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Introduction to Nanomaterials and Devices by Omar Manasreh

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Appendix G

Lattice Vibrations and Phonons

In semiconductor crystals, the atoms are tightly coupled to one another, and the binding energy is called cohesive energy, which is defined as the energy needed to separate the crystal into independent ions at a large distance from each other. The thermal kinetic energy of the atoms in the crystal is simply the vibrational energy of motion, which propagates in the crystal as waves. These waves are called acoustical or sonic waves. The quanta of these waves is called phonon. Phonons in semiconductors can absorb or scatter light in the infrared spectral region. To understand how the acoustical waves propagate in a solid, let us first consider a one-dimensional monatomic lattice, as shown in Fig. G.1. By including only the nearest-neighbor interaction and assuming that the vibrational amplitudes are smaller than the lattice spacing, one can write the force on the nth atom as

G.1 G.1

where m is the mass of the atom and γ is the force constant. For a solution having the character of a traveling wave, we have for the nth atom the following solution (Eq. G.1):

G.2 G.2

where k is the propagation constant, ω is the angular frequency, and xn = na (where a is the lattice constant). Similarly, the solutions for the nearest-neighbor atoms are

G.3

The dispersion ...

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