkl εhkl$⁠, vs pump-probe time delay. The observed increase in $Δεhkl·u2$ can be described by two exponentials of the form $A1(1−e−t/τ1)+A2(1−e−t/τ2).$ The determined fit parameters are summarized in Table I.

Insight into the mechanism by which energy is transferred to the lattice to and from electronic and spin degrees of freedom needs wavevector and time-resolved information about the evolution of the phonon populations after laser excitation. The primary source of lattice heating is the hot electron bath that gets excited directly by the laser. The subsequent energy transfer to phonons takes place via electron–phonon scattering events that show a pronounced wavevector dependence, i.e., Brillouin zone boundary phonons are often preferentially populated under the non-equilibrium conditions following ultrafast laser heating.²⁷ Diffuse electron diffraction has been used as a unique tool to directly probe the wavevector dependence of such transient phonon populations (e.g.,, Refs. 28–31). Here we extend diffuse scattering to the case of FePt with the aim of separating the non-equilibrium motion of Fe and Pt atoms for selected phonon modes.

From a diatomic chain model, it is straightforward to see that at the X-point of the Brillouin zone boundary, phonon modes have eigenvectors where either Fe or Pt atoms are at rest. It follows that the corresponding $ek,jPt, Fe$ shown in Eq. (3) must be zero. This is reproduced in the calculations shown in Figs. 3(b) and 3(c) that were performed following Ref. 27.

The data in Fig. 3(a) can be described by exponential increases of the form $A1−e−t/τ$⁠. We obtain for the X-points marked in the inset of Fig. 3(a) the fit parameters summarized in Table II, where the term $ε·eL, TPt,Fe2$ describes the square of directional cosines for phonons with longitudinal (L) and transverse (T) polarization at the two X-points. This demonstrates that measurements at the X_L and X_T points are sensitive mainly to longitudinal and transverse phonon polarizations, respectively.

Here we describe an extension of the average lattice expansion for FePt nanoparticles beyond what is shown in Fig. 1 and what has been reported so far.¹² We confine ourselves to normal incidence measuring the a, b lattice expansion depicted in Fig. 1(b). The interference of the scattering amplitudes from Fe and Pt atoms described in Eq. (1) will also allow us to corroborate if the observed lattice expansion is the same for both sublattices.

The situation is schematically depicted in Figs. 4(a) and 4(b), where the scattering amplitudes are illustrated for odd and even Bragg orders, respectively. If the lattice expansion is the same for Fe and Pt sublattices, the corresponding odd and even Bragg peaks will be shifted the same amount from the equilibrium lattice position marked by the gray dashed line. If, however, the Pt sub-lattice (orange dashed lines) expands more than the Fe sub-lattice (blue dashed lines), the intensity maxima of the interfering scattering amplitudes for odd and even Bragg orders [indicated by black vertical lines in Figs. 4(a) and 4(b)] will no longer match each other. For destructive interference at odd Bragg orders [Fig. 4(a)], the Bragg intensity maxima will shift to lower values of q/q₀ than for the constructive interference at even Bragg orders [Fig. 4(b)].

Although the shift between odd and even orders is relatively small, it can be clearly seen near the Bragg peak intensity maxima in Figs. 4(e) and 4(f). The peak intensity is normalized to remove Debye–Waller attenuation effects seen in Figs. 4(c) and 4(d) and described in Sec. III A. We can model the Bragg peak shift observed in Fig. 4(e) with a Pt sublattice expansion of 0.42% and an Fe expansion of 0.37% as shown in Fig. 4(f). An extended dataset together with the predicted intensity maxima is shown in Fig. 5. This leads to an average peak shift of all measured odd [Fig. 5(a)] and even [Fig. 5(b)] Bragg peaks.

There are several aspects related to FePt nanoparticles that make them unique candidates to study the non-equilibrium interplay between electronic, magnetic, and lattice degrees of freedom. While optical and x-ray pump-probe studies have focused on electron and spin thermalization times following laser heating,^12,33 our observation of a unit-cell volume expansion in Fig. 1(d) highlights the intricate link behind these processes. The observed timescale for this lattice expansion indicates that it is driven by acoustic strain waves. We have previously observed the longitudinal acoustic (LA) phonons that form the coherent phonon wave packets³⁴ driving this expansion in analogy to the THz strain wave propagation in ultrathin Fe films.³⁵ The lattice stress driving the expansion can be twofold: (1) electronic stress due to the non-equilibrium population of electronic levels and (2) increased mean square lattice displacements (“heating”) due to electron–phonon energy transfer following femtosecond laser excitation. We showed in Ref. 35 that (1) can displacively launch LA phonon wavepackets. Process (2) proceeds with the characteristic timescale of electron–phonon coupling involving mainly LA modes and to a lesser degree optical phonon modes.¹² Using the known speed of sound for LA phonons of 4.6 nm/ps³⁴ together with the average 7-nm nanoparticle diameter, we find that the lattice expansion rise time of ∼0.8 ps in Fig. 1(d) corresponds to the strain wave traversing about half of a nanoparticle. This is a reasonable estimate since such strain waves originate at the nanoparticle perimeter and propagate inwards leaving an expanded lattice behind.³⁵ 

