Phonons microscopic modes

Theoretically, the elasticity theory of continuous media may be used to study the long-wavelength modes. To determine the microscopic modes, numerical approaches are necessary. Most of them have used Born s model to estimate the inter-atomic forces. The semi-infinite crystals are modelled by thin films, whose thickness must be larger than the attenuation length of the surface modes. The complete MgO(OOl) phonon spectrum has been calculated, neglecting (Chen et al, 1977 Barnett and Bass, 1979) or taking into account (Lakshmi and de Wette, 1980) the surface relaxation. The same has been done for SrTiO3(001) (Prade et al, 1993). 

Fig. 4.1. Phonon dispersion curves in MgO(lOO) (according to Chen et ai, 1977). Hatched zones are the projection of the bulk modes. Surface modes S are indexed by n (1 < n < 7) the Rayleigh mode is Si the Fuchs and Kliewer modes have a frequency close to 12x10 rad s S3 is an example of a microscopic mode.

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. 

Chromatic aberrations do not arise in the acoustic microscope because in its usual mode of operation it may be considered essentially monochromatic. Even when it is necessary to take the spread of frequencies in the acoustic pulses into account, the media through which the waves pass are essentially non-dispersive in solids over the frequency range of interest the phonons are very near the centre of the first Brillouin zone where the dispersion relationship is linear, especially for sapphire. 

An effective Hamiltonian for a static cooperative Jahn-Teller effect acting in the space of intra-site active vibronic modes is derived on a microscopic basis, including the interaction with phonon and uniform strains. The developed approach allows for simple treatment of cooperative Jahn-Teller distortions and orbital ordering in crystals, especially with strong vibronic interaction on sites. It also allows to describe quantitatively the induced distortions of non-Jahn-Teller type. 

Localised phonon excitations are in principle best studied by neutral-atom scattering, or off specular HREELS, in order to reduce the strong dipole excitation of Fuchs-Kliewer modes. Two off specular HREELS measurements on MgO(lOO) have been reported [25, 68], however there is some disagreement concerning the energy and assignment of the substrate derived loss peaks. Since the microscopic surface modes are expected to be sensitive to the surface structure, it has been suggested [9] that the differences may be associated with differences in surface preparation. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

The bulk phonons are evaluated within a microscopic approach based on a force constants parametrization. We include central and angular forces in order to simulate the anisotropy of the electron gas produced by the presence of d levels. The surface phonons are evaluated, with these force constants, for a sufficiently thick slab in order to avoid interference effects between the modes of the -two surfaces. 

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

Figure 1.21 shows typical Raman spectra of the bulk and thin-film ZnO samples [123]. In this particular study, solid lines in both figures indicate Ei, Ai, and E2 phonon modes of ZnO. Dashed-dotted lines mark features observed at 332, 541, and 665 cm , which were assigned to possible multiple-phonon scattering processes [105, 108]. Dotted lines are related to the sapphire phonon mode frequencies (Aig mode 417 and 645 cm Eg mode 379,430,450, 578, and 751 cm ) [110]. For both samples Ai(LO) has not been observed and it was claimed that the scattering cross section for this mode is markedly smaller than that of the Ai(TO) mode due to the destructive interference between the Frbhlich interaction and the deformation potential contributions to the LO phonon scattering in ZnO [107]. Additionally, the occurence of the Ei(LO) mode in both samples was attributed to the breakdown of the selection rule due to the use of the Raman microscope. 

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