Wave-diffusion theory of spin transport in metals after ultrashort-pulse excitation

Spin and charge-current dynamics after ultrafast spin-polarized excitation in a normal metal are studied theoretically using macroscopic wave-diffusion equations for spin-resolved carrier and current densities. It is shown analytically how this set of equations yields a unified description of ballistic and diffusive properties of spin and charge transport, including the intermediate regime between these two limits. In the framework of the wave-diffusion approach, ultrafast excitation of spin-polarized carriers in thin gold films is modeled assuming slightly spin-dependent momentum relaxation times along with standard parameters (Fermi velocity, spin and momentum relaxation times). The unified treatment of diffusive and ballistic transport yields robust signatures in the spin and charge dynamics that are in qualitative agreement with recent experimental results [Melnikov et al., Phys. Rev. Lett. 107, 076601 (2011)]. The influence of boundary effects on the temporal signatures of spin transport is also studied.

Figure 1

Mean-square displacement ${\Delta }_z\left(t\right)$ of the spin density vs. time: transition from ballistic to diffusive spin transport.

Figure 2

Same as Fig. 1 on a longer time scale.

Figure 3

Schematic setup of the optical pump-probe experiment on a Fe/Au heterostructure.

Figure 4

Time evolution of the density $n_{+}(z_0,t)$ of majority electrons (a) and the total spin density $n_m(z_0,t)$ (b) at positions $z_0=25$ (solid), 50 (dashed), 100 (dotted), and 150 nm (dash-dotted) of an infinitely thick layer. The excitation current pulse $J_s$ is also shown in arbitrary units (thin red line).

Figure 5

Time evolution of the spin density of majority electrons $n_{+}(z=L,t)$ (a) and the total spin density $n_m(z=L,t)$ (b) for Au layer thicknesses of $L=25$ (solid), 50 (dashed), and 100 nm (dotted).

Figure 6

Snapshots of the spin-current density $J_{+}(z,t)$ (a) and of the spin density $n_{+}(z,t)$ (b) as a function of $z$ at times $t=0$, 50, 100, and 150 fs for a 100-nm gold slab.

Figure 7

Time evolution of the charge density $n(z,t)$ for layer thicknesses of $L=25$ (solid), 50 (dashed), 100 (dotted), and 150 nm (dash-dotted line).

Figure 8

Scaled dynamics of the density $n_{+}(z,t)$ of majority electrons at points $z=50$ (dashed), 100 (dotted), 150 (dash-dotted), and 300 nm (solid) of an infinitely thick layer.