Phase contribution of image potential on empty quantum well states in Pb Islands on the Cu(111) surface

We use scanning tunneling spectroscopy to explore the quantum well states in the Pb islands grown on a Cu(111) surface. Our observation demonstrates that the empty quantum well states, whose energy levels lie beyond 1.2 eV above the Fermi level, are significantly affected by the image potential. As the quantum number increases, the energy separation between adjacent states is shrinking rather than widening, contrary to the prediction for a square potential well. By simply introducing a phase factor to reckon the effect of the image potential, the shrinking behavior of the energy separation can be reasonably explained with the phase accumulation model. The model also reveals that there exists a quantum regime above the Pb surface in which the image potential is vanished. Moreover, the quasi-image-potential state in the tunneling gap is quenched because of the existence of the quantum well states.