Abstract
Magnetostriction, the strain induced by a change in magnetization, is a universal effect in magnetic materials. Owing to the difficulty in unraveling its microscopic origin, it has been largely treated phenomenologically. Here, we show how the source of magnetostriction—the underlying magnetoelastic stress—can be separated in the time domain, opening the door for an atomistic understanding. Xray and electron diffraction are used to separate the subpicosecond spin and lattice responses of FePt nanoparticles. Following excitation with a 50fs laser pulse, timeresolved Xray diffraction demonstrates that magnetic order is lost within the nanoparticles with a time constant of 146 fs. Ultrafast electron diffraction reveals that this demagnetization is followed by an anisotropic, threedimensional lattice motion. Analysis of the size, speed, and symmetry of the lattice motion, together with ab initio calculations accounting for the stresses due to electrons and phonons, allow us to reveal the magnetoelastic stress generated by demagnetization.
Introduction
The functional properties of materials often depend on the detailed and subtle interplay of electronic, spin and lattice degrees of freedom^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. The complexity of this interplay can lead to a variety of technologically useful behaviors. These effects include anomalous thermal expansion^{1,10,11}, optical switching of magnetization^{11,12,13,14,15,16,17}, and superconductivity^{18}. Understanding the details of this electron–spin–lattice interplay remains one of the most challenging scientific problems in condensed matter physics. A particularly perplexing aspect is the strong coupling of magnetic spin to electron and lattice degrees of freedom observed in magnetically ordered metals. In such metals, spin order can be extinguished on a timescale of order of 100 fs^{4,5,6,7,8}. This fast timescale implies strong coupling of spin to the electronic system, seemingly not limited by the requirement of angular momentum conservation that necessitates the involvement of the lattice. This observation poses an interesting question, how strong or fast are the interactions that govern the spin–lattice effect of magnetostriction.
Iron platinum (FePt) alloys are a particularly rich example of materials that exhibit such electron–spin–lattice interplay. In the L1_{0} crystal phase, FePt is ferromagnetically ordered with extremely high magnetocrystalline anisotropy, making FePt the material of choice for nextgeneration magnetic storage media^{19,20}. In addition, the FePt system displays magnetostriction, leading to anomalous thermal expansion with a caxis contraction in the paramagnetic phase^{21}. The anomalous thermal expansion leads to a temperaturedependent change in the tetragonal lattice distortion, a quantity that is instrumental for establishing the high magnetocrystalline anisotropy in FePt^{22}. Recent reports have shown that FePt also exhibits an alloptical magnetization reversal when subjected to ultrashort pulses of circularly polarized optical light^{13,14}. However, the mechanism of how light switches magnetization in FePt is not yet understood. Therefore, understanding the interplay of electrons, spins, and lattice in FePt is a necessary step toward unraveling the mysteries of its rich functional properties.
Experimentally, lattice stress is not directly observable. Instead, measurements can be made of lattice strain—the physical movement of atoms in response to the stress. The strain response is often complex in solids where macroscopic effects can constrain the motion. For this reason, we choose nanoparticle grains of FePt held in a freestanding carbon matrix as the medium to understand the stress. Previous studies of magnetic structural dynamics were performed on continuous thin films deposited on substrates^{23,24,25}. Our use of unconstrained FePt singlecrystal nanoparticles allows us to study the full threedimensional lattice motion on the natural timescale of the strain propagation through the nanoparticles^{26,27}. This approach separates the individual contributions from electrons, spins, and phonons to lattice stress in FePt particles via their different symmetry properties and temporal onsets, as illustrated in Fig. 1b. By employing ab initio calculations to capture the nonequilibrium stresses of electrons and phonons, we show that a large magnetoelastic stress term related to magnetostrictive spin–lattice coupling must contribute to the observed strain^{3}. This term dominates the anomalous lattice expansion of FePt under the nonequilibrium condition during the first few ps following fs laser excitation.
