Theory of laser-induced ultrafast superdiffusive spin transport in layered heterostructures

Femtosecond laser excitation of a ferromagnetic material creates energetic spin-polarized electrons that have anomalous transport characteristics. We develop a semiclassical theory that is specifically dedicated to capture the transport of laser-excited nonequilibrium (NEQ) electrons. The randomly occurring multiple electronic collisions, which give rise to electron thermalization, are treated exactly and we include the generation of electron cascades due to inelastic electron-electron scatterings. The developed theory can, moreover, treat the presence of several different layers in the laser-irradiated material. The derived spin-dependent transport equation is solved numerically and it is shown that the hot NEQ electron spin transport occurs neither in the diffusive nor ballistic regime, it is superdiffusive. As the excited spin majority and minority electrons in typical transition-metal ferromagnets (e.g., Fe, Ni) have distinct, energy-dependent lifetimes, fast spin dynamics in the femtosecond (fs) regime is generated, causing effectively a spin current. As examples, we solve the resulting spin dynamics numerically for typical heterostructures, specifically, a ferromagnetic/nonmagnetic metallic layered junction (i.e., Fe/Al and Ni/Al) and a ferromagnetic/nonmagnetic insulator junction (Fe or Ni layer on a large band-gap insulator as, e.g., MgO). For the ferromagnetic/nonmagnetic metallic junction where the ferromagnetic layer is laser-excited, the computed spin dynamics shows that injection of a superdiffusive spin current in the nonmagnetic layer (Al) is achieved. The injected spin current consists of screened NEQ, mobile majority-spin electrons and is nearly 90$\%$ spin-polarized for Ni and about 65$\%$ for Fe. Concomitantly, a fast demagnetization of the ferromagnetic polarization in the femtosecond regime is driven. The analogy of the generated spin current to a superdiffusive spin Seebeck effect is surveyed.

Figure 1

Sketch of the laser-induced electronic motion on the femtosecond timescale. After the laser excitation (1) the hot NEQ electron starts its motion in a random direction. It travels on a straight trajectory until it suffers a scattering (2) and is bounced back in any random direction. The electron can also undergo an inelastic scattering (3) where another electron is ousted from bands below the Fermi energy. While traveling the electron can eventually cross the interface between two materials (4). (a) shows the geometry for Eqs. (1) and (2) and for the calculation of the quasi 1D integrated flux ${\varphi }^{1\mathrm{D}}$ in Eq. (3). (b) depicts the geometry to compute the integrated flux ${\varphi }^{3\text{D}}$ in Eq. (6).

Figure 2

Time evolution of the anomalous diffusion coefficient $d_w$ as defined in Eq. (17), numerically computed for a particle with constant velocity and an elastic lifetime $\tau =10$ fs (full curve) or $\tau =40$ fs (dashed curve).

Figure 3

A totally reflecting surface causes an electron to elastically bounced back (a). This situation is equivalent to the case (b) where another electron is created in a mirrored auxiliary system and emitted in a mirrored direction. Since we assumed the source to emit isotropically, the mirrored source behaves as the real one but it is located at the mirrored position.

Figure 4

Analytically computed position-time intensity profile of the density of excited first-generation only electrons $n_{\delta }^{\left[1\right]}$ (left-hand panels), the change of density of electrons below the Fermi energy $n_{<E_F,\delta }$ (central panels) and the their sum (right-hand panels) for the case of a Dirac $\delta$ source in time and space. Bright areas indicate high intensities, dark areas low intensities. The depth $z$ dependence is on the abscissae, expressed in units of $v\tau$, while the time $t$ dependence is on the ordinates in units of $\tau$. The top row (a) refers to the case of a homogeneous material. The middle row (b) displays the case of a homogeneous material with a perfectly reflecting surface (the position of which is highlighted by a yellow thick line) at $z_S=v\tau$. The bottom row (c) shows the case of a heterojunction made of two different metallic materials. The area for $z>0.7v\tau$ represents the second material with $v_2=0.5v$ and ${\tau }_2=0.5\tau$. The interface between the two materials is highlighted by a thin broken white line.

Figure 5

The ab initio calculated electron lifetimes and velocities in Ni and Fe, taken from Ref. 65. Blue triangles up refer to spin-majority and red down triangles to spin-minority carriers. The blue and red lines show the extrapolation of the data as used in the present work for spin-majority and minority carriers, respectively. The top panels depict the lifetimes (a) and velocities (b) for Ni, the bottom panels (c) and (d) for Fe.

Figure 6

Comparison of ultrafast laser-induced spin-dynamics in Ni on different substrates. Panels on the left-hand side show the case of 10 nm of Ni grown on Al, computed for excitation by a laser of 1.5 eV photon energy and an average number of photons absorbed per Ni atom of 0.1. Panels on the right-hand side display the case of a 10-nm Ni film grown on an insulating substrate like MgO. For each case, following the panels in a counterclockwise order and starting from the one at the top left corner, we show the majority-spin electron density, the minority-spin electron density, the magnetization density, three snapshots at three different times and, finally, the normalized MOKE signal change (top-right panel).

Figure 7

Comparison of ultrafast laser-induced spin dynamics in Fe on different substrates. Panels on the left-hand side show the case of 10 nm of Fe grown on Al, computed for excitation by a laser of 1.5 eV photon-energy and an average number of photons absorbed per Fe atom of 1.0. Panels on the right-hand side display the computed electron and spin dynamics for the case of a 10-nm Fe film grown on an insulating substrate like MgO. The quantities displayed in each panel are the same as given in the caption of Fig. 6.