3.2 Phonon dispersion relations

Phonon dispersion overview

Phonon dispersion relations describe how the frequency of lattice vibrations depends on their wavevector. They're the vibrational analog of electronic band structures: just as $E(k)$ tells you about electron behavior, $\omega(k)$ tells you everything about how vibrations move through a crystal. From these relations you can extract sound velocities, thermal conductivity, heat capacity, and optical response.

Phonons as quasiparticles

Phonons are quantized normal modes of vibration in a crystal lattice, treated as quasiparticles. They represent collective excitations of atoms oscillating in a periodic structure, and they carry both energy ($E = \hbar\omega$) and crystal momentum ($p = \hbar k$). Their behavior is governed by Bose-Einstein statistics since phonons are bosons with no limit on occupation number.

Two fundamental types exist:

• Acoustic phonons arise from in-phase motion of atoms in the unit cell
• Optical phonons arise from out-of-phase motion, and require at least two atoms per unit cell

Lattice vibrations and phonons

Atoms in a crystal oscillate around their equilibrium positions. These vibrations can be decomposed into normal modes, each with a well-defined frequency $\omega$ and wavevector $k$. Phonons are the quantized excitations of these normal modes.

The phonon picture is powerful because it converts a many-body problem (billions of coupled oscillators) into a collection of independent quantum harmonic oscillators. Each mode can be occupied by an integer number of phonons, and interactions between modes (anharmonic effects) are treated as phonon-phonon scattering.

Acoustic vs optical phonons

Acoustic phonons correspond to atoms in the unit cell moving in phase with each other, much like a sound wave:

• They have a linear dispersion at long wavelengths (small $k$): $\omega \approx v_s |k|$, where $v_s$ is the speed of sound
• Three acoustic branches always exist in 3D (one longitudinal, two transverse)
• At $k = 0$, acoustic phonons have $\omega = 0$ (uniform translation of the crystal costs no energy)

Optical phonons involve atoms within the unit cell moving out of phase. They only appear when the basis contains more than one atom:

• They have higher frequencies than acoustic phonons, and their dispersion is relatively flat across the Brillouin zone
• At $k = 0$, optical phonons have a finite frequency, which is why they can couple to infrared light (in ionic crystals) or be detected via Raman scattering
• For a unit cell with $p$ atoms in 3D, there are 3 acoustic branches and $3p - 3$ optical branches

Phonon dispersion relations

The dispersion relation $\omega(k)$ encodes how phonon frequency varies with wavevector throughout the Brillouin zone. From it you can read off group velocities, identify bandgaps, and compute the density of states.

Dispersion relation definition

Mathematically, the dispersion relation is $\omega = \omega(\mathbf{k})$, relating the angular frequency to the wavevector. It's obtained by solving the equations of motion for the lattice with periodic boundary conditions.

For a simple monatomic chain with nearest-neighbor spring constant $C$ and atomic mass $M$, the result is:

$\omega(k) = 2\sqrt{\frac{C}{M}} \left|\sin\left(\frac{ka}{2}\right)\right|$

where $a$ is the lattice constant. This captures the key features: linear behavior near $k = 0$ and flattening at the zone boundary $k = \pi/a$.

Dispersion relations are plotted as $\omega$ vs. $k$ along high-symmetry directions in the Brillouin zone.

Wavevector and frequency

The wavevector $k$ describes the spatial periodicity and propagation direction of the phonon:

• Related to wavelength by $k = 2\pi/\lambda$
• The crystal momentum is $p = \hbar k$ (not true momentum, but conserved modulo a reciprocal lattice vector in scattering processes)

The frequency $\omega$ describes the temporal oscillation and sets the phonon energy:

• Energy per phonon: $E = \hbar\omega$
• Higher frequency means higher energy

Brillouin zones in dispersion

Dispersion relations are plotted within the first Brillouin zone, which is the Wigner-Seitz cell of the reciprocal lattice. This zone contains all physically distinct wavevectors because of the lattice periodicity: any $k$ outside the first zone is equivalent to one inside it (shifted by a reciprocal lattice vector $G$).

High-symmetry points label special locations in the zone:

• Γ: the zone center ($k = 0$)
• X, L, K, etc.: zone boundary points whose labels depend on the crystal structure (FCC, BCC, etc.)

Dispersion curves are plotted along paths connecting these points (e.g., Γ → X → L → Γ) to capture the most important features with minimal plotting.

