Electron Shockley surface states

Fig. 3. Electronic band structure of Cu projected on the (111) surface Brillouin zone. The cross-hatched region is the projected bulk continuum of states. The Shockley surface state derived from the s-p band (broken line) lies in the L-gap around the Fermi level.

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. 

One can readily verify that substitution of solution q.(2.156b) into Eq.(2.180) gives Eq.(2.159) for (7n. We will use this approadi to study again the equation for the Shockley surface states, but now in the limit of small coupling between the s and p electrons. The matrix elements of H are now ... 

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. 

Decay rates in a 3D electron gas model (Pegm, see Eq. (6.22)) of holes with the energy of the Shockley surface-state at P are also displayed. From Ref [37]. 

The Shockley surface states commonly found on many face-centered cubic (fee) (111) surfaces are occupied for ky = 0 (see also Chapter 5). The study of electron dynamics by time-resolved 2-PPE, however, is restricted to unoccupied states because the laser intensities required to induce a detectable change in the population of occupied states are close to the damage threshold of the sample. The Pd(lll) surface is one notable exception in which the Shockley surface state is found 1.35 eV above the Fermi energy Ep [75]. With time-resolved 2-PPE, the lifetime of this state was measured to 13 fs as shown in Figure 3.2.4.6 of Section 3.2.4.3... 

The effect of electron-defect scattering has been studied in photoemission by Kevan [71, 122]. For the Shockley surface state, an increase in the linewidth with... 

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. 

Little theoretical work has been done on the electronic structure of a solid with a free surface. The main contributions are those of Tamm (4), Shockley (5), Goodwin 6), Artmann (7), and Kouteck (5), and the main conclusion is that, in certain circumstances, surface states may exist in the gaps between the normal bands of crystal states. In this section we investigate the problem in the simplest way. The solid is represented by a straight chain of similar atoms, and its two ends represent the free surfaces. This one-dimensional model exhibits the essential features of the problem, and the results are easily generalized to three dimensions. 

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. 

By including terms to describe the rate of emission of electrons and holes from filled and empty surface states respectively, constraining the total number of surface states to Nt by N. + Ng = Nt and assuming that the states do not interact, the electron-hole recombination at. tile.surf ace has been analyzed -1— in analogy with Hall-Shockley-Read recombination. These methods are important in the study of semiconductor-... 

The 2D electron gas of the Shockley-type surface states plays also an important role in the substrate mediated interaction between adsorbed atoms. 

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. 

Problem 2.2. The electronic surface states are shown in Fig. 2.26 by dashed lines. The parabolic shape of the upper surface band implies that it is well described by the nearly-free electron approximation and hence this state is classified as a Shockley state. The lower surface state has the character of a surface resonance in the region where it intersects the electronic bulk band. An electronic transition between the surface states is possible if the upper state is unoccupied, i.e., it is located above the Fermi level. The minimum energy of such a transition is about 1.8 eV. 

Shockley states Electronic surface states which are obtained in a model implementing an inverted forbidden energy gap. 

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