Results
Spin and lattice response of FePt nanoparticles
In Fig. 1a, we show a TEM image of the freestanding sheet of FePt nanoparticles embedded in carbon, which is used in this study. The FePt nanoparticles are approximately cylindrically shaped with large surfaces that are free to respond to dynamic stresses along the surface normal. The particles are grown to have their magnetic easy axis, the crystallographic caxis, along the surface normal (see “Methods” section). We expect that a linear stress–strain surface response will dominate the motion for a time of τ = r_{np}/v_{s}, where r_{np} is the nanoparticle radius (mean r_{np} = 6.5 nm) and v_{s} is the speed of sound. A value of τ ≈ 3 ps is estimated for these FePt nanoparticles using the FePt sound velocity of 2.2 nm/ps^{22}.
We measured the spin and lattice responses separately on the same FePt–C samples (see “Methods” section), although using different experimental setups. Figure 2 shows the spin and lattice responses of a freestanding layer of FePt nanoparticles to fs laser excitation. We measured the spin response with ultrafast Xray resonant diffraction at the Fe L_{3}edge using circularly polarized Xrays^{15}. Figure 2b shows the measured Fe spin response following fs laser excitation (see schematic in Fig. 2a and “Methods” section). The data shown in Fig. 2b are the difference of two measurements of the resonant Xray diffraction intensity with the sample magnetized in opposite directions by applying an external magnetic field of ±0.4 T during the experiment. This field strength is significantly lower than the field required to switch the FePt nanoparticles when no laser pump is applied; the switching field without laser pump at room temperature is measured to be ±4 T^{28}. The large reduction of the switching field indicates that the magnetocrystalline anisotropy barrier between opposite magnetization directions is strongly reduced by laser excitation. The data in Fig. 2b are fitted with a double exponential^{6}, resulting in a time constant for demagnetization of 146 ± 15 fs, and a time constant of 16 ± 4 ps for the subsequent magnetization recovery. These results agree with optical fs laser experiments reported earlier^{6,7}. The fitted demagnetization amplitudes vs. laser pump fluence are shown in the inset of Fig. 2b.
We use ultrafast electron diffraction (UED) to directly measure the FePt nanoparticles’ lattice response^{29,30}. The nanoparticles were crystallographically aligned from their growth on a singlecrystal substrate; the substrate was subsequently removed (see “Methods” section). Therefore, the FePt diffraction pattern, shown in Fig. 2c, shows welldefined Bragg peaks. From timeresolved measurements of multiple Bragg diffraction peaks, the temporal evolutions of FePt lattice spacings along the a and c directions (the b direction evolves identically to the a direction) are extracted and shown as a function of time in Fig. 2d (see “Methods” section for details). We used a pump fluence of 5 mJ/cm^{2} that corresponding to a fs demagnetization of 30% of the initial magnetization (see inset of Fig. 2b). The measured change in lattice spacing along the c direction is clearly different from the change along the a and blattice directions. The lattice spacing in the a and b directions of FePt shows a rapid expansion, which peaks at ~3 ps. The expansion is followed by an oscillatory motion. The dynamics along the c direction is very different. The clattice spacing shows a rapid initial contraction on a similar timescale to the a and blattice expansions. This clattice motion also oscillates antiphase to the a and b directions. It then relaxes back toward the preexcitation value on a timescale of ~20 ps, whereas this does not occur in the a and b directions which show expansion.
Discussion
We are interested in understanding the dynamic response of the FePt lattice to the initial stresses generated during fs heating and demagnetization. To better visualize this initial lattice response, we replotted the FePt nanoparticle’s unit cell volume and its c–a lattice vectors for two laser fluences in Fig. 3. The nonequilibrium state of the system is modeled using thermal baths for the occurring degrees of freedom of the nanoparticle. To this end, we extend the threetemperature model of FePt^{31}, to include a temperature representing the carbon matrix (see “Methods” section). The fourtemperature model results are shown in Fig. 3a. The temporal evolution of the unit cell volume, V(t) = c(t).a(t)^{2} (note that a(t) = b(t)), is shown in Fig. 3b. This volume evolution shows that there is a clear crossover from rapidly increasing volume to a steadystate regime at around 3 ps; this crossover coincides with the first extremum of the oscillation in Fig. 2d. The 3 ps crossover can also be identified as the turning point of the a–c trajectory plot in Fig. 3b, which displays the relative lattice change in the a–c plane. The characteristic motions are clearly separated. Initially, the lattice system moves along a trajectory defined by the stresses at the nanoparticle surfaces toward a turning point at 3 ps time delay. This delay is defined by the time taken for the strain waves to reach the center of the nanoparticles. Following this time delay, the nanoparticles begin a damped ringing motion that brings them back to the quasiequilibrium state for c–a thermal expansion (shown in blue as a function of laser fluence in Fig. 3b).
