Phonon

A phonon is the quantum of a vibrational mode of a crystal lattice: a wave in which the atoms oscillate about their equilibrium positions with angular frequency $\Omega$ and wave vector $q$, carrying energy $\hbar\Omega$ and crystal momentum $\hbar q$. Phonons store and carry heat and exchange energy with electrons and light. Their energies are small on the scale of optical transitions: the zone-center optical phonon of silicon has a frequency of 15.6 THz and an energy of 64.5 meV, and the longitudinal optical (LO) phonon of GaAs about 36 meV. The thermal energy $k_BT$ at 300 K is 25.9 meV, so phonons are present in large numbers at room temperature and are frozen out progressively on cooling.

Acoustic and optical branches

A crystal with $p$ atoms in its primitive cell has $3p$ phonon branches. Three are acoustic: neighboring atoms move nearly in phase, the frequency rises linearly from zero as $\Omega = v_s q$ with $v_s$ the speed of sound, and at long wavelength these modes are ordinary sound waves, one longitudinal (LA) and two transverse (TA). The remaining $3p - 3$ branches are optical: atoms within the cell move against each other, and the frequency stays finite at $q = 0$. Silicon and GaAs both have two atoms per primitive cell, hence three acoustic and three optical branches. In a polar crystal such as GaAs the optical vibration carries a dipole, which splits the longitudinal (LO) from the transverse (TO) mode and couples the lattice to far-infrared light.

The occupation of a mode in thermal equilibrium follows the Bose-Einstein distribution,

$$\bar n = \frac{1}{e^{\hbar\Omega/k_BT} - 1}.$$

For the 64.5 meV silicon phonon at 300 K, $\bar n$ = 0.090; for the 36 meV GaAs LO phonon it is 0.33; for an 11 GHz acoustic phonon (45 µeV) it is about 570.

Phonons in optical processes

Momentum conservation decides most of the phonon's role in optics. A photon's wave vector is tiny on the scale of the Brillouin zone: at 1.1 µm it is $5.7 \times 10^{6}$ m⁻¹, while the conduction-band minimum of silicon lies near $0.85 \times 2\pi/a$, about $9.8 \times 10^{9}$ m⁻¹, some 1700 times larger.

• Indirect-gap absorption and emission. In silicon and germanium an electron cannot reach the conduction-band minimum by absorbing a photon alone; a phonon supplies the missing momentum, either absorbed from the lattice or emitted into it. The two-step process makes the absorption edge rise slowly, with thresholds at $E_g \pm E_\text{ph}$, and makes radiative recombination slow, as described under direct versus indirect bandgap.
• Raman scattering. Light exchanges energy with optical phonons near $q = 0$; the shift is 520 cm⁻¹ (15.6 THz) for crystalline silicon and peaks near 13.2 THz in silica fiber. The ratio of anti-Stokes to Stokes populations is $e^{-\hbar\Omega/k_BT}$, 0.083 for silicon at 300 K, which is the basis of Raman thermometry. See Raman scattering.
• Brillouin scattering. Light backscatters from acoustic phonons whose wavelength is half the optical wavelength in the medium, $\lambda/(2n)$ = 537 nm in silica at 1550 nm; with $v_s \approx$ 5960 m/s the shift $2nv_s/\lambda$ is about 11 GHz. See Brillouin scattering.
• Acousto-optic diffraction. In an acousto-optic modulator each diffracted photon absorbs or emits one phonon of the driven sound wave, shifting its frequency by the drive frequency.

Phonons and carriers in devices

Electrons and holes excited above the band edge lose their excess energy mainly by emitting phonons; in polar III-V materials LO-phonon emission dominates and brings hot carriers to the band edges within picoseconds. The energy appears as heat, which is part of the quantum defect heating of lasers. Non-radiative recombination through defects gives the full bandgap energy to the lattice by multiphonon emission. Phonon scattering also limits carrier mobility, raises the breakdown voltage of avalanche photodiodes with temperature, and homogeneously broadens transitions in gain media such as Nd:YAG (line broadening). In dielectrics heat is carried almost entirely by acoustic phonons, so thermal conductivity falls where they scatter from boundaries, alloy disorder and interfaces.

Pitfalls

Phonon momentum is crystal momentum, defined only modulo a reciprocal lattice vector. A single absorption threshold for silicon hides the separate phonon-emission and phonon-absorption edges resolved at low temperature. In amorphous silica $q$ is poorly defined, so its Raman band is broad.

Common questions

What is the difference between acoustic and optical phonons?

In acoustic modes the atoms of a cell move together and the frequency goes to zero at long wavelength; these are sound waves and carry most of the heat. In optical modes the atoms of a cell move against each other and the frequency stays at several THz at $q = 0$.

Why does silicon need a phonon to absorb light near its bandgap?

The valence-band maximum and conduction-band minimum of silicon are at different crystal momenta, and a photon carries almost none. A phonon provides the difference, so absorption near 1.12 eV is a second-order process and is weak.

Is a phonon a real particle?

It is a quasiparticle: the quantized excitation of a collective lattice mode, existing only inside the material. It obeys Bose-Einstein statistics, and its energy and crystal momentum enter conservation laws as a particle's would.

References: C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005); N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976); P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4th ed. (Springer, 2010); G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013).