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

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(latt) of attractive Gaussian-shaped core potentials of width sigma. The parameter sigma/a(latt) thus controls the overlap between adjacent core potentials and localization of the electronic eigenfunctions on the lattice points. Small values of sigma/a(latt) << 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 sigma/a(latt) greater than or similar to 0.4. By numerically solving the time-dependent Schrodinger 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 sigma/a(latt) = 0.3 from relatively large values below sigma/a(latt) similar to 0.2 to much smaller delays above sigma/a(latt) similar to 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.