Role of spin-orbit interaction in the ultrafast demagnetization of small iron clusters

The ultrafast demagnetization of small iron clusters initiated by an intense optical excitation is studied from the time-dependent spin density functional theory (TDSDFT). In particular we investigate the effect of the spin-orbit interaction on the onset of the demagnetization process. It is found that demagnetization occurs locally, in the vicinity of the atomic sites, and the initial rate of spin loss, coherent with the laser field, is proportional to the square of the ionic spin-orbit coupling strength $\lambda$. A simplified quantum spin model comprising spin-orbit interaction and a time-dependent magnetic field is found to be the minimal model able to reproduce our ab initio results. The model predicts the ${\lambda }^2$ dependence of the onset rate of demagnetization when it is solved either analytically for the small $t$ regime, or numerically integrated in the time domain. Our findings are supported by additional TDSDFT simulations of clusters made of Co and Ni.

Figure 1

(a) Typical electric field pulses used to excite the ${\mathrm{Fe}}_6$ cluster (cartooned as inset) and their corresponding fluences $\frac{c{\varepsilon }_0}{2}\int E_x^2\left(t\right)dt$. Time evolution of the total TDSDFT energy (b), total spin (c), and total KS angular momentum (d) of the cluster when subjected to each of the pulses in panel (a) with corresponding color code on all panels.

Figure 2

Contour plots of the time-averaged and $z$-averaged observables evaluated only within spheres of radius 0.85 Å around each atom: (a) variation of the spin density $S^z\left(\vec{r}\right)$ with respect to the ground state, (b) difference of the latter and its counterpart in the case of no SOC, (c) the effective magnetic field $B_{\mathrm{eff}}^z\left(\vec{r}\right)$ in the ground state $\left(t=0\right)$, and (d) the variation of $B_{\mathrm{eff}}^z\left(t\right)$ with respect to the ground state. See text for details.

Figure 3

(a) Evolution of global spin expectation value for different values of $\alpha =0,1,2,4$ in Eq. (4). (b) Same as in panel (a) but having the GS spin subtracted and $\mathrm{\Delta }{S}_{\mathrm{tot}}^z$ values multiplied by $\eta ={({\alpha }_{\max }/\alpha )}^2$. Panels (c) and (d) show the corresponding trajectories of the averaged over the nonoverlapping atom-centered spheres $B_{\mathrm{eff}}^z\left(t\right)$ [see Eq (9)].

Figure 4

Trajectories for $S^z$ corresponding to the model Hamiltonian of Eq. (10) in the case of (a) having an initial state with spin up or spin down or (b) different values of the SOC by a factor $\alpha$, such that $\lambda \to \alpha \lambda$ in Eq. (10), for an initial spin up. (c) Similarly to Fig. 2, the latter trajectories rescaled by $1/{\alpha }^2$. The shaded area is a reference for the temporal profile of $\vec{B}=[0,0,B\left(t\right)]$. The initial orbital momentum state is a linear combination of $l^z=0,1,2$ states.

Figure 5

(a) TDSDFT trajectories of the variation of $S_{\mathrm{tot}}^z\left(t\right)$ with respect to the GS spin for three different clusters: ${\mathrm{Fe}}_6,{\mathrm{Ni}}_6$, and ${\mathrm{Co}}_6$, all sharing the same ${\mathrm{Fe}}_6$ geometry [12]. (b) Closer view into the coherent part of the trajectories where a parabolic decay $[y=A{(t-t_0)}^2+A_0]$ is fitted for each trajectory (dashed curves). (c) The fitting coefficients $A$ from panel (b) versus the effective atomic SOC from Eq. (16) for each material.