3.3 Acoustic and optical phonons

Lattice vibrations

Lattice vibrations are the collective oscillations of atoms in a crystalline solid around their equilibrium positions. These vibrations govern key material properties like thermal conductivity, specific heat, and electrical conductivity, making them central to solid state physics.

This topic covers the two fundamental types of lattice vibrations: acoustic phonons (in-phase atomic motion, responsible for sound propagation) and optical phonons (out-of-phase motion, capable of interacting with light). Understanding the distinction between these two branches is essential for interpreting phonon dispersion relations and the thermal behavior of crystals.

Phonons

A phonon is a quantized mode of vibration in a crystal lattice. Rather than tracking the motion of every individual atom, you treat the collective vibrations as particle-like excitations (quasiparticles) that carry energy and crystal momentum. This simplification is what makes the analysis of lattice dynamics tractable.

Phonon properties depend on the interatomic force constants and the geometry of the lattice. Different crystals support different phonon spectra, which is why materials can have vastly different thermal and elastic behaviors.

Quantized lattice vibrations

Lattice vibrations are quantized: they can only take on discrete energy values. The energy of a single phonon is:

$E = \hbar \omega$

where $\hbar$ is the reduced Planck constant and $\omega$ is the angular frequency of the vibrational mode.

Because phonons are bosons, their thermal occupation follows the Bose-Einstein distribution. The average number of phonons in a mode at temperature $T$ is:

$\langle n \rangle = \frac{1}{e^{\hbar\omega / k_B T} - 1}$

At absolute zero, $\langle n \rangle = 0$ for all modes, meaning the lattice sits in its vibrational ground state (though zero-point energy still exists). As temperature rises, higher-frequency modes become progressively populated.

Acoustic phonons

Acoustic phonons are the low-frequency vibrational modes where neighboring atoms move approximately in phase with each other. Think of it as a wave where entire planes of atoms shift together in the same direction. These modes are directly responsible for sound propagation through solids.

Every crystal has exactly 3 acoustic branches (in 3D): one longitudinal (LA) and two transverse (TA). In the longitudinal mode, atoms oscillate along the direction of wave propagation. In the transverse modes, they oscillate perpendicular to it.

Long wavelength limit

When the phonon wavelength is much larger than the interatomic spacing ($\lambda \gg a$), the crystal behaves like a continuous elastic medium. In this regime, the dispersion relation becomes linear:

$\omega = c_s k$

where $c_s$ is the speed of sound and $k$ is the wave vector magnitude. This is the regime relevant to elasticity theory and everyday sound propagation.

Linear dispersion relation

The linear relationship $\omega = c_s k$ means acoustic phonons near the zone center propagate at a constant velocity regardless of frequency, just like sound waves in air. The slope of the dispersion curve gives the group velocity:

$v_g = \frac{d\omega}{dk} = c_s$

As $k$ approaches the Brillouin zone boundary, the dispersion curve flattens. The group velocity drops to zero at the zone edge, meaning those short-wavelength phonons form standing waves rather than propagating modes.

Debye model

The Debye model approximates the phonon spectrum by assuming the linear dispersion relation holds for all acoustic modes up to a maximum cutoff frequency $\omega_D$ (the Debye frequency). This cutoff ensures the total number of modes equals $3N$, where $N$ is the number of atoms.

The corresponding Debye temperature is:

$\Theta_D = \frac{\hbar \omega_D}{k_B}$

The model's key predictions:

• At low temperatures ($T \ll \Theta_D$), the specific heat follows a $T^3$ law
• At high temperatures ($T \gg \Theta_D$), it recovers the classical Dulong-Petit value of $3Nk_B$

Typical Debye temperatures: diamond has $\Theta_D \approx 2230$ K (very stiff lattice), while lead has $\Theta_D \approx 105$ K (soft lattice).

Speed of sound in solids

The speed of sound depends on the material's elastic stiffness and density:

$c_s = \sqrt{\frac{C}{\rho}}$

where $C$ is the relevant elastic modulus and $\rho$ is the mass density. For longitudinal waves, $C$ involves the bulk modulus (and shear modulus in bounded media); for transverse waves, $C$ is the shear modulus alone.

