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.

Figure 8 shows the onsets of noble metal Shockley surface states recorded with STM at low temperatures [58]. The widths of the onsets are inversely proportional to the lifetimes of the holes at the band minimum of the surface states. Intraband transitions within the 2D surface state... 

Figure 9.11 Depiction of an idealised chemical passivation of Shockley surface states. As the bonding strength between the surface and chemical group increases, the energies of the antibonding and bonding orbitals are shifted outside the bandgap.

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. 

Elementary quantum chemistry of the Shockley surface state. 

We will now consider the open infinite chain, simulating surface formation. If the ideal hybridization condition Eq.(2.128) is satisfied, dangling orbitals with orbital energies a = as = ap appear at the end of the chain. These are the Shockley surface states, because they are strictly localized on atoms 1 and N, The solution for the general case is found by solving the secular equations for the open chain. Equations (2.123) have to be solved with bonding conditions ... 

Figure 2.35. Shockley surface state or dangling bond in the almost ideal hybridisation limit.

Figure 2.37. Local densities of stalesl in several layers with respect to the surface as computed for Si (111). Note the dangling bond-state density (Shockley surface state) in the bulk bandgap.

We will first illustrate the use of the Green s functions to compute the LDOS pi E) of the open chain with only one atomic orbital per atom. Then we will introduce a convenient tool to work with Green s function, the resolvent method. It will be used extensively in the discussion of chemisorption. In this section we utilize it to describe Shockley surface states. 

THE RESOLVENT METHOD. APPLICATION TO SHOCKLEY SURFACE STATE. 

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 ... 

Figure 5.15 Squared wave functions for the Shockley surface state (n = 0) and the n = 1 image-potential surface state. The Shockley surface state is depicted for both a step and an image potential.

Figure 5.20 Sketch of the wave functions for the py (top) and the (bottom) Shockley surface state in the Y band gap of Cu(llO). The color change from purple to green symbolizes a change in sign, py is a-bonding from row to row and r-bonding along the rows, while pzs is tt-antibonding from row to row, and 7r-bonding along the rows.

Figure 5.21 Inverse photoemission spectra, showing the downshift of an unoccupied Shockley surface state at about 2.3 eV above Ef in the Y band gap of Ni(llO) induced by the adsorption of H. The peak just above Ef is due to the unoccupied uppermost part of...

An example of surface-state modification by adsorption as well as by reconstruction is depicted in Figure 5.25. In this diagram, the energy of the occupied Cu(llO) Shockley surface state close to the bottom of the projected bulk band gap at Y, as measured by PES, is depicted as a function of Na coverage. For adsorption at... 

Figure 5.25 Na-induced shift of the occupied Shockley surface-state band of Cu(llO) at Y for adsorption at lOOK and after annealing to 370 K. (From Ref [44].)...

Pauli repulsion becomes important when the wave functions of adsorbate and solid start to overlap. Compared to bulk states, Shockley surface states extend relatively... 

The contribution of the Shockley surface state to the physisorption bond on Pt(lll) can even be quantified [45] on adsorption of 0.33 ML of Xe, the occupied Pt(lll) surface-resonance band at F is shifted from —0.40 to —0.25 eV because of hybridi2ation with the occupied Xe 5p level. This translates into an energetic cost of r 17 meV/Pt-surface-atom or 50 meV/Xe-atom. On Pd where the Shockley surface state is a priori unoccupied, an upshift of the surface state does not affect the total energy balance and no surface-state-related destabilization occurs. Hence, approximately 60% of the Xe adsorption enthalpy difference of 80meV/atom between Pt and Pd stem from the surface state ... 

T= 18.5 K. Obviously, the oscillation period does not depend on the adatom type, but depends on the substrate, more precisely on the Fermi wavelength of the Shockley surface state. (From Ref [51].)... 

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. 

Separate contributions from intraband (within the surface state itself) and interband (between bulk states and the surface state) transitions to the decay of Shockley surface-state holes at the F point of the projected bulk band gap of the (111) surfaces of Cu, Ag, and Au are displayed in Table 6.2. We also show the... 

Table 6.2 Decay rates in millielectronvolts of the Shockley surface-state hole at the P point of the noble metal (111) surfaces. The decay rate Pe-e is decomposed into interband (rinter) and intraband (Pmtra) contributions. ...

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... 

---