Ionic surface phonons

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

We note that ionic crystals may have dielectric functions satisfying Eq. (4) for frequencies between their transverse and longitudinal optic phonon frequencies. SEW on such crystals are often called surface phonons or surface polaritons and the frequency range is the far IR. 

In the 2- 20 eV range, surface excitations related to the dielectric response function and free carrier density of the surface are observed. Clean-surface loss spectra can include features due to surface phonons, surface plasmons, interband transitions and surface optical phonons in ionic insulators. The probe depth for these phenomena in HREELS is about 10 nm. Transitions between surface states can also be observed in the loss spectrum. Some examples are given in the section related to hydrogen adsorption and surface states below. 

In dielectric layers, where a surface phonon mode may occur, or in ionic crystals, multiple scattering from the surface phonon mode can result in Poisson replicas of the no-loss peak. These modes are referred to as Fuchs- Kliewer modes they are a general feature of HREELS spectra of ionic and polar materials, and metal oxides. Ordered overlays on surfaces can also exhibit collective modes, but at submonolayer coverages the HREELS loss peaks are due almost exclusively to single oscillations of the fundamentals. Substrate (silver) phonon modes are shown at 10 meV (83 cm ) in Figure 7. 

Among surface phonon branches one can distinguish that of the surface optical phonon, a s(k ), which corresponds to surface vibrations of ionic crystals accompanied by an oscillating dipole moment. Its frequency in the long-wavelength limit (k 0) can be found, much in the same way as for the case of surface plasmons (see Section 2.1.3). The corresponding macroscopic electric field is determined by an electrostatic potential of the forms (2.55) and (2.56). As a result, cVs satisfies the equation e(o s) = —1, where e cv) is the dielectric function of the crystal. Taking the latter quantity in the form... 

Another effect of the electron-phonon interaction is a shift in the velocity of the electrons at the Fermi surface, in some ways analogous to the polaron effect in ionic crystals. Because of the wake of lattice distortion that accompanies the electron, its velocity is reduced, as it turns out, by a factor (I -t A) E (I- or a discussion of this effect, and references, sec Quinn, 1960, p. 58, oi Harrison, 1970, p. 418ff.) The reduction in velocity corresponds to a decrease in dE/dk at the Fermi surface and, therefore, to an increase in the density of stales by the same factor. We noted in Chapter 15 that the electronic specific heal is proportional to the density of states, so we may expect an enhancement of the experimental... 

Bulk rutile shows a high dielectric constant and low-fiequency ( soft ) phonons, but how is this manifest at the surface An FP study finds a soft, anisotropic and anharmonic surface mode (0.15 A ionic displacements at room temperature), which could accoimt for some of the discrepancy between SXRD and zero-temperature FP structures [60]. More generally, this illustrates that many oxide surfaces can be expected to show vibrational anisotropy [40], complicating the use of FP to provide quantitative predictions of structure at a given temperature. 

The high fraction of surface atoms in thin ionic slabs [141,173-175] and quantum dots causes the appearance of new modes on the low energetic side of the LO mode. These modes are due to vibrations at the crystallite surface and are therefore called surface optical (SO) phonons. In the case of the slabs, the calculation of SO modes corresponds to the problem of the IF modes. For quantum dots, the situation is different Here, the surface mode cannot be represented by a plane wave but by modes with spheroidal or torsional character, like the LF modes (see Sec. IILB.3). In samples containing semiconductor quan-... 

The microscopic modes have a penetration depth which is of the order of a few inter-plane spacings. Their description requires a precise account of the atomic structure. They are generally located in the gaps of the bulk phonon spectrum. However, at some positions in reciprocal space, they may become degenerate with the bulk modes, thus transforming into surface resonances. Lucas (1968) was one of the first authors to predict their existence in ionic crystals. 

Fuchs and Kliewer (1965) have predicted the existence of macroscopic surface optic modes in ionic crystals. We give here a simplified derivation of their result, based on the formalism of the dielectric constant. In the phonon frequency range, the bulk dielectric constant e( )) approximately varies with co as ... 

A. A. (1991) Validity of the dielectric approximation in describing electron-energy-loss spectra of surface and interface phonons in thin films of ionic crystals. Phys. Rev. B, 44, 6416-6428. 

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