Modeling by micromagnetic simulations

4ps to 7ps contribute to the measurement signal. This additional decay time for highly perturbed systems is also observed for thinner samples at moderate pump fluencies and high external fields. To model the time resolved spectra for various pump fluencies, the remagnetization process is reproduced using micromagnetic simulations using a oommf code[47].

4.3 Modeling by micromagnetic simulations

The micromagnetic simulations can be used to follow the magnetization dynam-ics in a given effective field using the Landau-Lifshitz-Gilbert equation. The sample characteristics are incorporated by taking the appropriate terms of the magnetic energy, such as the anisotropy constant, the demagnetization field and the Zeeman term, and by defining the Gilbert damping parameter. With an appropriate temporal and spatial resolution micromagnetic simulations can de-scribe the magnetization dynamics even in complicated structures. The details of the micromagnetic simulation principles can be found in [6, 42, 48, 49].

The demagnetization process itself is not included in the model. The highly non-equilibrium state resulting from the absorption of the pump pulse is represented

Figure 4.4: Micromagnetic simulation for a 50nm nickel film which is demagnetized 55%.

The evolution of the remagnetization is shown at 0ps, 0.7ps, 6.8ps and 16.7ps after excitation.

by the randomly oriented magnetic moments at the sample surface below a thickness equal to the optical penetration depth λ_opt. The excitation rate can be implemented into the simulations by introducing a starting demagnetization rate atτ=0. For τ >0 the simulations are driven in the corresponding effective field and no further heat transfer is introduced. Even though the temperatures, which the ferromagnet reaches during the absorption of the pump pulses, can approach the Curie temperature, the demagnetized surface layer describes the thermally induced demagnetization observed in the experiment well.

∆

τ

Figure 4.5: Micromagnetic simulation of the relaxation of theM^xmagnetization component for a 50nm nickel film with various demagnetization rates. The spectra are normalized and vertically shifted for clarity.

Fig. 4.4 shows the microscopical cut through the by-plane for the 50nm nickel sample at time delays of 0ps, 0.7ps, 6.8ps and 16.7ps after excitation by the pump pulse. The demagnetization profile at zero delay mirrors the penetra-tion profile of the laser light that decays exponentially on a length scale of λ_opt =25nm. The surface layer is characterized by the grainy profile and the gradual change of the demagnetization. As the time evolves, small domains form and spin waves emanate from the excited surface. At a time scale of 700fs, domains in the range of 3-6nm have formed already. The spin-wave front has penetrated the entire film thickness of 50nm after 6.8ps. The long wave-length excitations with a typical wavewave-length of 15-20nm dominate the upper part of the film, while the dominating wavelength is much smaller, on the order

4.3 Modeling by micromagnetic simulations

of 5nm, in the lower part. As the spin wave travels back and forth, standing waves are observed at a time delay of 16.7ps.

Fig. 4.5 plots the time resolved spectra of the magnetization component M_x versus delay time for various demagnetization rates. Similarly to the experi-mental results for different pump fluencies, the ferromagnetic order is recovered within a couple of ps at a moderate demagnetization rate. By increasing the demagnetization rate to more than 50%, much larger remagnetization times contribute to the signal. The time resolved spectra can be fitted with the dou-ble exponential function:

M

M₀ =ηexp (−τ /τ₁) +γexp (−τ /τ₂)

1 +γ , (4.1)

Where η denotes the degree of the demagnetization and γ denotes the contri-bution of each component. Tab. 4.2 lists the determined remagnetization times τ₁ and τ₂.

Table 4.2: Analysis of the restoration of theMxcomponent from the micromagnetic model using two characteristic remagnetization timescalesτ1 andτ1.

There is very good agreement with the experimentally determined values shown in Tab. 4.1. Onlyτ₁ is underestimated by the micromagnetic model. This could be due to the reduced exchange interaction during the highly non-equilibrium magnetization state shortly after excitation, which is not incorporated into the simulation.

To evaluate the remagnetization process and the additional slow remagnetiza-tion time, the magnetic modes can also be extracted from the micromagnetic simulations. Fig. 4.6 presents the Fourier spectrum of the relative change of the Mx magnetization component at different time delays. High spatial frequencies dominate the magnetization relaxation at τ =0ps. During the relaxation pro-cess, the center of the spectral weight moves towards lower spatial frequencies.

At 6.7ps the center is around kz/2π =0.1nm⁻¹, which corresponds to the spa-tial spin-wave period of 10nm. This frequency spectrum shift to lowerk-vectors for larger delay times describes the energy transfer from high-energy modes to low-energy modes within the spin-wave relaxation chain. In an ultrafast demag-netization experiment, the high-energy Stoner excitations are overpopulated by elementary relaxations of the hot electron system. These decay into short wave length spin excitations and gradually relax to lower spatial frequency excita-tions, towards the Kittelk= 0 precessional mode.

Figure 4.6: Cut through the micromagnetic simulation for a 50nm nickel film with a demagnetization of 55%. The corresponding Fourier transformation is presented as a function of the spatial frequency.

The slow recovery of magnetization for higher pump fluence is observed lately by Chantrell[50]. This approach is based on the Fokker-Plank equation which in-troduces a stochastic field term into the Landau-Lifshitz-Gilbert equation. The excitation mechanism is revealed by the increase in the electron temperature, which is directly related to the Gilbert damping parameter.