Phonon

A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because they contribute to many of the physical properties of materials, such as thermal and electrical conductivity. For example, the propagation of phonons is responsible for the conduction of heat in insulators, and the properties of long-wavelength phonons gives rise to sound in solids (hence the name phonon).

According to a well-known result in classical mechanics, any vibration of a lattice can be decomposed into a superposition of normal modes of vibration. When these modes are analysed using quantum mechanics, they are found to possess some particle-like properties (see wave-particle duality.) When treated as particles, phonons are bosons possessing zero spin.
The following article provides an overview of the physics of phonons.

Non-interacting phonons

Modelling a lattice

We begin our investigation of phonons by examining the mechanical systems from which they emerge. Consider a rigid regular (or "crystalline") lattice composed of N particles. We will refer to these particles as "atoms", though in a real solid they may actually be ions. N is some large number, say around 1023 (Avogadro's number) for a typical piece of solid.

If the lattice is rigid, the atoms must be exerting forces on one another, so as to keep each atom near its equilibrium position. In real solids, these forces include Van der Waals forces, covalent bonds, and so forth, all of which are ultimately due to the electric force; magnetic and gravitational forces are generally negligible. The forces between each pair of atoms may be characterized by some potential energy function V, depending on the separation of the atoms. The potential energy of the entire lattice is the sum of all the pairwise potential energies:

where ri is the position of the ith atom, and V is the potential energy between two atoms.

It is extremely difficult to solve this many-body problem in full generality, in either classical or quantum mechanics. In order to simplify the task, we introduce two important approximations. Firstly, we only perform the sum over neighbouring atoms. Although the electric forces in real solids extend to infinity, this approximation is nevertheles valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials V as harmonic potentials, which is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding V about its equilibrium value.)

The resulting lattice may be visualized as a system of balls connected by springs. Two such lattices are shown in the figures below. The figure on the left shows a cubic lattice, which is a good model for many types of crystalline solid. The figure on the right shows a linear chain, a very simple lattice which we will shortly use for modelling phonons. Other common lattices may be found in the article on crystal structure.

The potential energy of the lattice may now be written as

Here, ω is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. Ri is the position coordinate of the ith atom, which we now measure from its equilibrium position. The sum over nearest neighbours is denoted as "(nn)".

Kevin M Contreras H
Electrónica del Estado Sólido
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