Ultrafast demagnetizing fields from first principles

We examine the ultrafast demagnetization process of iron-based materials, namely, ${\mathrm{Fe}}_6$ clusters and bulk bcc Fe, with time-dependent spin-density functional theory (TDSDFT). The magnetization continuity equation is reformulated and the torque due to the spin-current divergence is written in terms of an effective time-dependent kinetic magnetic field, an object already introduced in the literature. Its time evolution, as extracted from the TDSDFT simulations, is identified as one of the main sources of the local out-of-equilibrium spin dynamics and it plays a major role in the demagnetization process in combination with the spin orbit interaction. Such demagnetization is particularly strong in hot spots where the kinetic torque is maximized. Finally, we find the rate of demagnetization in ${\mathrm{Fe}}_6$ to be strongly dependent on the direction of polarization of the exciting electric field and this can be linked to the out-of-equilibrium distribution of the kinetic field in two comparative cases.

Figure 1

(a) Typical electric-field pulse used to excite the ${\text{Fe}}_6$ cluster with the black arrow indicating the direction of the field. The fluence of this pulse is $580\mathrm{mJ}/{\mathrm{cm}}^2$. (b) Time evolution of the $z$ component of ${\mathbf{B}}_{\mathrm{kin}}$ and ${\mathbf{B}}_{\mathrm{X}}$ (exchange component of the field), with respect to their values at $t=0$ integrated over the system volume ${\mu }_B{\mathbf{B}}_{\mathrm{tot}}\left(t\right)={\mu }_B{\sum }_{\mathrm{I}}{\mathbf{B}}_{\mathrm{I}}\left(t\right)$. (c) Time evolution of the variation of the total magnetization $\mathrm{\Delta }{S}_z^{\mathrm{tot}}\left(t\right)={\sum }_{\mathrm{I}}\mathrm{\Delta }{S}_z^{\mathrm{I}}\left(t\right)$ with respect to its initial value. (d) Time evolution on atomic site 6 of the magnetization variation along $z$ and of the electron density variation with respect to its value at $t=0$ integrated inside a sphere of radius $R=0.9\text{Å}$.

Figure 2

Local spin dynamics of the ${\text{Fe}}_6$ cluster: (a) time evolution of the magnetization $S_z^{\mathrm{I}}\left(t\right)$ around the atomic centers and (b) time evolution of the $z$ component of ${\mathbf{B}}_{\mathrm{kin}}^{\mathrm{I}}\left(t\right)$. All the quantities are integrated inside a sphere of radius $R=0.9\text{Å}$ centered on the two atomic sites, where we have used $B_z^{\mathrm{I}}\left(t\right)=\frac{1}{|{\mathcal{S}}_{\mathrm{R}}^{\mathrm{I}}|}{\int }_{{\mathcal{S}}_{\mathrm{R}}^{\mathrm{I}}}{d}^{3}r{B}_z(\mathbf{r},t)$.

Figure 3

Contour plots of the time- and space-averaged (in the direction perpendicular to the plane spanned by atoms 1, 3, 5 and 6, as indicated on the plot) observables evaluated only within spheres of radius $R=1.0\text{Å}$ around each atom: (a) and (b) the temporal variation of the spin density $\mathrm{\Delta }{\mathbf{m}}^{(x,z)}(\mathbf{r},t)/\mathrm{\Delta }t$ for $\mathrm{\Delta }t=0.1\text{fs}$ and (c) and (d) the $x$ and $z$ components of the second term on the right-hand side of Eq. (22).

Figure 4

Evolution of the spin noncollinearity in the ${\text{Fe}}_6$ cluster. (a) Plot of $\mathrm{\Delta }\theta =\theta \left(t\right)-\theta \left(0\right)$ at site 1 for the ${\mathbf{B}}_{\mathrm{XC}}$ [or $\mathbf{S}\left(t\right)$] direction (black curve) and the ${\mathbf{B}}_{\mathrm{kin}}$ direction (red dashed curve). (b) Same quantities as in (a) but calculated at atomic site 6. (c) Plot of $\sqrt{{\overline{\mathcal{A}}}_1^2+{\overline{\mathcal{A}}}_2^2}$, where $\overline{\mathcal{A}}=\sqrt{{\sum }_{i=1}^3{\mathcal{A}}_i^2}$ and ${\mathcal{A}}_i$ is introduced in Eq. (23), compared to ${\overline{\mathcal{A}}}_3$ at atomic site 1. (d) Same quantities as in (c) but calculated at atomic site 6. The fields are measured within a sphere of radius $R=0.8\text{Å}$ centered on the atom center.

Figure 5

Global (a) spin and (b) energy variation in ${\mathrm{Fe}}_6$ for two different excitations differing only by the direction of polarization of the electric-field pulse. Cartoon of the cluster with labels of the relevant atoms and a reference frame are depicted as insets. The contour plots represent (c) and (e) the distribution of the angle between $\mathbf{m}$ and ${\mathbf{B}}_{\mathrm{kin}}$ in a plane through atoms 1, 3, 5, and 6 (as in Fig. 3) and (d) and (f) the perpendicular component of $\mathbf{m}$ with respect to ${\mathbf{B}}_{\mathrm{kin}}$ averaged over the time of the simulation, for the two different excitations (a), (c), and (d) $\mathbf{E}\parallel \mathbf{x}$ and (b), (e), and (f) $\mathbf{E}\parallel \mathbf{z}$.

Figure 6

Demagnetization of bcc Fe: (a) applied external electric field; (b) local value of $\mathrm{\Delta }{B}_{\mathrm{kin},z}^{\mathrm{I}}$ and $\mathrm{\Delta }{B}_{\mathrm{x},z}^{\mathrm{I}}$ around atom 1; (c) comparison between the value of the local magnetization $\mathrm{\Delta }{S}_{z,\mathrm{I}}\left(t\right)$ around atom 1 (black curve), the total magnetization integrated around the two sites $\mathrm{\Delta }{S}_{z,\mathrm{I}}\left(t\right)$ (red curve), and the total magnetization integrated inside the unit cell $\mathrm{\Delta }{S}_z\left(t\right)$ (green curve); and (d) local value of ${\sum }_{\mathrm{I}}\mathrm{\Delta }{B}_{\mathrm{kin},xy}^{\mathrm{I}}\left(t\right)$ (red curve) and of $\mathrm{\Delta }{S}_{xy}^{\mathrm{I}}\left(t\right)$, the noncollinear magnetization component for atom 1. All the local quantities are calculated inside a sphere of radius $R=1\text{Å}$ centered on site 1.