Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Non-metals: thermal phonons

As mentioned previously, metals have two modes of thermal conduction: electron based and phonon based. For non metals, there are relatively few free electrons, so the phonon method dominates.

Heat can be thought of as a measure of the energy in the vibrations of atoms in a material. As with all things on the atomic scale, there are quantum mechanical considerations; the energy of each vibration is quantised (and proportional to the frequency). A phonon is a quantum of vibrational energy, and by the combination (superposition) of many phonons, heat is observed macroscopically.

The energy of a given lattice vibration in a rigid crystal lattice is quantised into a quasiparticle called a phonon. This is analogous to a photon in an electromagnetic wave; thermal vibrations in crystals can be described as thermally excited phonons, which can be related to thermally excited photons. Phonons are a major factor governing the electrical and thermal conductivities of a material.

A phonon is a quantum mechanical adaptation of normal modal vibration in classical mechanics. A key property of phonons is that of wave-particle duality; normal modes have wave-like phenomena in classical mechanics but gain particle-like behaviour under quantum mechanics.

The energy of a phonon is proportional to its angular frequency ω:

$\varepsilon = (n + \frac{1}{2})\hbar \omega$

with quantum number n. The term $$\frac{1}{2}\hbar \omega$$ is the zero point energy of the mode. This is defined as the lowest possible energy that the system possesses and is the energy of the ground state.

If a solid has more than one type of atom in the unit cell, there will be two possible types of phonons: “acoustic” and “optical” phonons. The frequency of acoustic phonons is around that of sound, and for optical phonons, close to that of infrared light. They are referred to as optical because in ionic crystals they are excited easily by electromagnetic radiation.

If a crystal lattice is at zero temperature, it lies in its ground state, and contains no phonons. When the lattice is heated to and held at a non-zero temperature, its energy is not constant, but fluctuates randomly about some mean value. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. Because the temperature of the lattice generates these phonons, they are sometimes referred to as thermal phonons. Thermal phonons can be created or destroyed by random energy fluctuations.

It is accepted that phonons also have momentum, and therefore can conduct energy through the lattice. Unlike electrons, there is a net movement of phonons - from the hotter to the cooler part of the lattice, where they are destroyed. Electrons must maintain charge neutrality in the lattice, so there is no net movement of electrons during thermal conduction.

The following simulation shows schematic optical and acoustic phonons in a 2D lattice, and has the option to animate a 2D wavevector defined by clicking inside the green box.

## Umklapp scattering

When two phonons collide, the resulting phonon has the vector sum of their momenta. The way of treating particles moving in a lattice quantum mechanically under the reduced zone scheme (which is beyond the scope of this TLP but is explored in more depth in the Brillouin Zones TLP), leads to a conceptually strange effect. If the momentum is too great (outside the first Brillouin zone) then the resulting phonon moves in almost the opposite direction. This is Umklapp scattering, and is dominant at higher temperatures- acting to reduce thermal conductivity as the temperature increases. 