Metal surfaces electronic Shockley surface states

The reason for this behaviour is the presence of Shockley surface states [176] on the noble metal surfaces. On these surfaces, the Fermi energy is placed in a band gap for electrons propagating normal to the surface. This leads to exponentially decaying solutions both into the bulk and into the vacuum, and creates a two-dimensional electron gas at the surface. The gas can often be treated with very simple quantum mechanical models [177, 178], and much research has been done, especially with regards to Kondo physics [179, 180, 181]. There has also been attempts to do ab initio calculations of quantum corrals [182, 183], with in general excellent results. 

No "Jilt has so far been assumed that the semiconductor-electrolyte interphase does not contain either ions adsorbed specifically from the electrolyte or electrons corresponding to an additional system of electron levels. These surface states of electrons are formed either through adsorption (the Shockley levels) or through defects in the crystal lattice of the semiconductor (the Tamm levels). In this case—analogously as for specific adsorption on metal electrodes—three capacitors in series cannot be used to characterize the semiconductor-electrolyte interphase system and Eq. (4.5.6) must include a term describing the potential difference for surface states. 

After the first theoretical work of Tamm (1932), a series of theoretical papers on surface states were published (for example, Shockley, 1939 Goodwin, 1939 Heine, 1963). However, there has been no experimental evidence of the surface states for more than three decades. In 1966, Swanson and Grouser (1966, 1967) found a substantial deviation of the observed fie Id-emission spectroscopy on W(IOO) and Mo(lOO) from the theoretical prediction based on the Sommerfeld theory of metals. This experimental discovery has motivated a large amount of theoretical and subsequent experimental work in an attempt to explain its nature. After a few years, it became clear that the observed deviation from free-electron behavior of the W and Mo surfaces is an unambiguous exhibition of the surface states, which were predicted some three decades earlier. 

The required 2D nearly free electron gas is realized in Shockley type surface states of close-packed surfaces of noble metals. These states are located in narrow band gaps in the center of the first Brillouin zone of the (lll)-projected bulk band structure. The fact that their occupied bands are entirely in bulk band gaps separates the electrons in the 2D surface state from those in the underlying bulk. Only at structural defects, such as steps or adsorbates, is there an overlap of the wave functions, opening a finite transmission between the 2D and the 3D system. The fact that the surface state band is narrow implies extremely small Fermi wave vectors and consequently the Friedel oscillations of the surface state have a significantly larger wave length than those of bulk states. 

Since surface states with free-electron-like dispersion (Shockley type) have a low occupancy per surface unit cell and a low DOS at Ey, they are in general not considered to dominate the energetics of the surface, although - as discussed in Section 5.4.3 - situations exist where they can afiect the properties of the surface. On the contrary, metallic surface states derived from weakly dispersing bands (Tamm states) may have a high DOS at Ey and thus may influence the surface phase diagram considerably. As mentioned already in the case of quasi-2D states, transition metal surfaces are interesting in this respect, the question... 

The evaluation of the decay rate involves a double integral of the self-energy bracketed with the initial-state wave functions (see Eq. 6.2). For an efficient evaluation, a free-electron approximation parallel to the surface simpHfies the calculations considerably. Different effective masses may be used for bulk and surface states. The results of such calculations for the occupied Shockley surface state on noble metal (111) surfaces is shown in Table 6.2. 

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