Lattice dynamics and ultrafast energy flow between electrons, spins, and phonons in a 3d ferromagnet

The ultrafast dynamics of magnetic order in a ferromagnet are governed by the interplay between electronic, magnetic, and lattice degrees of freedom. In order to obtain a microscopic understanding of ultrafast demagnetization, information on the response of all three subsystems is required. A consistent description of demagnetization and microscopic energy flow, however, is still missing. Here, we combine a femtosecond electron diffraction study of the ultrafast lattice response of nickel to laser excitation with ab initio calculations of the electron-phonon interaction and energy-conserving atomistic spin dynamics simulations. Our model is in agreement with the observed lattice dynamics and previously reported electron and magnetization dynamics. Our approach reveals that the spin system is the dominating heat sink in the initial few hundred femtoseconds and implies a transient nonthermal state of the spins. Our results provide a clear picture of the microscopic energy flow between electronic, magnetic, and lattice degrees of freedom on ultrafast timescales and constitute a foundation for theoretical descriptions of demagnetization that are consistent with the dynamics of all three subsystems.

Figure 1

Details of the femtosecond electron diffraction experiment. (a) Schematic diagram of the experiment. The electrons in the sample are excited using a visible (vis) or infrared laser pulse. The excited electrons transfer energy to the spins as well as to the lattice, depending on the respective coupling strengths (black arrows). The lattice response is probed using an ultrashort electron pulse, which diffracts off the sample. Diffraction patterns are recorded in transmission. (b) Radial average of the diffraction pattern (solid blue curve) before laser excitation. The dashed black curve is a fit to the data (static fit). The background contribution obtained from the static fit was subtracted. (c) Differences of the radial averages at several pump-probe delays (solid curves) compared with the radial average before laser excitation. The dashed black curves show the fits to the data (dynamic fit). The details of the fits are described in the text.

Figure 2

Time evolution of the mean-square displacement (MSD) and the lattice temperature after laser excitation with 2300-nm light. In this measurement, the absorbed energy density was 1230 $\mathrm{J}/{\mathrm{cm}}^3$. The black dots show the experimental data (exp.), and the black curve is a fit with a single exponential function, convolved with a Gaussian (170 fs FWHM) to account for the time resolution. The gray shaded area represents the standard errors of the data points, obtained from the fit of the radial averages. The inset shows the time constants (fit result) for different excitation densities. The error bars represent the standard errors of the single-exponential fits. The dashed red line is a linear fit to the data $[\tau =a(T_{\mathrm{final}}-295\mathrm{K})+b]$, with $a=0.336\pm 0.06$ and $b=360\pm 20\mathrm{fs}$. The errors of $a$ and $b$ are the standard errors from the fit.

Figure 3

Temperature dependence of model parameters and schematic diagrams of the models. (a) Heat capacities of the electron (orange) and lattice subsystems (gray) as well as the combined heat capacity of electrons and spins (blue). Electronic and lattice heat capacities are calculated based on the spin-resolved DFT results. Since the magnetic contribution to the heat capacity cannot be calculated using DFT, we use the combined heat capacity of electrons and spins determined from experiments [33] for the s-TTM. The magnetic contribution peaks at the Curie temperature $T_{\mathrm{c}}$ (vertical dashed line). The light blue shaded area corresponds to the error estimate. (b) Electron-phonon coupling parameter $G_{\mathrm{ep}}$ as a function of electron temperature, obtained from the spin-resolved DFT calculations. The sum of majority and minority $G_{\mathrm{ep}}$ is shown. (c), (d), and (e) are schematic diagrams of the s-TTM, the regular TTM, and the ASD simulations, respectively (see text for details).

Figure 4

(a)–(e) Comparison of the experimental results with the regular two-temperature model (TTM) and the modified two-temperature model with infinitely strong electron-spin coupling (s-TTM). The lattice temperature predicted by the regular TTM (solid orange curves) and the s-TTM (solid blue curves) is displayed together with the experimental data for different energy densities (2300 nm excitation wavelength). (c) also shows the evolution of the electronic temperatures for the two models (dashed curves). The gray areas represent the standard errors of the experimental data points. Both the TTM and the s-TTM results for the lattice temperature are convolved with a Gaussian (150 fs FWHM) to account for the pulse duration of the electron pulse. Note that this is less than the convolution width for the single-exponential fits of Fig. 2 because the pump pulse duration of $\sim 80$ fs is already included in the TTM and s-TTM. The displayed energy densities correspond to the absorbed energy densities of the s-TTM.

