Projected band structure

The truncation of the lattice and/or the reconstruction and relaxation cause the electronic states at the surface or in the uppermost layers to be distinctly different from those of the bulk. Such new states are called surface states. Their wave functions decay exponentially on both sides of the surface. Since their k is imaginary, the surface band structure is defined in the surface Brillouin zone (SBZ), which is the projection of the 3-D Brillouin zone onto the surface plane. The projection of the bulk bands onto the SBZ is called the projected band structure. When the energy of a surface state is localized in a gap of the projected bulk structure (either an absolute gapp, i. e. one that extends throughout the whole SBZ, or a partial gap), one speaks of a true (or bona fide) surface state. When there is degeneracy (both in energy... 

Fig. 5.2-23 Theoretical projected band structures (shaded areas) along symmetry lines of the SBZ for Ag(lOO), Ag(llO) and Ag(lll). Surface states (solid lines) and resonances (dashed lines) are shown. SBZ in the insets [2.37-39]...

Fig. 5.2-26 Theoretical projected band structure for the unreconstructed Au(l 10) surface. Note that in the notation of this figure, X and M correspond to Y and S, respectively, in the SBZ of Fig. 5.2-18 [2.39]...

Fig. 5.2-28 Theoretical projected band structure shaded area) for Pd(lll), showing an empty surface state (near F) and various occupied surface states and resonances [2.42]...

Fig. 5.2-51 Surface phonon dispersion curves for Ag(l 11) measured by HATOF. dries are experimental points solid and dashed lines theoretical calculations. The hatched area represents the projected band structure [2.82].

Fig. 5.2-57 Surface phonon dispersion curves for NaCl(lOO). Theoretical projected band structure (hatched) and experimental points (dots) measured by HATOF. Solid and dashed lines represent theoretical surface states and resonances. Ordinates are given in units of 10 rad/s = 6.586 meV [2.88,89]...

Fig. 21. Calculated surface bands (solid lines) and resonances (dashed lines) for the 2 1 diamond (111) surface for fully relaxed Pandey chain model. The bulk projected band structure (shaded) and the experimental data of Ref. 55 (black dots) are shown for comparison. (from Ref. 56)...

Fig. 23. Surface bauds (dashed curves) and the projected band Structure for the Nb(OOl) surface. (from Ref. 77)... 

The calculated surface band structure for H on site C is shown in Fig. 25. The H adatoms induce extensive changes in the surface electronic structure of the clean Pd (111) indicating a strong surface chemical bond. The two most striking H-induced features are the narrow H-Pd bonding adsorbate band which appears about 2 eV below the Pd bulk d bands, and the 4 eV wide anti-bonding H-Pd band just above Ej, in a gap in the projected band structure... 

In principle, valence band XPS spectra reveal all the electronic states involved in bonding, and are one of the few ways of extracting an experimental band structure. In practice, however, their analysis has been limited to a qualitative comparison with the calculated density of states. When appropriate correction factors are applied, it is possible to fit these valence band spectra to component peaks that represent the atomic orbital contributions, in analogy to the projected density of states. This type of fitting procedure requires an appreciation of the restraints that must be applied to limit the number of component peaks, their breadth and splitting, and their line-shapes. 

Figure 39 Band structure of the (a) fully saturated and (b) partially saturated Si[i]-SiC>2(0 01) SL projected along the two symmetry directions of the 2D Brillouin zone of the (0 01) surface. K and M represent, respectively, the k-points in the corner and in the middle of the side of the 2D Brillouin zone. A self-energy correction of 0.8 eV has been added to the conduction states. Energies (in eV) are referred to the valence band maximum.

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.

In many cases there are electronic states with a strong weight in the surface layer, but which are not located in a gap of the projected bulk band structure. The electrons in these states can decay into bulk states much faster than those occupying pure surface states. These states are known as surface resonances. One of these cases occur in the Ru(0001) surface. 

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. 

The activation barriers AE for dissociation and recombination belong to the same realm of relative energies as AQAB. For this reason, we shall not discuss here purely numerical calculations of AE. Remarkably, many authors tried to conceptualize their computational results in terms of simple analytic models, which have no direct relation to the computations. For example, the effective medium theory (EMT) is a band-structure model with a complex and elaborated formalism including many parameters (154). Nevertheless, while reviewing the numerical EMT applications to surface reactions, Norskov and Stoltze (155) discussed the calculated trends in the activation energies for AB dissociation in terms of a one-parameter model (unfortunately, no details were provided) projecting A b to vary as NJ, 10 - Nd), where Nd is the d band occupancy [cf. Eqs. (21a)—(21c) of the BOC-MP theory]. 

Figure 14-7. Surface band structures for the configurations 1-1,1-2, F-l and F-2 at 0.125 ML. The shaded areas represent the projected bulk band structure, while surface states are shown as solid lines...

This is a band-structured matrix or a band matrix which is in this particular tridiagonal form also called the Jacobi matrix. Projecting both sides of Eq. (53) onto Pn+ fn+ I and using Eq. (59), we find ... 

Fig. 4.23. Results of band-structure calculations on TiO, using the SCF aug-mented-spherical-wave method. Shown are the total densities of states and the site-projected partial densities of states for Ti 3d, O 2s. and O 2p (normalized to one atom) (after Schwarz, 1987 reproduced with the publisher s permission).

Fig. 14 Band structure of a fully oxygen defective (1 x 1) MgO(lOO) surface along the three symmetry lines J-F-M of the 2D Brillouin Zone, as obtained through the FP-LMTO calculation (Full Potential- Linear MufiSn-Tin Orbital method). The dashed horizontal line represents the Fermi level, black dots (st indicate the energy positions of the filled (empty) Bloch states at F calculated in a (2v x 2- /2) supercell. The dashed line in the gap of the projected bulk bandstructure gives the dispersion of the F, centre band. The dashed-dotted line is used for the surface conduction band of lowest energy (from Ref. 69).

The spectra of metallic crystals can be solved in k-space. Theoretical calculations of the partial and projected density of states of the crystal band structure were reported by several groups to interpret the XANES of metals " >. We discuss here the band structure approach developed by Muller et al. by which it is possible... 

XANES spectra of different systems have been interpreted with the band structure aproximation As an example for a transition metal we discuss here palladium absorption edges. The comparison between the K and Lj edge of Pd metal with the theoretical band approach is shown in Fig. 21. We can observe that the K and Li edges present the same spectral features and therefore contain identical information. In fact, the selection rule for electronic transitions selects the same I = 1 projected density of states. Because the L, edge occurs at lower energy a better instrumental energy resolution is obtained and the structures are better resolved. 

---