Displacement surface phonons

They found a whole bunch of soft phonons, which are primarily horizontally polarized, near the zone boundaries between M and X. The most unstable mode they observed is the Mj phonon, the displacement pattern of which is shown in Fig. 40 note the similarity between this pattern and the reconstruction model in Fig. 39. According to Wang and Weber, these soft phonons are caused by electron-phonon coupling between the surface phonon modes and the electronic 3 surface states at the Fermi surface. They attributed the predominant Ms phonon instability to an additional coupling between d(x — y ) and d(xy) orbitals of the Zj states. 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

The situation at surfaces is more complicated, and richer in information. The altered chemical environment at the surface modifies the dynamics to give rise to new vibrational modes which have amplitudes that decay rapidly into the bulk and so are localized at the surface [33]. Hence, the displacements of the atoms at the surface are due both to surface phonons and to bulk phonons projected onto the surface. Since the crystalline symmetry at the surface is reduced from three dimensions to the two dimensions in the plane parallel to the surface, the wavevector characterizing the states becomes the two-dimensional vector Q = qy). (We follow the conventional notation using uppercase letters for surface projections of three-dimensional vectors and take the positive sense for the z-direction as outward normal to the surface.) Thus, for a given Q there is a whole band of bulk vibrational frequencies which appear at the surface, corresponding to all the bulk phonons with different values of (which effectively form a continuum) along with the isolated frequencies from the surface localized modes. 

Surface phonon bands along symmetry lines of the SBZ are given for fee metals in Figs. 5.2-49-5.2-55 and in Table 5.2-20. In all figures the horizontal axis is the reduced wave vector, expressed as the ratio to its value at the zone boundary. Table 5.2-21 gives the surface Debye temperatures for some fee and bcc metals, as well as the amplitudes of thermal vibrations of atoms in the first layer p as compared with those of the bulk pb-In the harmonic approximation, the root mean square displacement of the atoms is proportional to the inverse of the Debye temperature. 

The results obtained above for a ID chain of atoms can be qualitatively extended to the case of a 3D crystal terminated by a 2D surface, much as it has been demonstrated before for electronic surface states. One can imagine a 3D solid as being constructed of an infinite number of linear chains parallel to each other. Due to the interatomic interactions the phases of vibrations in different chains are correlated to each other. A surface phonon then can be described by a wave vector ky parallel to the surface. The atomic displacements are now represented by the vectors... 

The first mention of surface phonons is due to Lord Rayleigh (1885), who predicted the existence of a surface acoustic mode with a sound velocity lower than in the bulk. He proved this result, using elasticity theory, by representing the semi-infinite sofid by a continuous and isotropic medium (Landau and Lifehitz, 1967). Considering an infinitesimal volume element, he wrote a Fourier component of its displacement u q, co), in the following form ... 

The p(2x2) 0/ Ni(lll) system is reconstructed with a twist deformation of three of the top layer nickel atoms and a vertical displacement of all of the atoms in the top layer of the unit cell (LEED [90Gri]). The oxygen coverage is 0.25 ML. A schematic view of the structure of the p(2x2) overlayer is reported in Fig. 44. The oxygen lifts three of the nickel atoms away from their original bulk positions, while the fourth relaxes towards the second layer Ni atoms. The surface phonon dispersion measured by HREELS is reported in Fig. 45. Five optical modes are observed. The modes at 67 and 71 meV are assigned to oxygen adsorbate vibrations, while the lower modes lie within the bulk bands. The open (filled) circles... 

It is often useful to characterize bulk and surface phonons with respect to their polarization. At the surface, phonons are classified according to the movement of the atoms with respect to the sagittal plane, the plane that is spanned by the surface normal and the direction of k. The displacements of the atoms can be within the sagittal plane with a dominant vertical (along the surface normal) or a dominant in-plane component, which are transverse and longitudinal modes. Alternatively, modes are called shear horizontal if the movements are perpendicular to the sagittal plane. 

Figure 8.2.25 Atomic displacement pattern of the Si(100)-(2xl) surface phonon modes at the r point. Modes that are odd with respect to the surface Cs mirror plane are presented in top view (left column). The even...

Figure 9.48 Displacement patterns of a few selected optical surface phonon modes at the center of the Brilloiun zone r of the Si(lll)-(2xl). (Figure adapted from Ref. [87].)...

The volume integral will give a higher order term in k, so for now, we focus on the surface integral. The displacement due to the phonon is conveniently expanded in terms of the spherical waves e " =... 

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

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