To understand the initial expansion trajectory in Fig. 3b, we conducted ab initio calculations of the electron and phonon stress contributions^{32,33}. The electronic stress σ^{e} is evaluated from a calculation of the electronic Grüneisen parameter along different symmetry directions (see “Methods” section). We determined that the electronic stress is anisotropic with \(\sigma _{a,b}^e = 2\sigma _c^e\), but has a positive electronic pressure along all crystallographic axes. Next, to treat the phonon stress contribution, we calculated the outofequilibrium behavior of the lattice from the phonon populations, assuming independent phonon modes and including phonon–phonon interactions (see “Methods” section). By UED measurement of the Debye–Waller effect, we experimentally determine that these modes are populated exponentially with a time constant of 2.7 ps^{34}. The ab initio calculated modedependent contributions to the phonon stress are shown in Fig. 4a. As with the electronic stress, we find that the nonequilibrium phonon stress is highly anisotropic with \(\sigma _{a,b}^{\rm{ph}} = 7.3\sigma _c^{\rm{ph}}\), but it remains positive for all crystallographic directions. Consequently, to explain the strong negative strain observed along the caxis, a further stress contribution arising from the magnetic system must be considered.
Any ab initio calculation of the magnetic stress relies on the details of the coupling between spins and lattice, and, spins and electrons. For ultrafast demagnetization, these details remain unknown. Instead, we attempt a more robust approach. We determine the structural ground state of FePt in the ferromagnetic (FM) and paramagnetic (PM) phases using spinpolarized density functional theory in the local density approximation (see “Methods” section for details). The calculated total energies for constant volume are shown as a function of the c/a ratio in Fig. 4b, and are in good agreement with the experimental values. The difference between the two structural energy minima for FM and PM phases are characterized by a −0.53% clattice contraction and +0.25% alattice expansion. It is assumed that the magnetic stress vector is characterized by a straightline trajectory between these two phases, i.e., \(\sigma _{a,b}^m \approx  0.47\sigma _c^m\). This stress trajectory is represented by the green arrow in Fig. 3c, and shows the required negative stress along the caxis. We note that this would be the trajectory predicted for a 100% demagnetization, under static equilibrium conditions, and in our experiment we have neither 100% demagnetization nor equilibrium conditions.
To understand the relative sizes of the different stress contributions in the experiment, we can consider the FePt lattice response in terms of a simple coupled harmonic oscillator with a twodimensional displacement coordinate: Q_{i} (i = a, c)^{35}. The lattice motion in Fig. 2d is modeled by:
where β is a phenomenological damping constant and, Λ_{ij} is a matrix whose diagonal terms describe the frequency, ω, of the coherent breathing mode, while its offdiagonal terms represent the elastic coupling between j = ab, clattice strains. Equation (1) is characterized by two solutions, with symmetric (Q_{a,b} ~ Q_{c}) and antisymmetric (Q_{a,b} ~ −Q_{c}) eigenvectors and eigenfrequencies ω_{1} = ±(Λ_{aa} + 2Λ_{ac})^{1/2} and ω_{2} = ±(Λ_{aa})^{1/2}, respectively (we assume here that Λ_{aa} = Λ_{cc}). Figure 2d shows that the antisymmetric solution is favored. We note that the symmetry of the resulting lattice strain amplitude will depend on the symmetry of the driving stress terms as previously discussed. We describe the driving stresses in Eq. (1) as:
In the following paragraphs, we detail the form of the stresses assumed in the harmonic oscillator model
As discussed in the “Methods” section, the electronic stress \(\sigma _i^e\) is determined by the electronic temperature and electronic heat capacity of the system: \(\sigma _i^e =  \gamma _i^eC_e\left( {T_e} \right)\delta T_e\). The transient electronic temperature is determined by the fourtemperature model, while our ab initio electronic structure calculations determine the ratio of the coefficients to be: γ_{ab} = 2γ_{c}. The stress thus has the form: \(\sigma _i^e = \sigma ^e\left( {\overleftarrow {ab} + 0.5\overleftarrow {c} } \right)T_e(t)\delta T_e(t)\). A single scaling constant, σ^{e}, is used to characterize the transient electronic stress.
