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Hybrid modes exist as a consequence of acoustic and optical waves having to satisfy the boundary conditions at an interface or at a surface. The author begins the description of hybrid modes in nanostructures with an account of modes in a non-polar, free-standing slab. This chapter includes long-wavelength assumption decouples acoustic and optical modes; isotropy decouples LO and TO modes; s and p modes; acoustic hybrid modes: Love waves, Lamb waves, guided modes, Rayleigh waves; the boundary condition u = 0 for optical modes; the sTO mode; double hybrid: LO and pTO modes; and energy normalization.

This chapter provides a classical account of the elasticity theory of acoustic modes in the diamond lattice. It contains the mathematical definitions of stress and strain and their mutual relation in terms of elastic constants. It also discusses quantization and connection rules in nanostructures.

Pristine graphite crystallizes according to the D46h space group. There are twelve modes of vibration associated with the three degrees of freedom of the four atoms in the primitive cell. The hexagonal Brillouin zone and the phonon dispersion curves of pristine graphite, calculated by Maeda et al. (1979), are shown in Figure 4.1. The zone-center (Γ point) modes are labeled as three acoustic modes (A2u + Elu), three infrared active modes (A2u + Elu), four Raman active modes (2E2g), and two silent modes (2Blg). The first calculation of phonon dispersion for the stage-1 compounds KC8 and RbC8 was presented by Horie et al. (1980) on the basis of the model of Maeda et al. (1979) for the lattice dynamics of pristine graphite. Although the calculated phonon energies do not agree well with the experimental data, the model has most of the ingredients for describing the lattice dynamics of stage-1 GICs. A simple review of their work is presented as follows. The primitive cell of KC8, having a p(2 × 2)R0° superlattice, contains 16 carbon atoms and two K atoms. Note that only an αβ stacking sequence is assumed here (see Section 3.6.1). The primitive translation vectors are given by t1 (0, a, 0), t2 = (−√3a/2, a/2, 0), and t3 = (−√3a/4, −a/4, c), where a = 2aG = 4.91 Å and c = 5.35 × 2 = 10.70 Å. The corresponding Brillouin zone is shown in Figure 4.2b. The phonon dispersion for KC8 has been calculated by Horie et al. (1980) on the basis of the Born-von Karman force constant model. This dispersion curve is compared with that of pristine graphite by folding the dispersion curves of graphite into the first Brillouin zone of KC8. Since the side of the Brillouin zone in KC8 is not flat in two directions, as shown in Figure 4.2b, it is a little difficult to transfer the information on the dispersion curves in the first Brillouin zone of graphite into the Brillouin zone of KC8. For simplicity, nevertheless, we assume that the side of the Brillouin zone in KC8 is flat like that of graphite.

The single heterostructure is one of the most technologically versatile devices, being the structure of field-effect transistors and Schottky-effect devices, to say nothing of its capability of exhibiting the quantum Hall effect at low temperatures. Here, the focus is on a heterostructure composed of III–V compounds such as AlAs/GaAs at room temperature and above, where optical waves are readily excited. This chapter covers: hybrid model for optical modes: LO, pTO, IF; ionic displacement and associated electric fields; fields in the barrier layer – remote phonons; mechanical boundary condition u = 0; energy normalization; reduced boundary conditions; acoustic hybrids: sTA, pTA, pLO; and interface acoustic modes.

An immediate problem that presented itself in the physics of nanostructures, formed from two semiconductors, was to describe conditions at their interface. In principle, band-structure and lattice-dynamical calculations could be performed to give the eigenvalues and eigenstates of the whole heterostructure, but any result along these lines would refer only to the special case considered, and would provide little in the way of providing criteria for predicting the properties of the huge number of structures that were conceivableThis chapter will outline classical connection rules for acoustic modes, detailed connection rules for optical modes, and electromagnetic interface conditions.

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G₂ in the G₁-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.