In anisotropic crystals, the LA and TA branches can have quite different velocities depending on the propagation direction. For example, in silicon along the [100] direction, the longitudinal sound velocity is about 8430 m/s while the transverse velocity is about 5840 m/s.

Optical phonons

Optical phonons are the high-frequency vibrational modes where atoms within the same unit cell move out of phase with each other. If you have a diatomic basis (like NaCl or GaAs), the two sublattices oscillate in opposite directions.

A crystal with $p$ atoms per unit cell has $3p - 3$ optical branches and 3 acoustic branches, for a total of $3p$ branches. So a monatomic lattice ($p = 1$) has no optical phonons at all, while a diatomic lattice ($p = 2$) has 3 optical branches.

Atoms vibrating out of phase

In optical modes, the counter-motion of neighboring atoms (or ions) creates a time-varying electric dipole moment within each unit cell. This oscillating dipole is what allows optical phonons to couple directly to electromagnetic radiation.

You can picture the motion as a periodic stretching and compression of the bonds between the two sublattices. In ionic crystals like NaCl, the positive Na$^+$ ions move one way while the negative Cl$^-$ ions move the other, producing a strong oscillating polarization.

Non-zero frequency at Brillouin zone center

A defining feature of optical phonons: they have a finite frequency even at $k = 0$. At the zone center, the two sublattices oscillate against each other with uniform amplitude throughout the crystal.

This contrasts sharply with acoustic phonons, where $\omega \to 0$ as $k \to 0$ (uniform translation costs no energy). For optical modes, even the $k = 0$ vibration involves relative displacement of atoms within the unit cell, which stretches bonds and costs energy. The zone-center optical frequency depends on the interatomic force constant $\kappa$ and the reduced mass $\mu$ of the basis atoms:

$\omega_{\text{opt}}(k=0) = \sqrt{\frac{2\kappa}{\mu}}$

Infrared activity of optical modes

Because optical phonons generate oscillating dipole moments, they can absorb infrared photons whose frequency matches the phonon frequency. This is the basis of infrared (IR) spectroscopy of crystals.

Selection rules matter here:

• In crystals lacking inversion symmetry, modes can be both IR-active and Raman-active
• In centrosymmetric crystals, the mutual exclusion rule applies: a mode is either IR-active or Raman-active, but not both
• A mode is IR-active if it transforms as a polar vector (changes the dipole moment)
• A mode is Raman-active if it transforms as a symmetric tensor (changes the polarizability)

LO-TO splitting

In polar crystals, the longitudinal optical (LO) and transverse optical (TO) phonons have different frequencies at $k = 0$. This LO-TO splitting arises because the longitudinal mode generates a macroscopic electric field (via the oscillating polarization), which adds an extra restoring force that the transverse mode doesn't experience.

The relationship between the two frequencies is captured by the Lyddane-Sachs-Teller (LST) relation:

$\frac{\omega_{LO}^2}{\omega_{TO}^2} = \frac{\varepsilon_0}{\varepsilon_\infty}$

where $\varepsilon_0$ is the static dielectric constant and $\varepsilon_\infty$ is the high-frequency dielectric constant. The larger the ratio $\varepsilon_0 / \varepsilon_\infty$, the greater the splitting. In GaAs, for example, $\omega_{LO} \approx 292$ cm$^{-1}$ and $\omega_{TO} \approx 269$ cm$^{-1}$.

Phonon dispersion relations

Phonon dispersion relations plot $\omega$ vs. $k$ for all phonon branches along high-symmetry directions in the Brillouin zone. They encode the complete vibrational spectrum of a crystal: the interatomic force constants, the masses, and the lattice geometry all leave their fingerprint on the shape of these curves.

Acoustic vs optical branches

For a crystal with $p$ atoms per unit cell in 3D:

• 3 acoustic branches (1 LA + 2 TA)
• $3p - 3$ optical branches

The gap between the top of the acoustic branches and the bottom of the optical branches (if one exists) is sometimes called the phonon band gap. Not all crystals have such a gap; it depends on the mass ratio and force constants.

Brillouin zone boundaries

At the zone boundary ($k = \pi/a$ for a 1D chain), the phonon wavelength equals twice the lattice spacing. Here, the waves form standing waves rather than propagating modes, and the dispersion curves flatten ($v_g = 0$).