The magnetic stress, \(\sigma _i^m\), is predicted by our FM and PM structuralgroundstate calculations to have the trajectory \(2.12\sigma _{ab}^m =  \sigma _c^m\). We assume this to be the case, and, further, assume that the stress must be proportional to the square of the change in magnetization, due to time reversal symmetry. Therefore, the evolution of this stress, is determined by the measured change in the FePt magnetization, having the form: \(\sigma _i^m = \sigma ^m\left( 0.47\overleftarrow{ab}  \overleftarrow {c} \right)({\mathrm{\Delta }}M(t)/M_0)^2\), where M(t) is defined by the biexponential fit to Fig. 2a. Again, a single parameter, σ^{m}, is used to characterize the strength of this stress component.
The stresses due to phonons are dominated by the lowenergy modes due to the larger atomic displacement per unit energy associated with these modes. For this reason, the phonon stress is not adequately modeled by the lattice temperature—a measure of the energy in the lattice—but must also account for the phonon thermalization time. The attenuation of the Bragg reflections due to the Debye–Waller effect is measured to determine a lattice equilibration time. It is found that the meansquare atomic displacements increase with an exponential time constant τ_{l} of 2.7 ps. Again, we use our ab initio theory calculations to determine the relative strength of the phonon stresses in the a,b and c directions (\(\sigma _{ab}^{{\rm{ph}}} = 7.3\sigma _c^{{\rm{ph}}}\)). The phonon stress thus has the form: \(\sigma _i^{{\rm{ph}}} = \sigma ^{{\rm{ph}}}\left( \overleftarrow {ab} + 0.14\overleftarrow {c} \right)\left[ {1  \exp \left( {  \frac{t}{{\tau _l}}} \right)} \right].\)
Finally, we note that the carbon matrix must act to compress the nanoparticle in the a, b direction. The stress is assumed to evolve with the temperature of the carbon bath: \(\sigma _{ab}^{{\rm{carb}}} = \sigma ^{{\rm{carb}}}T_{{\rm{carb}}}(t)\). No stress from the carbon is considered in the c direction, as nothing restrains the motion in this direction.
Using the above stress models, a leastsquares nonlinear curve fitting is made to the experimental data. The best fit to the experiment is shown in Fig. 2d. The form of these stresses present and the fitting values obtained are presented in Table 1. The results indicate that a nonzero magnetoelastic stress does develop within the FePt nanoparticle on the timescale of the ultrafast demagnetization. It is primarily this stress that drives in the anisotropic lattice displacement, which proceeds as a strain wave moving inwards from the nanoparticple’s boundary.
Our results demonstrate how the stress contributions for different degrees of freedom can be seperated in the time domain by measuring nanometer sized particles. In particular, we uncovered the existence of the magnetostrictive driving force for strongly anisotropic lattice motion in FePt nanoparticles. Magnetoelastic stress builds up on the subps timescale, characteristic of ultrafast demagnetization. On the ps timescale, stress from transiently populated phonons takes over and results in a lattice tetragonality. Studies by Lukashev et al. have shown that a reduction in tetragonality favors a reduced FePt magnetocrystalline anisotropy barrier^{36}. We speculate that this reduced barrier results in reduced magnetic coercivity, which allows laserexcited FePt nanoparticles to reverse their magnetization in a magnetic field of 0.4 T. This observation provides a new avenue towards manipulating the magnetocrystalline anisotropy in future laserassisted magnetic data storage technologies and opens the possibility of using this ultrafast magnetostriction for new types of THz frequency magnetostrictive actuators.