Reduced vs extended schemes

• Reduced scheme: All branches are folded back into the first Brillouin zone. This is the standard representation and emphasizes that wavevectors differing by a reciprocal lattice vector are physically equivalent. Each distinct phonon mode appears as a separate branch.
• Extended scheme: Branches are plotted beyond the first zone, showing the dispersion as a continuous curve. This is helpful for visualizing how the monatomic-chain dispersion "folds" into multiple branches when you go to a larger unit cell (e.g., a diatomic chain).

The reduced scheme is used in nearly all published phonon dispersion diagrams.

Dispersion curves

Dispersion curves are the graphical representation of $\omega(k)$. Reading them correctly gives you direct access to phonon velocities, bandgaps, and mode character.

Dispersion curve characteristics

• Multiple branches appear: 3 acoustic + $(3p - 3)$ optical for $p$ atoms per unit cell
• The slope at any point gives the group velocity: $v_g = d\omega/dk$
• Flat regions (zero slope) mean zero group velocity and a high density of states at that frequency
• Gaps between acoustic and optical branches represent frequency ranges with no allowed phonon modes (phonon bandgaps). These gaps are larger when the mass difference between atoms in the basis is larger.

High symmetry points and directions

Dispersion curves are plotted along paths connecting high-symmetry points in the Brillouin zone. The specific labels depend on the crystal structure:

• FCC: Γ, X, W, L, K
• BCC: Γ, H, N, P

These points are chosen because they capture symmetry-enforced degeneracies and extrema in the dispersion. The path Γ → X → W → L → Γ for an FCC crystal, for example, reveals the full character of all branches.

Acoustic branch dispersion

Acoustic branches start at $\omega = 0$ at the Γ point and increase with $|k|$:

• Near Γ, the dispersion is linear: $\omega = v_s |k|$. The slope gives the speed of sound in that direction and polarization.
• Longitudinal acoustic (LA) branches have higher velocity than transverse acoustic (TA) branches because longitudinal stiffness exceeds shear stiffness in most materials.
• At the zone boundary, acoustic branches flatten out and reach their maximum frequency.

These branches dominate low-temperature thermal transport because they're the most populated modes at low $T$.

Optical branch dispersion

Optical branches sit at higher frequencies and are relatively flat:

• Their flatness means optical phonons have low group velocities and contribute little to heat transport.
• The high density of states near optical branch frequencies makes them important for specific heat at intermediate temperatures.
• In polar materials (e.g., NaCl, GaAs), the LO (longitudinal optical) and TO (transverse optical) frequencies split at $k = 0$ due to the long-range electric field associated with LO modes. This is the LO-TO splitting, and its magnitude reflects the ionicity of the bonding.

Degenerate modes in dispersion

Degeneracy occurs when two or more branches share the same frequency at a given $k$:

• Along high-symmetry directions, the two TA branches are often degenerate (same frequency, different polarization directions).
• At high-symmetry points, additional degeneracies can arise from the point group symmetry of the crystal.
• Perturbations like applied strain, electric fields, or symmetry-lowering phase transitions can lift degeneracies, causing mode splitting that's observable in spectroscopy.

Experimental measurement of dispersion

Three main techniques probe phonon dispersion, each with different strengths in terms of the $(\omega, k)$ range they can access.

Inelastic neutron scattering

Inelastic neutron scattering (INS) is the workhorse technique for full dispersion mapping:

1. A monochromatic neutron beam hits the crystal sample.
2. Neutrons exchange energy and momentum with phonons in the lattice.
3. The scattered neutrons are analyzed for their final energy and direction.
4. Conservation of energy and momentum gives the phonon $\omega$ and $k$ directly: $\hbar\omega = E_i - E_f$ and $\hbar\mathbf{k}_{phonon} = \hbar(\mathbf{k}_i - \mathbf{k}_f) - \hbar\mathbf{G}$.

INS can map the entire Brillouin zone and is sensitive to both acoustic and optical modes. The main limitation is that it requires large single crystals and access to a neutron source (reactor or spallation facility).

Inelastic x-ray scattering

Inelastic x-ray scattering (IXS) works on a similar principle but uses high-energy x-ray photons instead of neutrons:

• X-rays interact with the electron density rather than nuclei directly.
• Modern synchrotron sources achieve meV energy resolution, sufficient to resolve phonon energies.
• IXS works well on small samples and materials containing heavy elements (where neutron absorption can be a problem).
• It's particularly good for probing phonons near the zone center and for high-frequency optical modes.