Several notable features appear at zone boundaries:

• Acoustic and optical branches can approach each other, sometimes nearly touching
• Degeneracies can occur or be lifted depending on the crystal symmetry
• The density of states tends to peak near these flat regions (van Hove singularities)

Experimental techniques for measuring dispersion

Inelastic neutron scattering (INS) is the most comprehensive technique. Neutrons have wavelengths comparable to interatomic spacings and energies comparable to phonon energies, making them ideal probes. By measuring the energy and momentum change of scattered neutrons, you can map out the full dispersion relation across the entire Brillouin zone.

Inelastic X-ray scattering (IXS) uses high-energy synchrotron X-rays. It requires very high energy resolution but works well for small samples and high-pressure environments where neutron scattering is impractical.

Raman and infrared spectroscopy probe optical phonons near $k \approx 0$ only (because photon momentum is negligible compared to the Brillouin zone size). They're useful for identifying zone-center optical frequencies and symmetries but cannot map the full dispersion.

Phonon-phonon interactions

In a perfectly harmonic crystal, phonons would propagate forever without scattering. Real crystals have anharmonic interatomic potentials, meaning the restoring force isn't perfectly proportional to displacement. This anharmonicity causes phonons to scatter off each other, which is the dominant mechanism limiting thermal conductivity at moderate to high temperatures.

Anharmonic effects

The harmonic approximation keeps only the quadratic term in the Taylor expansion of the interatomic potential. Anharmonic corrections come from the cubic, quartic, and higher-order terms.

Consequences of anharmonicity:

• Phonons acquire finite lifetimes (they decay into other phonons)
• Phonon frequencies shift with temperature
• Thermal expansion occurs (a purely harmonic crystal would not expand)
• Thermal resistance exists (a harmonic crystal would have infinite thermal conductivity)

Anharmonic effects grow with temperature because larger thermal vibrations sample the non-quadratic parts of the potential more strongly.

Phonon scattering and lifetimes

Phonon-phonon scattering conserves energy but handles crystal momentum in two distinct ways:

• Normal (N) processes: Total crystal momentum is conserved ($\mathbf{k}_1 + \mathbf{k}_2 = \mathbf{k}_3$). These redistribute momentum among phonons but don't directly create thermal resistance.
• Umklapp (U) processes: The resulting wave vector falls outside the first Brillouin zone and gets folded back by a reciprocal lattice vector ($\mathbf{k}_1 + \mathbf{k}_2 = \mathbf{k}_3 + \mathbf{G}$). These processes transfer momentum to the lattice and are the primary source of thermal resistance from phonon-phonon scattering.

Umklapp processes require phonons with large enough wave vectors, so they freeze out exponentially at low temperatures (roughly as $e^{-\Theta_D / bT}$ where $b$ is a numerical factor). This is why thermal conductivity rises sharply as temperature drops.

Thermal conductivity of solids

The kinetic theory expression for lattice thermal conductivity is:

$\kappa = \frac{1}{3} C_v v_g \ell$

where $C_v$ is the volumetric specific heat, $v_g$ is the average phonon group velocity, and $\ell$ is the phonon mean free path.

At high temperatures, Umklapp scattering dominates and $\kappa$ typically decreases as $1/T$. At very low temperatures, the mean free path is limited by sample boundaries or defects rather than phonon-phonon scattering, and $\kappa$ drops as $T^3$ (following the specific heat).

Diamond has exceptionally high thermal conductivity (~2200 W/mK at room temperature) because of its stiff bonds, light atoms, and high Debye temperature, all of which suppress Umklapp scattering.

Electron-phonon interactions

Electrons in a crystal don't just see a static periodic potential. The lattice vibrates, and those vibrations modulate the potential that electrons experience. This electron-phonon coupling affects carrier mobility, enables conventional superconductivity, and governs indirect optical transitions.

Polaron concept

A polaron forms when an electron (or hole) distorts the surrounding lattice, and that distortion in turn creates a potential well that the carrier drags along as it moves.

• Large polarons: The lattice distortion extends over many unit cells. The carrier remains mobile but has an enhanced effective mass. This is the typical situation in weakly polar semiconductors.
• Small polarons: The distortion is confined to roughly one unit cell. The carrier becomes essentially self-trapped and moves by thermally activated hopping rather than band transport. This occurs in materials with strong electron-phonon coupling, such as transition metal oxides.

