Ultrafast time-resolved structural changes of thin-film ferromagnetic metal heated with femtosecond optical pulses

The fundamental interactions between the electron, spin, and lattice degrees of freedom after femtosecond laser pulse excitation is of crucial importance for the determination of the properties of magnetic materials,¹ including both complex oxide materials and simple magnetic metals such as nickel and iron. Upon femtosecond laser excitation, the photon energy is initially deposited into the free electrons within the skin depth of metal. Subsequently, the hot free electrons equilibrate thermally within hundreds of femtoseconds through electron-electron scattering, followed by electron-phonon coupling that occurs on the order of picoseconds and eventually the electron and lattice systems reach a new equilibrium state.² For magnetic materials, however, in addition to the above-mentioned thermalization dynamics, the electron-spin and spin-lattice coupling are also involved and contribute to the demagnetization processes.

The ultrafast time-resolved magneto-optical Kerr effect (Tr-MOKE) experiments revealed that the ultrafast laser-induced demagnetization of nickel, heated above the Curie temperature, required a few hundred femtoseconds, suggesting that the spin dynamics occurred faster than the electron-lattice interaction³ and it triggered a large field of ultrafast magnetizations studied by Tr-MOKE.^4–8 As a recent example, the combined Tr-MOKE and time-and-angle resolved photoelectron spectroscopy (Tr-ARPES) reported that the out-of-equilibrium phase transition occurred within 20 fs during the ultrafast demagnetizations of nickel.⁹ In addition to these and studies based on optical probes,¹⁰ a femtosecond electron probe has also been applied for measuring the electronic Grüneisen parameter of polycrystalline nickel films.¹¹ 

The different combinations of pump and probe pulses selected from ultrafast optical, electron, x-ray, and THz pulses provide unprecedented means for excellent ultrafast time-resolved investigation of the diverse dynamics of magnetic materials.^12,13 In this paper, we present ultrafast time-resolved x-ray diffraction studies which reveal the transient structural changes of single crystal nickel (111) films illuminated by 100 fs, 800 nm laser pulses. We specifically chose a thick, 150 nm ferromagnetic nickel (111) single crystal rather than 10–20 nm thin crystals that are typically used in ultrafast studies. We revealed both the ultrafast heating within the skin depth and the heat transfer from the surface (skin) layer to the bulk of the crystal. Blast force formation, propagation of sonic waves, lattice compression, and coherent phonon generations are experimentally observed in this study. The femtosecond to picosecond temperature evolution of electron, spin, and lattice systems is also presented by numerical simulations based on a three-temperature model (TTM). It requires tens of picoseconds for the entire 150 nm thick sample to reach a thermal equilibrium state after femtosecond laser excitation.

The table-top ultrafast time-resolved x-ray diffraction has been detailed in our previous publications.¹⁴ Essentially, the subpicosecond hard x-ray pulses were generated by intense laser-plasma interactions^15,16 while the relative delay time between the pump laser pulse and probe x-ray pulse was precisely controlled by a linear translation stage. The 100 mJ, 100 fs, 800 nm, 10 Hz laser beam emitted from a Ti:sapphire laser system were split into two parts. About 80% the laser pulse energy was focused onto a moving copper wire, which was situated in a vacuum chamber, to generate subpicosecond, 8.04 keV hard x-ray pulses. The remaining part of the laser output was attenuated and focused onto the single crystal samples to initiate transient structural changes. The excitation area on the sample was about 2 mm × 3.5 mm. The probing hard x-ray was collimated by a 200 µm vertical slit and impinged onto the Ni (111) single crystal sample at a reflecting diffraction configuration. The diffracted x-ray beam was collected by a 2 K × 2 K x-ray CCD camera, and the time-dependent structural changes were extracted through the diffraction patterns recorded at each delay time. The 100 nm and 150 nm thick Ni (111) single crystal samples were grown on sapphire substrates by thermal evaporation at a base temperature of 500 °C and a deposition rate of 0.5 Å/s. The samples were inspected by a commercial CW x-ray machine which recorded a strong Ni (111) diffraction line at 22.30°, which suggested that the crystal was of very good quality. As shown in Fig. 1, the CCD detector is large enough to record the diffraction from both excited (S, signal) and non-excited (R, reference) areas of the sample. The signal and reference areas were vertically integrated to obtain their one-dimensional intensity plot and then fitted to a Gaussian function in order to determine the changes of the x-ray rocking curves as a function of laser excitation.

In the experiments described here, the excitation laser fluence was well below the damage threshold of single crystal Ni films. Because the laser excitation area was much larger than the probe x-ray spot size and significantly larger than the Ni (111) film thickness, a one-dimensional approximation along the film thickness direction was employed. Upon femtosecond laser irradiation, the photon energy was initially deposited into the free electrons within the skin depth of the Ni crystal. As the surface-layer electron temperature increased to hundreds or thousands of degrees, within the duration of the laser pulse, the energy began to be transferred from the surface electrons to the spin and lattice subsystems through electron-spin and electron-lattice interactions. Eventually, the entire crystal reached a new elevated equilibrium temperature.