Methods
Sample preparation and characterization
The single crystalline L1_{0} FePt grains were grown epitaxially onto a singlecrystal MgO(001) substrate by cosputtering Fe, Pt and C^{28}. This resulted in FePt nanoparticles of approximately cylindrical shape with heights of 10 nm and diameters in the range of 8–24 nm, with an average of 13 nm. The FePt nanoparticles form with a and b crystallographic directions oriented parallel to the MgO surface. The space inbetween the nanoparticles is filled with glassy carbon, which makes up 30% of the film’s volume. Following the sputtering process, the MgO substrate was chemically removed and the FePtC films were floated onto copper wire mesh grids with 100μmwide openings.
We performed electron diffraction from individual FePt nanoparticles using transmission electron microscopy (TEM). A highresolution TEM image is shown in Fig. 1a. Supplementary Fig. 1a shows an FePt singleparticle diffraction pattern. Fitting the peak shape, shown in Supplementary Fig. 1b, allows us to accurately determine the Bragg peak position. The room temperature equilibrium value of the tetragonal distortion is determined to be: c/a = 0.972 ± 0.003. Diffraction patterns were taken at two sample temperatures, 300 K and 500 K, allowing us to measure the static lattice expansion along c and aaxes. The (400)type Bragg peaks were used to probe the aaxis expansion. We deduce a + 0.3% aaxis lattice expansion between 300 and 500 K. The determination of the caxis lattice change is more difficult due to drifts of the sample positions at elevated temperatures. These data are plotted together with longdelay pump probe data in Supplementary Fig. 1c.
Resonant magnetic Xray scattering from granular FePt
The dynamic magnetic response of the FePt nanoparticles to laser pulses of 50 fs duration and 800 nm central wavelength was measured at the SXR instrument of the Linac Coherent Light Source (LCLS) at SLAC. The average optical absorption of these nanoparticles was previous calculated to be 21%^{16}. We probed the magnetization changes using circularly polarized Xrays, with the Xray energy tuned to the Fe 2p–3d resonance (708 eV photon energy)^{15}. Scattered Xrays were measured in transmission geometry by a pnCCD Xray camera (illustrated in Fig. 2a). Radial scattering profiles, after azimuthal angular averaging of the pnCCD patterns, display a peak due to the interparticle scattering. Supplementary Fig. 2 shows typical radial scattering profiles after azimuthal angular averaging of the pnCCD patterns and the demagnetization time constants at individual wavevectors, q. The dashed (solid) lines in Supplementary Fig. 2b correspond to magnetic diffraction profiles obtained at positive (negative) pump probe time delays. The time constants for ultrafast demagnetization, shown in Supplementary Fig. 2a, were extracted from double exponential fits to the time delay traces at the respective wavevectors, q.
Figure 2b displays the difference in Xray scattering yield with the sample magnetization aligned in externally applied magnetic fields of ±0.4 T. This difference signal is proportional to the average particle magnetization along the Xray incidence direction, switched by the external field. Note that 0.4 T is significantly lower than the static coercive field of 4 T in FePt nanoparticles, but allows a nearly 90% reversal of magnetization after laser excitation. We found that fs laser excitation, with the fluences shown in Fig. 2b, enabled a magnetization reversal similar to heatassisted magnetic recording. We also measured the ultrafast magnetization dynamics of MgO supported FePt nanoparticles and found identical results to the freestanding FePt granular films shown in Fig. 2b.