The unit cell volume increase by more than 0.4% [see Fig. 1(d)] could also significantly alter the electronic structure and affect electron–phonon coupling and energy transfer as observed previously for Ni.²⁹ We should, therefore, look for evidence in this direction for the FePt nanoparticle system. An obvious candidate is the fast timescale observed in the longitudinal phonon populations probed in Fig. 3(a) at the X_L-point. We determined the rise time constant to 0.47 ± 0.06 ps (see Table II); however, a saturation-like leveling off is observed at longer times closer to that also evident in the unit cell expansion data [Fig. 1(d)]. It is, therefore, conceivable that the two effects are linked. Inspection of the phonon dispersions close to the X-point in Figure 3(b) shows that two longitudinal modes can be detected in our UED geometry, one optical (LO) and one acoustic (LA) mode. They both reach relatively similar frequencies at the X-point. However, for the LO mode, only Fe atoms vibrate while the LA mode is characterized by only Pt vibrations [see Fig. 3(c)]. From our measurements alone, we cannot differentiate if one of the two modes is preferentially occupied; however, calculations in Ref. 27 favor a stronger electron–phonon coupling and, thus, mode occupation for the LO mode. Such an assignment with the correspondingly stronger Fe vibration amplitude would also agree with the observed initial increase in the mean square displacements especially for Fe atoms in Fig. 2(c) that occur on a similar timescale.

On longer timescales beyond 1 ps, we observe a slower increase in the mean square displacements in Fig. 2(c) as well as the population of a X-point phonon mode with transverse polarization in Fig. 4(a). Both could be related and are caused by a reduced electron–phonon coupling, possibly in part due to the unit cell expansion. While we cannot rule out from Fig. 3 that the LO mode becomes populated and contributes to the diffuse scattering signal, the larger density of states observed for the TO mode seems to give this mode the preference to contribute at least to the mean square displacement increase in this time range.

The observed unequal expansion of Fe and Pt sub-lattices may be traced back to the strain-induced lattice expansion of FePt nanoparticles. Strain waves propagate with the speed of sound for LA phonons. In FePt, this is about 4.6 nm/ps,³⁴ which reproduces the observed oscillation period of the volume-conserving breathing mode in Fig. 1(d) for 7–8 nm diameter particles. However, the first such oscillation cycle will be influenced and is in fact driven by electronic and magnetic stresses.^12,13,35

Such stresses are related to the non-equilibrium heating of electrons and reduction of the magnetic order that occurs on timescales of just a few 100 fs, i.e., within the transit time of LA strain waves through the nanoparticles. While a detailed modeling of these processes is beyond the scope of the present paper, it is straightforward to imagine that this can lead to an inhomogeneous lattice expansion across a nanoparticle around the 1 ps pump-probe delay time where the maximum a,b-axis lattice expansion is observed [see Figs. 1(d) and 4(g)]. The stronger average lattice expansion for the Pt sublattice (of 0.42%) compared to that of Fe (0.37%) is in line with a larger weight of Pt than Fe to the eigenvectors of this LA mode throughout the Brillouin zone [Fig. 3(c)].

In this work, we have developed a novel approach to study the element specific lattice dynamics of FePt nanoparticles. It is based on using the constructive and destructive interference effects present in multi-atomic lattices. We show that these effects can be utilized based on the simultaneous access to multiple Bragg peaks in ultrafast relativistic electron diffraction. We demonstrated that the method allows us to separate Fe and Pt mean square displacements and corroborated the assignment with diffuse electron diffraction measurements. We identified an inhomogeneous ultrafast lattice expansion that is larger for the Pt than the Fe sublattice, possibly driven by coherent longitudinal acoustic phonon wave packets.

The UED work was performed at the LCLS MeV-UED, which is operated as part of the Linac Coherent Light Source at the SLAC National Accelerator Laboratory, supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. D.T and H.A.D. acknowledge support by the Swedish Research Council (VR). H.A.D. acknowledges support by the Knut and Alice Wallenberg Foundation (KAW). X.J.W. and K.S.T. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Centre (CRC) 1242 (project No. 278162697, project C01 Structural Dynamics in Impulsively Excited Nanostructures). P.M.O. acknowledges support through the Knut and Alice Wallenberg Foundation (Grant Nos. 2022.0079 and 2023.336) and VR.

The authors have no conflicts to disclose.

Diego Turenne: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Igor Vaskivskyi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Klaus Sokolowski-Tinten: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Xijie J. Wang: Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Alexander H. Reid: Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Xiaozhe Shen: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Ming-Fu Lin: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Suji Park: Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Stephen Weathersby: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Michael Kozina: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Matthias C. Hoffmann: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Jian Wang: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Jakub Sebesta: Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Yukiko K. Takahashi: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Oscar Grånäs: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Peter M. Oppeneer: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Hermann A. Durr: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author.