Weak-coupling approach to the semi-infinite Hubbard model: non-locality of the self-energy

The Hubbard model on a semi-infinite three-dimensional lattice is considered to investigate electroncorrelation effects at single-crystal surfaces. The standard second-order perturbation theory in the interaction U is used to calculate the electronic self-energy and the quasi-particle density of states (QDOS) in the bulk as well as in the vicinity of the surface. Within a real-space representation we fully account for the non-locality of the self-energy and examine the quality of the local approximation. Numerical results are presented and discussed for the three different low-index surfaces of the simple-cubic lattice. Compared with the bulk significant differences can be found for the top-layer local self-energy, the imaginary part of which is energetically narrowed and has a reduced total weight. The non-local parts of the self-energy Σij(E) decrease with increasing distance between the sites i and j. At the surface and for the three-dimensional bulk their decrease is faster than for a two-dimensional lattice. For all surfaces considered the effects of the non-local parts of the self-energy on the QDOS are found to be qualitatively the same as for the bulk: The weight of the quasi-particle resonance at the Fermi energy is lowered while the high-energy charge-excitation peaks become more pronounced. The main structures in the layer-dependent spectra are already recovered within the local approximation; taking into account the nearest-neighbor non-local parts turns out to be an excellent approximation. Due to the reduced coordination number for sites at the very surface, the top-layer QDOS is narrowed. Contrary to the the free (U = 0) system, quasi-particle damping results in a comparatively weak layer dependence of the QDOS generally. Pronounced surface effects that are related to surface Friedel oscillations show up as a fine structure within the resonance around the Fermi level.