Surface Brillouin Zones

Fig. 25. Structure factor integrated over 2.3% of the surface Brillouin zone (radius of 5 mesh lengths in Fig. 24) vs. TtoclesL plotted with the rescaled energy (crosses) for the J i x overlayer on the triangular lattice. Rescaling involves multiphc-ation by a negative number and shifting by a constant. Temperature is measured in units of

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

See Surface Brillouin zone Cantilever 314—317 fahrication 316 requirements 314 Charge-density contours 117 Chemical hond 13, 172... 

See Surface Brillouin zone Scanning tunneling microscope 1 concentric-tube 111 low-temperature 275 pocket-size 270 schematic diagram 1 single-tube 273... 

See Scanning tunneling spectroscopy Superconductors 332—334 Surface Brillouin zone 92 hexagonal lattice 133 one-dimensional lattice 123, 128 square lattice 129 Surface chemistry 334—338 hydrogen on silicon 336 oxygen on silicon 334 Surface electronic structures 117 Surface energy 96 Surface potential 93 Surface resonance 91 Surface states 91, 98—107 concept 98... 

Figure 2.13. Schematic N 2px, N 2py and Cu 3d orbital plots at the different high symmetry points in the surface Brillouin zone. Note that the degeneracy is lifted at the X point where both N 2px and N 2py orbitals are shown. From Ref. [3].

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 25. Measured band structure in the range of occupied and unoccupied levels for CO(2 x I)p2mg/Ni(l 10). The wave vector K is determined along the two orthogonal directions in the surface Brillouin zone as shown at the top and its energy dependence according to = (2wcfr kin)l/ sin i .

Fig. 9.19. Calculations used in the construction of an effective Hamiltonian for the W(OOl) surface (adapted from Roelofs et al. (1989)). Panel (a) shows the surface Brillouin zone while panels (b)-(e) show the energy associated with various displacement patterns, with the solid lines corresponding to the energetics of the effective Hamiltonian while the discrete points are the energies associated with first-principles total energy calculations.

Figure 25.9 The band structure of hydrogen atoms adsorbed on a Ni(lOO) surface (E((t) relation) alongthe high-symmetry directions of the surface Brillouin zone, f-X and f-M. Only the states belonging to the A, representation of the C, point group are shown. After Puska et al. [46],...

A particularly useful variety of UPS is angle-resolved photoelectron spectroscopy (ARPES), also called angle-resolved ultraviolet photoelectron spectroscopy (ARUPS) [, 62]. In this technique, measurements are made of the valence band photoelectrons emitted into a small angle as the electron emission angle or photon energy is varied. This allows for the simultaneous determination of the kinetic energy and momentum of the photoelectrons with respect to the two-dimensional surface Brillouin zone. From this information, the electronic band structure of a single-crystal material can be experimentally determined. 

Figure 8. The evolution of surface phonon dispersion curves for a monatomic fee (111) surface in slab dynamics calculations as a function of the number of layers in the slab. The surface localized modes, marked by arrows in the last panel (iV = 15), lie below the bulk bands Mross the entire surface Brillouin zone and appear between the bands in the small gap near K in the TK region and in the larger gap in the MK region. (Reproduced from Fig. 1 of Ref. 24, with permission of Elsevier Science Publishers.)...

Figure 14. (a) Schematic of the direct lattice for the (001) surface of the rocksalt structured crystal of Fig. 13 with the bulk unit cell indicated by dashed lines and a surface unit cell by solid lines. The high-symmetry directions for this surface, <100> and <110>, are indicated with arrows. (b) The reciprocal lattice corresponding to the direct lattice in panel a is shown with the first surface Brillouin zone (SBZ) indicated by Mlid lines. The shaded triangular region, FXM, is the irreducible portion of the SBZ, from which the entire SBZ can be constmcted by synunetry. 

Fig. 4.6 Two-dimensional band dispersion along the T-A/direction of the Gd surface Brillouin zone. Reprinted from [7], Copyright (1998), with permission from Elsevier...

F. 21. Dispersion of CO vibrational frequencies for the main directions of the surface Brillouin zone, measured under non-specrdar conditions for the well ordered (2xl)p2mg CO structure [90Voi]. 

Vibration frequencies and phonon dispersion See Figs. 20 - 23. Table 13. Perpendicular vibration frequencies /zcoi and characteristics of the phonon dispersion curves for the noble gas monolayers. The sound velocities c/ and c, were obtained from the initial slope of the dispersion curves for the longitudinal (L) and shear-horizontal (SH) modes, respectively. Where complete or partial dispersion curves are available, oidy the value at the boundary of the surface Brillouin zone is indicated. Abbreviations used F, M, K high syrtunetry points of the 2D adlayer Brillouin zone (BZ) [001], [110] and [112] crystallographic directions of the substrate surface. All data were obtained using inelastic He-atom scattering. (Ad. = adsorbate) ... 

Fig. 14. Dispersion curves for the 5p Xenon levels along the TKMT direction of the surface Brillouin zone of a Xe layer adsorbed on Pt(lll). Circles refer to the...

Fig. 6. Inverse photoemission spectra recorded on single-domain K/Si( 100)2x1 (approx. 0.5 ML coverage) recorded for different incidence angles 0 along the F —/direction of the surface Brillouin zone. U2 corresponds to the unoccupied counterpart of F2 from Figure 5. From [91Joh].

Fig. 7. Experimental Fermi surface map of the K/Si(100)2xl system at saturation coverage (hv = 21.2 eV). The surface Brillouin zones have been drawn superimposed. Bright areas denote regions where a band crosses the Fermi energy. From [98Mar].

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... 

H,L,K,K surface Brillouin zone as indicated in V critical exponent for finite size scaling... 

As can be seen from table 1, most of the band structure calculations are for the bulk (for reviews of the bulk electronic structure see Barrett 1992 and Liu 1978). Often these calculations are along the F-A direction of the bulk Brillouin zone (as indicated in the diagram of the surface and bulk Brillouin zone in fig. 3). There are a few calculations that do consider the electronic structure of the surface. Of particular interest are surface states states that occur in a gap of the bulk band structure when projected on to the surface Brillouin zone. Surface resonances, i.e. states where there is considerable weight... 

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