Effect of wave-function localization on the time delay in photoemission from surfaces

Abstract

We investigate streaking time delays in the photoemission from a solid model surface as a function of the degree of localization of the initial-state wave functions. We consider a one-dimensional slab with lattice constant ${a}_{\mathrm{latt}}$ of attractive Gaussian-shaped core potentials of width $\ensuremath{\sigma}$. The parameter $\ensuremath{\sigma}/{a}_{\mathrm{latt}}$ thus controls the overlap between adjacent core potentials and localization of the electronic eigenfunctions on the lattice points. Small values of $\ensuremath{\sigma}/{a}_{\mathrm{latt}}\ensuremath{\ll}1$ yield lattice eigenfunctions that consist of localized atomic wave functions modulated by a ``Bloch-envelope'' function, while the eigenfunctions become delocalized for larger values of $\ensuremath{\sigma}/{a}_{\mathrm{latt}}\ensuremath{\gtrsim}0.4$. By numerically solving the time-dependent Schr\"odinger equation, we calculate photoemission spectra from which we deduce a characteristic bimodal shape of the band-averaged photoemission time delay: as the slab eigenfunctions become increasingly delocalized, the time delay quickly decreases near $\ensuremath{\sigma}/{a}_{\mathrm{latt}}=0.3$ from relatively large values below $\ensuremath{\sigma}/{a}_{\mathrm{latt}}\ensuremath{\sim}0.2$ to much smaller delays above $\ensuremath{\sigma}/{a}_{\mathrm{latt}}\ensuremath{\sim}0.4$. This change in wave-function localization facilitates the interpretation of a recently measured apparent relative time delay between the photoemission from core and conduction-band levels of a tungsten surface.