Attosecond dynamics of electron scattering by an absorbing layer

Abstract

Attosecond dynamics of electron reflection from a thin film is studied based on a one-dimensional jellium model. Following the Eisenbud–Wigner–Smith concept, the reflection time delay $\Delta {\tau }_{\text{r}}$ is calculated as the energy derivative of the phase of the complex reflection amplitude $r$. For a purely elastic scattering by a jellium slab of a finite thickness $d$ the transmission probability $T$ oscillates with the momentum $K$ in the solid with a period $\pi /d$, and $\Delta {\tau }_{\text{r}}$ closely follows these oscillations. The reflection delay averaged over an energy interval grows with $d$, but in the limit of $d\to \infty$ the amplitude $r$ becomes real, so $\Delta {\tau }_{\text{r}}$ vanishes. This picture changes substantially with the inclusion of an absorbing potential $-i{V}_i$: As expected, for a sufficiently thick slab the reflection amplitude now tends to its asymptotic value for a semi-infinite crystal. Interestingly, for $V_i\ne 0$, around the $T\left(E\right)$ maxima, the $\Delta {\tau }_{\text{r}}\left(E\right)$ curve strongly deviates from $T\left(E\right)$, showing a narrow dip just at the $\Delta {\tau }_{\text{r}}\left(E\right)$ maximum for $V_i=0$. An analytical theory of this counterintuitive behavior is developed.