Tuning the laser-induced ultrafast demagnetization of transition metals

The ultrafast demagnetization (UFD) dynamics of itinerant ferromagnets is theoretically investigated as a function of the characteristics of the initial laser excitation. A many-body $pd$-band Hamiltonian is considered which takes into account hybridizations, Coulomb interactions, spin-orbit interactions, and the coupling to the laser field on the same electronic level. In this way, a fruitful connection is established between the nonadiabatic quantum dynamics and the well-known equilibrium statistical mechanics of itinerant-electron ferromagnets. The time evolution during and after the pulse absorption is determined exactly by performing numerical Lanczos propagations on a small cluster model with parameters appropriate for Ni. The most relevant laser parameters, namely, the fluence $F$, wave length $\lambda$, polarization $\widehat{\varepsilon }$, and pulse duration ${\tau }_p$ are varied systematically. The results show how $F$, $\widehat{\varepsilon }$, and ${\tau }_p$ allow one to control the total absorbed energy, the spectral distribution of the initial excitation, and the subsequent magnetization dynamics. The calculations show that reasonable changes in these parameters do not affect the UFD dynamics qualitatively and have only a minor influence on the timescale ${\tau }_{\text{dm}}$ which characterizes the initial demagnetization. In contrast, our model predicts that the degree of demagnetization $\mathrm{\Delta }{S}_z/S_z^0$ achieved for $t\gtrsim {\tau }_{\text{dm}}$ correlates well with the average number of electrons excited by the laser or average number of absorbed photons $n_{\text{ph}}$, which can be tuned by varying the fluence, spectral distribution and polarization of the laser pulse. The theoretical results are discussed by comparing them with available experiments. From a fundamental perspective, the robustness of the ultrafast demagnetization effect is theoretically demonstrated, as a phenomenon reflecting the intrinsic dynamics of the metallic $3d$ valence electrons. A wide variety of well-focused possibilities of tailoring the efficacy of the ultrafast demagnetization process is thereby opened.

Figure 1

Time dependence of the spin magnetization in an equilateral triangle with $N_e=4$, 5, and 7 electrons, after excitation using a linearly polarized $5\mathrm{fs}$ laser pulse with wave length $\lambda =1051$, 556, and 849 nm, respectively. The considered laser fluences $F$ are indicated together with the corresponding absorbed energies per atom $\mathrm{\Delta }E$. The left insets show the demagnetization times ${\tau }_{\text{dm}}$ as a function of $F$, as obtained from exponential fits to $S_z\left(t\right)$ given by the dashed curves in the main panels. The right insets show, for all considered values of $F$, the nearly identical time dependencies of the corresponding scaled spin magnetization $S_{\text{sc}}\left(t\right)=[S_z\left(t\right)-S_z^{\infty }]/\mathrm{\Delta }{S}_z$. In the bottom panel the amplitude $E\left(t\right)$ of the triggering electric field is illustrated.

Figure 2

Time dependence of the spin magnetization in a rhombus having $N_e=5$ valence electrons. The excitation at $t=0$ corresponds to a linearly polarized $5\mathrm{fs}$ laser pulse with wave length $\lambda =385\mathrm{nm}$. See the caption of Fig. 1.

Figure 3

Spectral density $D_{\mathrm{\Psi }}\left(\epsilon \right)$ of the excited many-body state $\left|\mathrm{\Psi }\right(t)\rangle$ after the laser-pulse passage ($t\ge 15$ fs) as a function of excitation energy $\epsilon =E-E_0$. Results are given for the equilateral triangle having $N_e=4$ and representative laser fluences $F$. The laser wave length is $\lambda =1051\mathrm{nm}$ ($\hslash \omega =1.18$ eV). The discrete spectral lines have been broadened with a finite width $\delta =20$ meV for the sake of clarity. The ground-state energy and the excitation energies corresponding to the absorption of one and two photons are indicated by the dashed lines.

Figure 4

Correlation between the laser-induced degree of demagnetization $\mathrm{\Delta }{S}_z/S_z^0$ at long times $t\gg {\tau }_{\text{dm}}$ and the absorbed energy $\mathrm{\Delta }E$ (right) or the average number of absorbed photons $n_{\text{ph}}=\mathrm{\Delta }E/\hslash \omega$ (left), where $\omega$ is the angular frequency of the exciting laser. The crosses are obtained from the exact calculated time evolution for (a) a rhombus with $N_e=5$ electrons, and an equilateral triangle having (b) $N_e=4$, (c) 5, and (d) 7 electrons. The laser fluence is always $F=40\text{mJ}/{\text{cm}}^2$, while the different photon energies $\hslash \omega$ are indicated in eV. The dashed straight lines on the left panels are fits to the approximate linear dependence of $\mathrm{\Delta }{S}_z/S_z^0$ on $n_{\text{ph}}$.

Figure 5

Time dependence of the average (a) spin moment $S_z$ and (b) orbital moment $L_z$ following the excitation with a ${\tau }_p=5\mathrm{fs}$ laser pulse having a wave length $\lambda =1051\mathrm{nm}$ and a polarization $\widehat{\varepsilon }$ which is in-plane linear ($\sigma =0$), right circular ($\sigma =+$) or left circular ($\sigma =-$). The full curves are obtained from the exact time evolution of the equilateral triangle model with $N_e=4$ electrons. The dashed curves in (a) are exponential fits to $S_z\left(t\right)$ with a common demagnetization time ${\tau }_{\text{dm}}=19\mathrm{fs}$. The inset in (a) shows the degree of demagnetization $\mathrm{\Delta }{S}_z/S_z^0$ as a function of the average number of absorbed photons $n_{\text{ph}}=\mathrm{\Delta }E/\hslash \omega$, together with the same linear approximation (dashed line) as the one found in Fig. 4 left.

Figure 6

Time dependence of the average (a) spin moment $S_z$ and (b) orbital moment $L_z$ corresponding to an excitation with a ${\tau }_p=20\mathrm{fs}$ laser pulse, having a wave length $\lambda =1051$ nm and a polarization $\widehat{\varepsilon }$ which is in-plane linear ($\sigma =0$), right circular ($\sigma =+$) and left circular ($\sigma =-$). See also the caption of Fig. 5.

Figure 7

Time dependence of the average spin moment $S_z$ in an equilateral triangle with $N_e=4$ electrons resulting from laser-pulse excitations having wave length $\lambda =1051\mathrm{nm}$ and pulse durations ${\tau }_p=1–50\mathrm{fs}$. The full curves are obtained from the exact time evolutions, while the dashed curves are the corresponding exponential fits to $S_z\left(t\right)$. The laser fluences are such that the average number of absorbed photons per atom is $n_{\text{ph}}/N_a=\mathrm{\Delta }E/\left(N_a\hslash \omega \right)=0.22$ for all ${\tau }_p$. The inset shows the demagnetization time ${\tau }_{\text{dm}}$ and the long-time demagnetization $\mathrm{\Delta }{S}_z/S_z^0$ as a function of ${\tau }_p$.