Separating caxis and a, baxis motion in UED data
The dynamic response of the FePt lattice was measured by ultrafast electron diffraction in a transmission geometry with 3 MeV electrons from the SLAC ultrafast electron diffraction source^{30}. The FePt nanoparticles were dynamically heated with a 1.55 eV, 50 fs optical laser pulse. To meet the Bragg condition for different lattice reflections, the film was rotated around axes normal to the probe beam. Due to geometrical restrictions, rotation angles were limited to 45^{o} from normal incidence. Measurements were made at different incidence angles to access Bragg peaks with projections along the caxis; these data are displayed in Supplementary Fig. 3. The time evolution data for Bragg peak positions (hkl) shows large differences for peaks with different “ l ” indexes, i.e., different projections along the outofplane (caxis) direction. Peaks with the same relative projection along the caxis, l^{2}/(h^{2}+k^{2}+l^{2}), showed the same temporal response. Measurements of multiple Bragg reflections with different caxis projections were used to extrapolate the motion along the caxis (Supplementary Fig. 3b). The extrapolated motion for the FePt caxis is shown in Fig. 2d, together with the directly measured aaxis (& b) motion.
Ab initio calculations of electronic and phononic stress
The ultrafast laser excitation of the FePt nanoparticles is followed by a dynamical response of the lattice, which can be characterized by magnetic, electronic and phononic stresses. To achieve a complete determination of the latter two, we used classical kinetic theory along with the Fermi’s golden rule of scattering theory to derive novel outofequilibrium rate equations for the electronic and phononic distributions^{37}. We solve those equations by using input from firstprinciples calculations, where the incorporation of a temperature and modedependent electronphonon coupling and anharmonicities through phonon–phonon interaction are essential to attain a full solution of the time evolution.
The electronphonon coupling was computed as response function within the density functional perturbation theory. We used a 16 × 16 × 16 kpoint grid in the Brillouin zone (BZ) for the selfconsistent calculations and a 4 × 4 × 4 grid for the phonon properties. Supplementary Fig. 4 shows the calculated modedependent phonon lifetimes due to electronphonon coupling along highsymmetry lines in the BZ. We can observe that the smallest lifetimes correspond to the optical phonon modes, while, the lifetime of acoustic phonon modes increases exponentially for kpoints approaching the Γ point. The computed lifetimes are used to describe the energy rate flowing from the electronic subsystem to the lattice after laser excitation; thus, they are needed to achieve a dynamical description of the system.
Subsequently, we obtained the induced stresses directly from the ab initio phononic and electronic distributions via a proportionality relation. We determined the contribution along the different realspace directions by the mode and branch dependence of the Grüneisein parameter and the phononic distribution.
Calculations of the phonon–phonon interaction
The phonon linewidths due to phonon–phonon interaction were determined using manybody perturbation theory in a thirdorder anharmonic Hamiltonian, which included up to threephonon scatterings^{38}. Under these conditions, the phonon linewidth computation reduces to knowing the thirdorder anharmonic interatomic force constants, which can be determined from density functional theory calculations. To obtain the anharmonic phonon properties, we performed calculations with the finite displacement method using the PHONOPY package^{39}, with ABINIT employed as the ab initio code to calculate the pairwise and cubic interatomic force constants. To evaluate these, we employed supercells consisting of 4 × 4 × 4 (128 atoms) and 3 × 3 × 3 (54 atoms) primitive cells, respectively. Due to the high symmetry of the tetragonal system, only 4 and 316 sets of frozen phonon structures were needed to calculate the dynamical matrices and the phonon linewidths. During post processing, the phonon frequencies and lifetime were sampled on a 100 × 100 × 100 qmesh. Calculated phonon lifetimes were typically larger than several ps.
Supplementary Fig. 5 shows the calculated modedependent inverse phonon–phonon lifetimes as a function of the phonon frequency, along with an average inverse lifetime. From this plot, we can see that the phonon lifetimes for lowenergy modes are of the order of 20 ps or larger, while only for LO modes the lifetimes are of the order of 10 ps or smaller. This calculation not only emphasizes the relatively short lifetime of the excited optical modes, which relax rapidly through decay into acoustic modes, it also provides us with quantitative, physical insights into the phonon thermalization process.
Electronic and phononic stresses
Instantaneous changes in the electron and phonon populations induced by a laser pulse or any other interaction induce a stress in the material leading, in the case of equilibrium distributions, to thermal expansion of the material. Microscopically, the induced stress can be written as:
where n_{e} and n_{L} are the electron and phonon populations, respectively, E_{k} and ℏω_{k} are the kdependent electron and phonon energies and γ_{e} and γ_{L}(k) are the corresponding electronic and phononic Grüneisen parameters, respectively^{40}.
