Physics of Surfaces and Interfaces[Chapter-7-Vibrational Excitations at Surfaces]英文精品课件

of motion for a particular atom n in the Cartesian direction D is
M (n)u&&D (n) ¦) (n,D; m, E )uE (m) 0 ,
(7.2)
m,E
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7. Vibrational Excitations at Surfaces
Many aspects of the chemistry of adsorbates were revealed through the vibration spectra of the adsorbed species on surfaces, as the vibration frequency spectrum is characteristic for the strength and the type of the bonds. The development of experimental techniques for studying surface vibrations was therefore a major step forward in the understanding of surface chemistry. Of fundamental interest are the vibrational excitations of clean, two-dimensional periodic surfaces, the surface phonons. The general basis for the consideration of vibrational excitations is the Born-Oppenheimer approximation [7.1]. According to this approximation, the electronic eigenstates follow the moving atoms adiabatically, which means than the electron energy levels change with the atom positions, but the electrons remain in the same eigenstates. The total energy of the solid as a function of the atom position therefore plays the role of a potential for the atom motion. We begin this section by looking into the consequences of the Born-Oppenheimer approximation for periodic surfaces.
in which M(n) is the mass of the atom n. The system of equations (7.2) technically couples all atom motions to each other. In reality, the coupling vanishes rapidly for larger distances. For an approximate picture of the vibration spectrum the consideration of nearest and possibly next-nearest neighbors often suffices. For a three-dimensional periodic structure, the system of equations (7.2) is separated into a subsets of 3s equations, with s the number of atoms in the unit cell, by the ansatz
) (^r(n)`) )0 (^r0 (n)`) ¦ĭ(n,D; m, E )uD (n)uE (m)...Fra bibliotek(7.1)
n,m,D ,E
Here, uD(n) is the deviation of the atom n from the equilibrium position in the Cartesian direction D. The first derivatives vanish, as the expansion is around the global minimum of the potential. If higher order terms are neglected, the equation
7.1 Surface Phonons of Solids
7.1.1 General Aspects
The potential ) for the atom motion depends on the coordinates {r(n)} of all atoms in the solid. The potential can be expanded into a Taylor series around the equilibrium positions denoted as r0(n).
uD (n) u0,D ei Ȧ(q)t qr0 (n)
(7.3)
The quantized plane wave solutions of the type (7.3) are the phonons of the 3Dsolid. A flat surface/interface breaks the 3D-translational symmetry of the solid, which gives rise to solutions of (7.2) that are localized at the surface/interface in the sense that the vibrational amplitude decays in an essentially exponential manner away from the surface/interface. These modes are called surface (interface) modes, or surface (interface) phonons. A schematic overview over the spectrum of eigenmodes at a surface is shown in Fig. 7.1. The frequencies are displayed as function of the component of the wave vector parallel to the surface q||. In this projection, bulk modes form a continuum because their wave vector perpendicular to the surface remains arbitrary. The continuum of bulk phonons is shown as a shaded area in Fig. 7.1. The frequency of a surface phonon is uniquely determined by q||. Surface phonons have therefore a defined dispersion branch in this graph. They are plotted as solid lines. Genuine surface modes can exist only if no bulk phonons of the same symmetry are present in the same Zq|| range. In addition to the surface phonons, so-called surface resonances exist.