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 alatt of attractive Gaussian-shaped core potentials of width σ . The parameter σ/alatt thus controls the overlap between adjacent core potentials and localization of the electronic eigenfunctions on the lattice points. Small values of σ/alatt � 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 σ/alatt 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 σ/alatt = 0.3 from relatively large values below σ/alatt ∼ 0.2 to much smaller delays above σ/alatt ∼ 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.