The peak shifts of the 150 nm and 100 nm Ni (111) x-ray diffraction rocking curves are shown in Fig. 2, for the pump fluences at 13.3 mJ/cm² and 16.7 mJ/cm², respectively. For both sets of data, a few picoseconds after the laser excitation, a negative x-ray diffraction peak shift is clearly observable. After that, the peak shift raises up from the negative maximum to positive as shown in Fig. 2, followed by damping oscillations that last up to 100 ps. The negative peak shifts during the first few picoseconds after time zero (excitation) indicate that the distance between lattice planes decreased. Such, observed, contractions of the crystal planes are due to the generation of compression waves which were theoretically predicted and experimentally observed.^17–19 Owing to the hot electron temperature gradient, a blast force is generated within the crystal surface layer, which propagates through the bulk crystal. As a consequence, the crystal surface skin layer(s) are initially denser, closer together immediately after laser irradiation, which corresponded to a reduced lattice plane distance. The contraction begins as the hot electron temperature gradient is established. In addition, owing to the fact that the contraction is within the about 10 nm skin depth of the crystal while the x-ray pulse probes the entire 100 nm thick samples, the observed negative shift accounts for less than ∼15% of the entire shift of the x-ray diffraction signal; therefore, as shown in Fig. 2, the negative shift is proportionally small. After the initial contraction, the electron-spin, electron-lattice, and spin-lattice coupling contributes to the establishment of a new thermal equilibrium state for the entire crystal. Therefore, the thermal energy transfers from within the skin depth to the bulk and consequently heats the entire bulk of the single crystal. The elevated crystal temperature induces thermal expansion, as illustrated by the increased peak shifting of the x-ray diffraction rocking curve in Fig. 2. In addition, the damping oscillations are also distinct from the diffraction peak shift, which indicates that coherent phonons are generated. Similar oscillations have been widely observed previously in time-resolved diffraction experiments that employ electron and x-ray pulses. Such studies include the ultrafast time-resolved electron diffraction studies on nanometer thick Al, Ni crystals^11,20 and the time-resolved x-ray diffraction studies on single crystals such as Au, Cu, and Ge.^21–23 In these previous studies, the time periods of the observed damping oscillations are determined by the thickness of the crystal and are independent on the pump laser fluences, polarization directions, and incident angles. The damping oscillations discussed here are generally explained by the Fermi-Pasta-Ulam anharmonic chain model^24,25 that may be simplified to a one-dimensional standing wave that travels between the surface and bottom of the single crystal. Thus, the oscillation period, T, can be calculated using the longitudinal velocity of the acoustic wave, ν, in the solid state nickel crystal through T = 2L/ν, where L is the crystal thickness. Given that the longitudinal sound velocity in nickel is 6040 m/s,²⁶ the oscillation periods predicted by the standing wave mode are 33 ps and 50 ps, which agree well with the experimentally observed periods of 35 ps and 46 ps for the 100 nm and 150 nm Ni (111) films, respectively, used in current experiments. The recorded damping of those oscillations is attributed to the dissipation of the energy into the sapphire substrates. The damping requires about 125 ps while the diffraction peak shifts remain unchanged up to the longest delay time, 225 ps, of our experimental observations.

Differentiating on both sides of the first-order Bragg diffraction equation, 2d sin θ = λ, where d, θ, and λ are the lattice plane distance, Bragg angle, and x-ray wavelength, respectively, the following relation is obtained: Δd/d = −Δθ/tan θ. The change in the lattice plane distance, Δd, is associated with the change in the crystal temperature. Therefore, denoting the thermal expansion coefficient of nickel as α, the lattice temperature change, ΔT_l, was determined by the following relation:

Assuming a purely thermal heating upon the femtosecond laser excitation, the lattice temperature change depends linearly on the laser intensity absorbed, I_abs, and the shifts of the x-ray rocking curve,

Increasing the pump laser intensity while keeping it below the damage threshold, it is expected that the x-ray diffraction peak will shift linearly. As the excitation laser intensity was increased from 3.3 mJ/cm² to 13.3 mJ/cm², at the representative delay time of 180 ps, the diffraction peak shifted linearly as shown in Fig. 3 for the 150 nm thick Ni (111) crystal sample. The diffraction peak shifts reach almost a plateau at 180 ps. The linear relation with a slope of 0.92 shown by the log-log plot in Fig. 3 agrees well with Eq. (2) and indicates that the nickel single crystal heated by an 800 nm femtosecond laser can be described as a pure thermal process.

