Attosecond dynamics of electron scattering by an absorbing layer

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 $\Delta\tau_{\rm R}$ 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 $\Delta\tau_{\rm R}$ 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 $\Delta\tau_{\rm R}$ vanishes. This picture changes substantially with the inclusion of an absorbing potential $-iV_{\rm i}$: As expected, for a sufficiently thick slab the reflection amplitude now tends to its asymptotic value for a semi-infinite crystal. Interestingly, for $V_{\rm i} \ne 0$, around the $T(E)$ maxima, the $\Delta\tau_{\rm R}(E)$ curve strongly deviates from $T(E)$, showing a narrow dip just at the $\Delta\tau_{\rm R}(E)$ maximum for $V_{\rm i}=0$. An analytical theory of this counterintuitive behavior is developed.