Assuming that femtosecond electron–electron scatterings keep the electron distribution in a pseudoequilibrium state, we can assess the thermal pressure due to excited conduction electrons by considering the electronic stress,
where C_{e} is the specific heat capacity (per unit volume) of the electronic system with temperature T_{e}. Thermal effects can be taken into account^{41}, and the specific heat and electronic Grüneisen parameter can be computed. Since the specific heat can be assumed to be isotropic, it is the electronic Grüneisen parameter that determines the anisotropic behavior of the electronic stress. To determine the anisotropic behavior, we have calculated the electronic Grüneisen parameter along different symmetry directions using the equation:
where g(E_{F}) is the electronic density of states.
Having calculated the electronic Grüneisen parameter along the c and the aaxis, we find that its value along the aaxis is a factor ~2 larger than along the caxis (2.59 against 1.40). From these results, we infer that the electronic Grüneisen parameter has a nonnegligible contribution along the caxis and that a positive electronic thermal pressure along all crystallographic directions is expected.
For the lattice system, we compute the temporal and spatial evolution of the phonon population with a stateoftheart methodology that goes beyond the conventional twotemperature model. Here, the outofequilibrium behavior of the lattice system is correctly treated by assuming independent phonon modes and by including the phonon–phonon interaction. In this manner, the modedependent lattice stress can be computed for nonequilibrium phonon populations, giving the results shown in Fig. 4a. Here it is important to emphasize that the phonon stress has been calculated for all realspace directions, but only the projections along the a and caxis have been plotted. In addition, and to weight the contribution of the different strain waves, only strain waves with group velocities larger than 1000 m/s have been considered. The large anisotropy in the phononic system shown in Fig. 4a (phonon stress along the aaxis being seven times larger than the phonon stress along the caxis) has its origin in the combination of two main quantities. The first one is the large anisotropy of the Grüneisen parameter for phonon modes with group velocity larger than 1000 m/s, and the second one is the different change in phonon population for different modes. Thus, we find that phonon modes propagating along the caxis either have a small Grüneisen parameter, a small change in the phonon population or very small group velocities, and would therefore not contribute to the lattice expansion at short timescales.
Ab initio calculations of FePt groundstate properties
The groundstate properties of ferromagnetic FePt have been calculated using spinpolarized density functional theory (DFT) in the local density approximation (LDA) as implemented in the code ABINIT^{42}.
The electron–ionic core interaction on the valence electrons was represented by projectoraugmented wave potentials (PAW)^{43}, and the wave functions were expanded in plane waves with an energy cutoff at 29 Hartree and a cutoff for the double grid of 31 Hartree, which was sufficient to converge the total energy for a dense kpoint sampling. Reciprocal space sampling was performed using the MonkhorstPack scheme with a kpoint mesh of 32 × 32 × 32. After optimization, the lattice parameters for the FePt L1_{0} (P4/mmm) structure were found to be a = 3.857 Å and c = 3.761 Å, which is comparable to the experimental values a = 3.852 Å and c = 3.713 Å. The resulting value of c/a = 0.975 is in good agreement with experiment (c/a = 0.964). The local magnetic moments of the Fe atoms in the ordered FePt structures in the ferromagnetic phase are 3.065 μ_{B}.
For the paramagnetic (PM) state, we adopt the disordered local moments (DLM) approach, which states that paramagnetism can be modeled as a state where atomic magnetic moments are randomly oriented (i.e., noncollinear magnetism). The DLM approach can be simplified by considering only collinear magnetic moments when the spinorbit coupling is not taken into account. Hence, the problem of modeling paramagnetism becomes a problem of modeling random distributions of collinear spin components. It can be solved by using special quasirandom structures (SQS). A SQS is a specially designed supercell built of ideal lattice sites to mimic the most relevant pair and multisite correlation functions of a completely disordered phase (PM order in our case). As PM simulation cell, we adopted an extended lattice cell of 32 atoms (16 FePt unit cells). The disordered local moment approach (or SQS) provides a better description of the PM phase than the spinnonpolarized calculation, even without SOC, and gives good results for magnetostriction (even without SO)^{44}.