The laser heating process of the Ni (111) single crystal is simulated using a three-temperature model, which describes the temporal evolutions of electron, spin, and lattice subsystems. As discussed before, the three-dimensional laser excitation can be simplified as a one-dimensional energy flow along the thickness direction of the crystal bulk. Therefore, the time dependent electron, spin, and lattice temperature at a given depth, z, under the sample surface are denoted as T_e(z, t), T_s(z, t), and T_l(z, t), respectively. The femtosecond pump laser pulse, S(z, t), serves as the femtosecond heat source that initially deposits the energy onto electrons. The evolution of the electron, spin, and lattice temperature is described by the following three coupled equations:

Because the laser excitation fluence is well below the damage threshold of the Ni (111) single crystal, the electron-lattice coupling constant, electron-spin coupling constant, and spin-lattice coupling are taken as constants g_el = 8.0 × 10¹⁷ W m⁻³ K⁻¹, g_es = 6.0 × 10¹⁷ W m⁻³K⁻¹, and g_sl = 0.3 × 10¹⁷ W m⁻³ K⁻¹, respectively.³ The electron heat capacity depends linearly on the temperature for such fluences: C_e = γT_e, in which γ = 6.0 × 10³ J m⁻³ K⁻² and the electron thermal conductivity, κ_e, was obtained from the Drude model.^27–29 The spin and lattice heat capacity, C_e and C_e, respectively, were obtained from previous studies,^30–33 and a fifth order Padé approximation may be used to express the data in analytical forms during the numerical simulation.²¹ 

The two-dimensional plots that represent the spatiotempral evolution of the electron, spin and lattice temperature are presented in Fig. 4, which also includes a line plot of the temperature evolutions within the surface layers of the 150-nm Ni (111) single crystal film. After the 13.3 mJ/cm² laser irradiation, the electron temperature reaches over 700 K within ∼1 ps, which is followed by the thermalization between electron, spin, and lattice subsystems. Subsequently, about 25 ps are required for the entire, 150 nm thick, single crystal sample to equilibrate through electron-electron, spin-electron, spin-lattice, and phonon-phonon interactions.

Because a thick sample (150 nm) was used in this study, the recorded x-ray data actually consist of two stages: the ultrafast heating within the surface layer (skin depth) and the heat transfer from the surface layer to the bulk of the crystal. For the surface layer, the lattice heating is fast because the laser excites homogeneously and simultaneously the entire skin and the electron-phonon coupling occurs within a few picoseconds, which is depicted in Fig. 4(d) that describes the surface layer temperature evolution of the crystal. During this short period of time, the bulk of the crystal remains, essentially, at the base temperature and contributes to a “DC component” to the observed change of the x-ray diffraction lines. Therefore, a time constant of ∼5.7 ps, obtained from exponential fitting of the diffraction peak shifts in Fig. 2, is dominated by the temperature change within the surface layer. This time constant generally agrees with the surface layer heating up time revealed in the simulation. For the heat transfer from the surface layer to the bulk of the crystal, the surface layer can be considered as a heat reservoir from which the energy is carried into the bulk of the crystal through thermal diffusion. This energy transfer takes longer time than the couplings within the surface layer. Therefore, as revealed in the simulation, it takes more than 25 ps before the entire crystal reaches an equilibrium temperature. In the simulation, the equilibrium temperature of the entire sample after laser excitation is about 48 K, which agrees with the value of 50 K calculated by applying the linear thermal expansion coefficient of 13 × 10⁻⁶ K⁻¹ of nickel to the diffraction data shown in Fig. 2. However, the linear thermal expansion coefficient of the nanocrystal and its bulk may be different.³⁴ Because this three-temperature model describes only the temperature evolution, not the motion of each individual atom, the lattice oscillations observed in our experiments are not included in the simulation.

Utilizing subpicosecond 8.04 keV hard x-ray pulses generated from a table-top system, the transient lattice motion of single crystal nickel (111) films excited with femtosecond laser pulses was obtained by analyzing the x-ray diffraction pattern recorded at each delay time. Coherent phonons were observed in nickel single crystal films of different thicknesses and the experimentally observed oscillation periods agree well with the one-dimensional standing wave model. The lattice contractions due to the formation of the blast force were also experimentally observed and the developments of sonic waves within the film thickness were monitored in real-time. Owing to the fact that a thick sample was used in our studies, in addition to the ultrafast heating within the skin depth, the heat transfer process from within the skin depth to the bulk of the crystal proceeding via thermal diffusion is also discussed. After femtosecond laser excitation, the spatiotempral distribution of electron, spin, and lattice temperatures within the 150 nm thick nickel single crystal was also simulated using the three-temperature model (TTM).

We gratefully acknowledge the partial support from the Welch Foundation (Grant No. 1501928), the Air Force Office of Scientific Research (Grant No. FA9550-18-1-0100), and the Texas A&M University TEES funds.