Figure 5

Atomistic spin dynamics (ASD) simulations and comparison with the experiment. (a)–(e) show the comparison between ASD simulations (solid red curves) and experiments (black dots) for different absorbed energy densities. The energy densities are slightly different compared with Fig. 4 due to the different spin heat capacity in the ASD simulations. In the simulations, the pump pulse has a FWHM of 80 fs. The results for the lattice temperature are convolved with a Gaussian (150 fs FWHM) to account for the pulse duration of the electron pulse. The electron-lattice interaction in the simulations is described based on spin-resolved DFT calculations, without fit parameters. (c) additionally displays the evolution of the electronic (solid blue curve) and the spin temperature (solid green curve). (f) displays the magnetization dynamics predicted by the ASD simulations (solid green curve), normalized to the magnetization at $T_{\mathrm{s}}=0$ K, as well as experimental results from Ref. [22] for the same absorbed energy density (dashed black curve). (g) compares the evolution of the electronic temperature in the ASD simulations (solid blue curve) to experimental data from Ref. [26] (black dots). In this case, we assumed a shorter pump pulse duration in the simulations (30 fs FWHM). Note that the sample geometry (film thickness and substrate) was different in the measurements from Refs. [22, 26]. The gray shaded areas in (a)–(e) represent the standard errors of the data points. The gray shaded area in (g) represents the errors of the experimental data points from Ref. [26].

Figure 6

Microscopic energy flow between electronic, magnetic, and lattice degrees of freedom according to the atomistic spin dynamics (ASD) simulations. (a) shows how the additional energy after laser excitation is distributed between the three different subsystems as a function of time. The black curve corresponds to the total additional energy in the material, demonstrating that energy is conserved in the model. (b) visualizes the energy flow during the demagnetization. There is a large energy flow from the electrons to the spin system as well as energy flow from electrons to the lattice. (c) shows the energy flow during remagnetization. Energy flows back from the spins to the electrons. In addition, energy flows from the electrons to the lattice, such that the electron as well as the spin energy decreases while the lattice energy increases.

Figure 7

Nonequilibrium spin dynamics. (a) Magnetization dynamics from the ASD simulations for several excitation densities of our experiments. (b) Energy content of the spin system as a function of pump-probe delay for several excitation densities (solid curves). For comparison, the dashed curves show the energy content of a hypothetical, thermalized spin system with the magnetization dynamics from the ASD simulations [shown in (a)].

Figure 8

Spin-polarized electronic density of states (DOS) and electron-phonon coupling of nickel, calculated using spin-resolved DFT. (a) Electronic DOS (e-DOS). The Fermi level is marked with a dashed black line. (b) Electron-phonon coupling parameter $G_{\mathrm{ep}}$ as a function of electron temperature. The majority (majo.) $G_{\mathrm{ep}}$ (blue), the minority (mino.) $G_{\mathrm{ep}}$ (red), and their sum (black) are displayed.

Figure 9

Experimental and theoretical results regarding electron and phonon thermalization. (a) Time constants of electron-lattice equilibration for different excitation wavelengths, obtained by single-exponential fits of the experimental data. The gray dots are the same data as in the inset of Fig. 2, shown again for comparison. The error bars represent the standard errors from the single-exponential fits. (b) Comparison of two-temperature model (TTM) results with nonthermal lattice model (NLM) results for two different fluences. Experimental data for 2300 nm excitation wavelength are also shown. The gray shaded areas represent the errors of the experimental data. The inset shows the Eliashberg function ${\alpha }^{2}F$ (solid curves, sum of majority and minority Eliashberg function) and the phonon DOS (vDOS, dashed curves) projected onto the three phonon branches (TA1, TA2, LA).

Figure 10

Atomistic spin dynamics (ASD) simulation results for equilibrium and nonequilibrium conditions. (a) Comparison between experimentally measured equilibrium heat capacity (black circles) and the simulated equilibrium heat capacity (yellow curve). The experimentally measured spin heat capacity corresponds to the heat capacity of electrons and spins [33] minus the electronic heat capacity from our DFT calculations. (b) Comparison between the experimentally measured magnetization curve as a function of temperature from Ref. [81] (black circles) and the simulation (yellow curve). (c) Experimentally measured lattice dynamics (black dots) and ASD simulation results (solid curves) for different values of the Gilbert damping parameter $\alpha$. The absorbed energy density is 540 $\mathrm{J}/{\mathrm{cm}}^3$. The gray shaded area represents the errors of the experimental data. (d) Magnetization dynamics predicted by the ASD simulations for different values of $\alpha$ (solid curves). The dashed black curve corresponds to the experimental magnetization dynamics for the same absorbed energy density of 540 $\mathrm{J}/{\mathrm{cm}}^3$ from Ref. [22]. (e) Evolution of the electronic temperature for different values of $\alpha$ according to the ASD simulations. Note that in addition to $\alpha$, the initial rise in the electronic temperature also depends on the pump pulse duration (here, 80 fs FWHM).