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

We investigated 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 [1]. We consider a 1D slab with lattice constant $a_{latt}$ of attractive Gaussian-shaped core potentials of width \textit{$\sigma $}. The parameter \textit{$\sigma $ / a}$_{latt}$ thus controls the localization of the electronic eigenfunctions. Small values of \textit{$\sigma $ / a}$_{latt} \quad <<$1 yield lattice eigenfunctions that consist of localized atomic wave functions modulated by a ``Bloch-envelope'' function, while the eigenfunctions become delocalized for larger values of \textit{$\sigma $ / a}$_{latt}$ $>$ 0$.$4. From calculated photoemission spectra we deduced a characteristic \textit{bimodal }shape of the band-averaged photoemission time delay: as the slab eigenfunctions become increasingly delocalized, the time delay quickly decreases near \textit{$\sigma $ / a}$_{latt}$ = 0$.$3. This change in wave-function localization facilitates the interpretation of a recently measured apparent relative time delay [2] between the photoemission from core and conduction-band levels of a tungsten surface [3]. \\[4pt] [1] C.-H. Zhang and U. Thumm, Phys. Rev. A \textbf{84}, 065403 (2011).\\[0pt] [2] C.-H. Zhang and U. Thumm, Phys. Rev. A \textbf{84}, 033401 (2011).\\[0pt] [3] C.-H. Zhang and U. Thumm , Phys. Rev. Lett. 102, 123601 (2009).