Phase analysis of image states and surface states associated with nearly-free-electron band gaps

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Abstract
Surface-state occurrence on low-index faces of metals is analyzed with use of a combination of elementary multiple-reflection theory and elementary nearly-free-electron (NFE) theory. Five NFE gaps on Cu are considered in detail, three associated with the bulk L gap at Γ¯:Cu(111), Y¯:Cu(110), and X¯:Cu(001), and two associated with the bulk X gap at Γ¯:Cu(001) and X¯:Cu(110). The predicted surface states, of both the image-potential-induced and crystal-induced kinds, are all intimately related through the phases of the multiple-reflection approach, and their energies are in reasonable accord with photoemission and inverse-photoemission measurements. An empirical curve for the energy dependence of the barrier phase change spanning the changeover region between steplike and imagelike behavior is proposed. Effective masses, surface corrugation, and the effects of work-function variation are also discussed.