Ultrafast dynamics of electrons excited by femtosecond laser pulses: Spin polarization and spin-polarized currents

Laser radiation incident on a ferromagnetic sample produces excited electrons and currents whose spin polarization must not be aligned with the magnetization—an effect due to spin-orbit coupling that is ubiquitous in spin- and angle-resolved photoemission. In this paper, we report on a systematic investigation of the dynamics of spin polarization and spin-polarized currents produced by femtosecond laser pulses, modeled within our theoretical framework evolve. The spin polarization depends strongly on the properties of the laser pulse and on the sample composition, as shown by comparing results for Cu(100), Co(100), and a Co/Cu heterostructure. We find a transition from coherence before the laser pulse's maximum to incoherence thereafter. Moreover, the time dependence of the spin-polarization components induced by spin-orbit coupling differ significantly in Cu and Co: in Cu, we find long-period oscillations with tiny rapid modulations, whereas in Co prominent rapid oscillations with long-period ones are superimposed. The pronounced spatial dependences of the signals underline the importance of inhomogeneities; in particular, magnetic/nonmagnetic interfaces act as a source for ultrafast spin-polarization effects. Our investigation provides detailed insight into electron dynamics during and shortly after a femtosecond laser excitation.

Figure 1

Geometry of a Co/Cu heterostructure. The fcc film consists of 40 layers stacked in the $x$ direction, with 20 layers of Co atoms (cyan spheres) and 20 layers of Cu atoms (magenta spheres). The Co magnetic moments point along the $z$ direction (black arrows). The film is infinite in the $y$ and $z$ directions but finite in the $x$ direction. Sites with intense color forming a zigzag chain belong to one unit cell of the slab. A laser pulse impinges with a polar angle ${\vartheta }_{\mathrm{ph}}$ of $45{}^{\circ }$ within the $xz$ plane onto the sample.

Figure 2

Photoinduced spin polarization and currents for a Cu(100) sample excited by $p$-polarized light. (a) Site-averaged spin polarization $S^y\left(t\right)$ (black) and electric field of the laser pulse (orange; schematic). (b) Local spin polarization $s_l^y\left(t\right)$ for selected sites, as indicated. (c) Currents $j_{kl}\left(t\right)$ between neighboring sites $l\to k=l+1$ for selected site pairs as indicated. (d) Currents $j_{kl}\left(t\right)$ and (e) spin-resolved currents $j_{kl}^y\left(t\right)$; their magnitude is indicated by color bars with the same range (red is positive; blue is negative). Data in [(c)–(e)] are in arbitrary units. Vertical dashed lines at $t=0\mathrm{fs}$ mark the maximum of the laser pulse.

Figure 3

Photoinduced spin polarization and current of a Cu(100) film excited by $s$-polarized light. (a) $s_l^z\left(t\right)$ for selected sites as indicated. Sites 8 (18) and 31 (21) are equivalent. (b) Currents $j_{kl}\left(t\right)$ displayed as a color scale (red is positive, and blue is negative; in arbitrary units). Dashed arrows serve as guides to the eye. Vertical dashed lines at $t=0\mathrm{fs}$ indicate the maximum of the laser pulse.

Figure 4

Photoinduced spin polarization $S^{\mu }\left(t\right)$ and spin-resolved currents $j_{kl}^{\mu }\left(t\right)$ for a Cu(100) film excited by circularly polarized light with helicity ${\sigma }^{+}$ (top row) and for fcc Co(100) excited by $p$-polarized light (bottom row). (a) Site-averaged spin polarization $S^{\mu }\left(t\right)$ for Cu(100) ($\mu =x,y,z$). [(b)–(d)] Spin-resolved currents $j_{kl}^{\mu }\left(t\right)$ displayed as a color scale, as in Fig. 2. (e) $S^x\left(t\right)$ and $S^y\left(t\right)$ for Co(100). [(f)–(h)] The same as [(b)–(d)] using the same color scale. Dashed vertical lines indicate the maximum of the laser pulse at $t=0\mathrm{fs}$.

Figure 5

Laser-driven precession of the spin polarization in Co(100) excited by $p$-polarized light. The color scale visualizes the time evolution from $t=-20\mathrm{fs}$ (dark blue) to $t=0\mathrm{fs}$ (orange). (a) Correlation of $S^y\left(t\right)$ and $S^x\left(t\right)$ using data presented in Fig. 4. (b) and (c) show $S^x\left(t\right)$ and $S^y\left(t\right)$ versus the electric field $E_{\mathrm{ph}}\left(t\right)$ of the laser pulse, respectively.

Figure 6

Photoinduced spin polarization and currents of a Co/Cu(100) heterostructure excited by $p$-polarized light. (a) Component $S^x\left(t\right)$ of the site-averaged spin polarization (black) decomposed into that in the Co region (cyan) and that in the Cu region (magenta). The latter are normalized with respect to $N_{\mathrm{site}}$ [see Eq. (7)]. The maximum of the laser pulse at $t=0\mathrm{fs}$ is marked by the vertical dashed line. (b) The same as (a), but for $S^y\left(t\right)$. (c) Currents $j_{kl}\left(t\right)$ depicted as a color scale (red is positive; blue is negative). The Co/Cu interface is identified by the horizontal dashed line. (d) The same as (c), but for spin-resolved currents $j_{kl}^z\left(t\right)$. Arrows serve as a guide to the eye.