Page 67 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
2.4 The Vibrating Uniform Lattice
The elastic, spring-like nature of the interatomic bonds, together with the
massive atoms placed at regular intervals; these are the items we isolate
for a model of the classical mechanical dynamics of the crystal lattice
(see Box 2.3 for brief details on Lagrangian and Hamiltonian mechan-
ics). Here we see that the regular lattice displays unique new features
unseen elsewhere: acoustic dispersion is complex and anisotropic, acous-
tic energy is quantized, and the quanta, called phonons, act like particles
carrying information and energy about the lattice.

2.4.1 Normal Modes
As we saw in Section 2.3, an exact description of the forces between the
atoms that make up a crystal are, in general, geometrically and mathe-
matically very complex. Nevertheless, certain simplifications are possi-
ble here and lead both to an understanding of what we otherwise observe
in experiments, and often to a fairly close approximation of reality. We
assume that:
• The atoms that make up the lattice are close to their equilibrium posi-
tions, so that we may use a harmonic representation of the potential
binding energy about the equilibrium atom positions. The spatial gra-
dient of this energy is then the position-dependent force acting on the
atom, which is zero when each atom resides at its equilibrium posi-
tion.
• The lattice atoms interact with their nearest neighbors only.
• The lattice is infinite and perfect. This assumption allows us to limit
our attention to a single Wigner-Seitz cell by assuming translational
symmetry.
• The bound inner shell electrons move so much faster than the crystal
waves that they follow the movement of the more massive nucleus
that they are bound to adiabatically.

64 Semiconductors for Micro and Nanosystem Technology