Raman spectroscopy for phonons

Raman spectroscopy measures zone-center ($k \approx 0$) optical phonons:

1. Monochromatic laser light illuminates the sample.
2. Most photons scatter elastically (Rayleigh scattering), but a small fraction exchange energy with phonons.
3. The frequency shift of the scattered light equals the phonon frequency: Stokes (phonon created) or anti-Stokes (phonon absorbed).

Raman is fast, non-destructive, and works on tiny samples. The trade-off is that it only probes $k \approx 0$ because photon wavevectors are negligible compared to the Brillouin zone size. Selection rules based on crystal symmetry determine which modes are Raman-active.

Phonon group velocity

The group velocity $v_g = \nabla_k \omega(\mathbf{k})$ tells you how fast and in what direction energy carried by a phonon wavepacket travels. It's the key quantity connecting dispersion relations to thermal transport.

Group velocity from dispersion

• In 1D: $v_g = d\omega/dk$, which is simply the slope of the dispersion curve.
• In 3D: $\mathbf{v}_g = \nabla_{\mathbf{k}} \omega(\mathbf{k})$, a vector that can point in a different direction than $\mathbf{k}$ in anisotropic crystals.
• Near the Γ point for acoustic branches, $v_g$ equals the speed of sound.
• At zone boundaries and near flat optical branches, $v_g \to 0$, meaning those phonons carry energy very slowly.

Phonon propagation and transport

The lattice thermal conductivity in the kinetic theory picture is:

$\kappa = \frac{1}{3} \sum_s \int C_s(\mathbf{k}) \, v_{g,s}^2(\mathbf{k}) \, \tau_s(\mathbf{k}) \, \frac{d^3k}{(2\pi)^3}$

where the sum runs over branches $s$, $C_s$ is the mode heat capacity, $v_{g,s}$ is the group velocity, and $\tau_s$ is the relaxation time.

High group velocity and long relaxation times both increase thermal conductivity. Scattering processes that limit $\tau$ include:

• Phonon-phonon scattering: Normal (N) processes conserve crystal momentum; Umklapp (U) processes do not, and are the primary source of thermal resistance
• Defect and impurity scattering: disrupts phonon propagation, especially for short-wavelength modes
• Boundary scattering: dominates at low temperatures or in nanostructures

Phonon focusing effects

In anisotropic crystals, the group velocity direction $\mathbf{v}_g = \nabla_{\mathbf{k}} \omega$ doesn't generally point along $\mathbf{k}$. This means phonons with different wavevectors can have group velocities pointing in similar directions, concentrating energy flow along certain crystallographic axes.

This phonon focusing leads to highly directional heat transport. It's been directly imaged in experiments where a point heat source on one face of a crystal produces a non-uniform pattern on the opposite face. Materials like silicon and germanium show strong focusing effects along $\langle 100 \rangle$ directions.

Phonons in thermal properties

Phonon dispersion relations underpin the thermal behavior of solids. The density of states derived from the dispersion determines heat capacity, and the group velocities determine how efficiently heat moves through the material.

Phonon density of states

The phonon density of states $g(\omega)$ counts the number of phonon modes per unit frequency interval. It's computed from the dispersion by:

$g(\omega) = \sum_s \int \frac{d S_\omega}{(2\pi)^3 |\nabla_{\mathbf{k}} \omega_s|}$

where the integral is over the constant-frequency surface $S_\omega$ in $k$-space.

Key features of the DOS:

• At low frequencies, $g(\omega) \propto \omega^2$ in 3D (the Debye behavior, from the linear acoustic dispersion)
• Van Hove singularities appear where $\nabla_{\mathbf{k}} \omega = 0$ (flat regions of the dispersion), causing peaks or kinks in the DOS
• The total number of modes equals $3pN$, where $p$ is the number of atoms per unit cell and $N$ is the number of unit cells

Phonon heat capacity

Phonons dominate the heat capacity of solids, especially at low and intermediate temperatures.

The heat capacity is calculated by weighting each mode's contribution using the Bose-Einstein distribution:

$C_V = \sum_s \int \hbar\omega_s \frac{\partial n_{BE}}{\partial T} g_s(\omega) \, d\omega$

where $n_{BE} = 1/(e^{\hbar\omega/k_BT} - 1)$.