The polaron binding energy and spatial extent depend on the electron-phonon coupling strength, often characterized by the Fröhlich coupling constant $\alpha$ for polar materials.

Superconductivity mediated by phonons

In conventional (BCS) superconductors, phonons mediate an attractive interaction between electrons:

1. An electron moves through the lattice and attracts nearby positive ions toward it, creating a region of excess positive charge.
2. A second electron is attracted to this positive charge concentration.
3. The net effect is an attractive interaction between the two electrons, mediated by the exchange of virtual phonons.
4. This attraction binds electrons into Cooper pairs, which condense into a macroscopic quantum state with zero resistance below the critical temperature $T_c$.

The BCS theory predicts $T_c$ depends on the electron-phonon coupling constant $\lambda$ and the Debye frequency. Stronger coupling and higher phonon frequencies generally lead to higher $T_c$, though conventional phonon-mediated superconductors rarely exceed ~40 K.

Phonon-assisted optical transitions

In indirect band gap semiconductors like Si and Ge, the conduction band minimum and valence band maximum occur at different $k$-points in the Brillouin zone. A photon alone cannot bridge this gap because photons carry negligible crystal momentum.

A phonon must participate to conserve both energy and crystal momentum:

$\hbar\omega_{\text{photon}} = E_g \pm \hbar\omega_{\text{phonon}}$

The $+$ sign corresponds to phonon absorption, the $-$ sign to phonon emission. This two-particle process has a much lower probability than a direct transition, which is why indirect gap semiconductors like silicon are poor light emitters compared to direct gap materials like GaAs.

Applications of phonons

Phonon physics underpins a wide range of technologies and material design strategies. Controlling how phonons propagate, scatter, and interact with electrons is central to thermal management, energy conversion, and emerging acoustic devices.

Heat capacity of solids

At low temperatures ($T \ll \Theta_D$), the phonon contribution to heat capacity follows the Debye $T^3$ law:

$C_v = \frac{12\pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3$

At high temperatures ($T \gg \Theta_D$), all modes are fully excited and the heat capacity saturates at the Dulong-Petit limit: $C_v = 3Nk_B$, or about 25 J/(mol·K) per mole of atoms. The crossover between these regimes occurs around $T \sim \Theta_D$.

Thermal expansion

A purely harmonic potential is symmetric, so the average atomic position wouldn't shift with temperature. Thermal expansion exists because real potentials are anharmonic: the repulsive wall at short distances is steeper than the attractive tail at long distances. As temperature rises, the average bond length increases.

The volumetric thermal expansion coefficient $\beta$ is related to the Grüneisen parameter $\gamma$:

$\beta = \frac{\gamma C_v}{B V}$

where $B$ is the bulk modulus and $V$ is the volume. Materials with large Grüneisen parameters (strong anharmonicity) expand more.

Thermoelectric effects

Thermoelectric efficiency is characterized by the dimensionless figure of merit:

$ZT = \frac{S^2 \sigma T}{\kappa}$

where $S$ is the Seebeck coefficient, $\sigma$ is the electrical conductivity, and $\kappa$ is the total thermal conductivity (electronic + lattice contributions).

The challenge is that $\sigma$ and the electronic part of $\kappa$ are linked (Wiedemann-Franz law), so the main strategy for improving $ZT$ is reducing the lattice thermal conductivity without harming electrical transport. Approaches include:

• Nanostructuring to scatter phonons at grain boundaries
• Introducing heavy rattler atoms in cage-like structures (e.g., skutterudites, clathrates)
• Alloying to create mass disorder that scatters short-wavelength phonons

Phononic crystals and metamaterials

Phononic crystals are periodic structures engineered to create band gaps for phonons, analogous to photonic crystals for light. Within the phononic band gap, vibrations at those frequencies cannot propagate through the structure.

Design involves choosing materials with contrasting densities and elastic constants, arranged in a periodic pattern with a lattice constant matched to the target wavelength range.

Applications include:

• Acoustic waveguides and filters
• Vibration isolation platforms
• Thermal management (blocking specific phonon frequencies to reduce thermal conductivity)

Phononic metamaterials go further by achieving properties not found in natural materials, such as negative effective mass density or negative elastic modulus, enabling effects like acoustic cloaking and sub-wavelength focusing.