The thuscalculated PM phase shows an expansion of 0.25% in the a,bdirection and a reduction of −0.53% in the clattice direction when moving from the ferromagnetic to the paramagnetic phase. Thus, the computed c/aratio of paramagnetic FePt becomes 0.964. The expected change in the lattice constants between the FM and PM phases under static conditions is indicated by the dashed green arrow in Fig. 3c.
Fourtemperature model of nonequilibrium state
The microscopic threetemperature model (M3TM) has been developed to understand the nature and evolution of the nonequilibrium state in laserexcited ferromagnets^{45}. This model has be recently applied to FePt:Cu by Kimling et al^{31}. Here we adapt the model of Kimling et al. to our sample of FePt nanoparticles in a carbon matrix. We model the system as four coupled thermal baths: the FePt electronic bath; the FePt spin bath; the FePt phonon bath, and the carbon matrix bath. Each bath has its own associated temperature. Table 2 summarizes the parameters used in the model, the heat capacities of the four baths and relative coupling strengths between the baths. We note that the phonon bath is approximated by a constant heat capacity over the modeled range for simplicity; this is justified by a Debye model of C_{ph}, which suggests that only a negligible 3% change heat capacity would occur in this range. The four equations of the model are:
The model is solved numerically for an excitation of the electronic temperature with a duration of 50 fs. The solution obtained is shown in Fig. 3a.
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files. All other relevant data supporting the findings of this study are available on request.
Change history
07 March 2018
“The technical support from SLAC Accelerator Directorate, Technology Innovation Directorate, LCLS laser division and Test Facility Division is gratefully acknowledged. We thank S.P. Weathersby, R.K. Jobe, D. McCormick, A. Mitra, S. Carron and J. Corbett for their invaluable help and technical assistance. Research at SLAC was supported through the SIMES Institute which like the LCLS and SSRL user facilities is funded by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DEAC0276SF00515. The UED work was performed at SLAC MeVUED, which is supported in part by the DOE BES SUF Division Accelerator & Detector R&D program, the LCLS Facility, and SLAC under contract Nos. DEAC0205CH11231 and DEAC0276SF00515. Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DEAC0276SF00515.”
and
“Work at BNL was supported by DOE BES Materials Science and Engineering Division under Contract No: DEAC0298CH10886. J.C. would like to acknowledge the support from National Science Foundation Grant No. 1207252. E.E.F. would like to acknowledge support from the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES) under Award No. DESC0003678.”
This has been corrected in both the PDF and HTML versions of the Article.
16 October 2020
The original version of this Article was updated after publication following an error that resulted in the ORCID ID of L. Wu being also assigned to J. Wu.
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Acknowledgements
P.M, K.C., and P.M.O. acknowledge support from the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (grant No. 2015.0060), the RöntgenÅngström Cluster, the European Union’s Horizon 2020 Research and Innovation Programme under Grant agreement No. 737709 (FEMTOTERABYTE) and the Swedish National Infrastructure for Computing (SNIC).
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A.H.R., H.A.D. conceived the experiment, Y.K.T., E.E.F., O.H. grew and characterized the samples, A.H.R., X.S., T.C., R.K.L., T.V., N.H., R.C., J.W., X.W., H.A.D. performed the electron diffraction experiments and analysis, A.H.R., E.J., T.C., P.G., T.L., Z.C., D.H., G.D., W.F.S., H.O., J.S., H.A.D. performed the Xray experiments and analysis, A.H.R., J.L., Y.Z., J.C. performed the simulations, P.M., K.C., P.O. performed ab initio calculations, A.H.R., H.A.D. coordinated writing of the paper with contributions from all coauthors.
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Reid, A.H., Shen, X., Maldonado, P. et al. Beyond a phenomenological description of magnetostriction. Nat Commun 9, 388 (2018). https://doi.org/10.1038/s41467017027307
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