Two important limits:

• Low $T$: Only low-frequency acoustic modes are excited. $C_V \propto T^3$ (Debye model), because $g(\omega) \propto \omega^2$ and only modes with $\hbar\omega \lesssim k_BT$ contribute.
• High $T$: All modes are fully excited, each contributing $k_B$ to the heat capacity. $C_V \to 3pNk_B$ (Dulong-Petit law).

Thermal conductivity contributions

In non-metallic solids, phonons are the dominant heat carriers. The thermal conductivity depends on three factors from the dispersion: group velocity, density of states, and scattering rates.

Scattering processes that limit thermal conductivity:

• Umklapp scattering: phonon-phonon processes where crystal momentum is not conserved (involves a reciprocal lattice vector). This is the dominant intrinsic resistance mechanism at moderate to high temperatures.
• Normal scattering: conserves crystal momentum and doesn't directly cause resistance, but redistributes phonons among modes.
• Boundary scattering: important in nanostructures and at very low temperatures where phonon mean free paths become comparable to sample dimensions.
• Isotope/defect scattering: mass disorder scatters short-wavelength phonons efficiently (Rayleigh-type $\omega^4$ dependence).

Engineering thermal conductivity involves manipulating these factors: nanostructuring introduces boundary scattering, alloying adds mass disorder, and superlattices can create phonon bandgaps that block certain modes.

Phonons in different materials

The details of phonon dispersion vary significantly across material classes, reflecting differences in bonding, mass, and structure.

Phonon dispersion in metals

In metals, conduction electrons screen the interatomic forces and modify the phonon dispersion. A distinctive feature is the Kohn anomaly: a sharp dip or kink in the phonon dispersion at wavevectors $k = 2k_F$ (where $k_F$ is the Fermi wavevector). This occurs because the electronic screening response has a singularity at this wavevector.

Electron-phonon coupling in metals is also responsible for:

• Conventional superconductivity (BCS theory): Cooper pairs form through phonon-mediated electron-electron attraction
• Electrical resistivity at finite temperature: electrons scatter off thermally populated phonons

INS and IXS are the primary tools for measuring phonon dispersion in metals.

Semiconductors and insulators

Phonon dispersion in semiconductors directly affects electronic and thermal performance:

• Electron-phonon scattering by optical phonons (especially LO modes in polar semiconductors like GaAs) limits electron mobility at room temperature.
• The thermal conductivity of semiconductors like silicon and diamond is dominated by acoustic phonons with long mean free paths.
• Alloying (e.g., $Si_{1-x}Ge_x$) introduces mass disorder that strongly scatters phonons, reducing thermal conductivity. This is exploited in thermoelectric materials.

Raman spectroscopy is widely used to characterize optical phonons in semiconductors, providing information about composition, strain, and crystal quality.

Effects of material anisotropy

Anisotropic crystals (e.g., hexagonal structures, layered materials) have direction-dependent phonon dispersion:

• Group velocities and thermal conductivity differ along different crystallographic axes.
• Layered materials like graphite have strong in-plane bonding and weak van der Waals interlayer coupling, leading to very different phonon velocities parallel vs. perpendicular to the layers.
• This anisotropy produces phonon focusing effects and directional thermal transport.

Dispersion in 2D materials

Two-dimensional materials like graphene and transition metal dichalcogenides (TMDs) have distinctive phonon properties:

• Flexural (ZA) modes: out-of-plane acoustic vibrations with a quadratic dispersion ($\omega \propto k^2$) near Γ, unlike the linear dispersion of in-plane acoustic modes. These modes strongly influence thermal conductivity in suspended 2D sheets.
• Graphene has an exceptionally high in-plane thermal conductivity (~3000-5000 W/mK) due to its stiff sp2 bonds and high acoustic phonon group velocities.
• In TMDs (e.g., $MoS_2$), the heavier atoms and more complex unit cell produce many optical branches, and interlayer coupling in few-layer samples causes branch splitting observable by Raman spectroscopy.

Phonon dispersion in 2D materials is probed by Raman spectroscopy, IXS, and electron energy loss spectroscopy (EELS). Understanding these dispersions is central to engineering thermal management and optoelectronic performance in 2D material devices.